VII International Conference GEOMETRY AND TOPOLOGY OF …im0.p.lodz.pl/.../07_BedlewoEhresmann2005/folder05abcde.pdf · 2008-09-18 · Singular foliations, cohomology theories for

VII International Conference

GEOMETRY AND TOPOLOGY OFMANIFOLDS

The Mathematical Legacy of Charles Ehresmannon the occasion of the hundredth anniversary of his birthday

Under the auspices of Prof. Jan KrysinskiRector of the Technical University of ×ódz

B¾edlewo, Poland, May 8 �15, 2005

http://im0.p.lodz.pl/konferencje/bedlewo05

×ÓDZ�WARSZAWA 2005

Contents

Foreword 4

Organizers and Scienti�c Committee 5

List of participants 7

Titles of minicourses 13

Titles of lectures 13

The Mathematical Legacy of Charles Ehresmann, Abstracts 16R. WOLAK, CHARLES EHRESMANN . . . . . . . . . . . . . . . . . . . . . . 17A. C. EHRESMANN (the widow), How Charles Ehresmann�s vision of Geom-

etry developed with time. (Read by Ronnie Brown) . . . . . . . . . . . . . . 21R. BROWN, Groupoids, Local-to-global, Higher dimensions: Three themes in the

work�of Charles Ehresmann . . . . . . . . . . . . . . . . . . . . . . . . . . 26J-C. HAUSMANN, Robot arms and Moebius transformation (the snake charmer

algorithm). (Joint work with Eugenio Rodriguez) . . . . . . . . . . . . . . 27A. KOCK, Pregroupoids and their enveloping groupoids . . . . . . . . . . . . . . 28I. KOLÁµR, Functorial prolongations of Lie groupoids . . . . . . . . . . . . . . . 29P. LIBERMANN, C. Ehresmann concepts in Di¤erential Geometry . . . . . . . 30C-M. MARLE, C. Ehresmann concepts in Di¤erential Geometry . . . . . . . . 31G. MEIGNIEZ, The nature of �brations . . . . . . . . . . . . . . . . . . . . . . 32P. MOLINO, Speech read by Jean Pradines . . . . . . . . . . . . . . . . . . . . 33B. MONTHUBERT, Groupoids and Index Theory in the singular manifolds setting 34J. PRADINES, In Charles Ehresmann�s footsteps: from Group Geometries to

Groupoid Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35A. RODRIQUES, Contact and equivalence of submanifolds of homogeneous spaces 37W. TULCZYJEW, Modernization of Ehresmann jet theory . . . . . . . . . . . 40W. WEN-TSUN, Speech read by Jean Pradines . . . . . . . . . . . . . . . . . . 41M. ZISMAN,Quasi-commutative cochains in algebraic topology (After Max Karoubi) 42

Abstracts of minicourses 43A.S.MISHCHENKO, K-theory over C*-algebras . . . . . . . . . . . . . . . . . 44

1 Some elementary and evident examples 44

2 Almost �at bundles from the point of view of C*-algebras 45

3 Twisted K-theory due to M.Atiyah and G.Segal 45

Abstracts of lectures 47K. ABE, On the structure of the groups of di¤eomorphisms of manifolds with

boundary and its applications. (Joint work with Kazuhiko Fukui) . . . . . 48A. AROUCHE , Restriction properties of equivariant K-theory rings . . . . . . 48B. BALCERZAK, Secondary characteristic classes for extensions of anchored

Leibniz algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49M. BOBIENSKI, On SO(3) geometry in dimension �ve. (Joint work with P.

Nurowski) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49B. BOJARSKI, G. KHIMSHIASHVILI, Riemann-Hilbert problems in loop spaces 50M. BORODZIK, H. ·ZO×¾ADEK, Classi�cation of complex plane a¢ ne algebraic

curves with zero Euler Characteristic . . . . . . . . . . . . . . . . . . . . . 51M. CAPPELLETTI MONTANO, On Legendrian foliations on almost S-manifolds 51G. CICORTAS, Categorical sequences and relative categories . . . . . . . . . . . 52J. EICHHORN, Absolute and relative characteristic numbers for open manifolds,

their application to bordism theory and the Novikov conjecture . . . . . . . 53A. ERMOLITSKI, On universality of Kählerian manifold . . . . . . . . . . . . 53J. GAO, A Parameter related to W � Topology in Banach Spaces . . . . . . . . . 54J. GRABOWSKI, N. PONCIN, On the Chevalley-Eilenberg cohomology of some

in�nite-dimensional algebras of geometric origin . . . . . . . . . . . . . . . 55S. HANSOUL, Existence of natural and projectively equivariant quantizations . 55J. HUEBSCHMANN, Strati�ed Kähler structures on adjoint quotients . . . . . 56J. JANY�KA, Higher order Utiama�s theorem . . . . . . . . . . . . . . . . . . . 57R. LEANDRE, Stochastic Poisson-Sigma model . . . . . . . . . . . . . . . . . . 57J. LECH, T. RYBICKI, Perfectness at in�nity of di¤eomorphism groups on open

manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58P. LECOMTE, From a Euclidian space to Cartan geometries: how to rebuild

di¤erential operator from their principal symbol? . . . . . . . . . . . . . . . 58S. MAKSYMENKO, Homotopy types of stabilizers and orbits of Morse mappings

of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59P. MORMUL, Nilpotent algebras hidden in special multi-�ags . . . . . . . . . . 61R. NEST, The geometry of the calculus of Fourier integral operators . . . . . . 69I. NIKONOV, On Hopf-type cyclic cohomology with coe¢ cients. (Joint work with

G.I.Sharygin) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70A. PANASYUK, Algebraic Nijenhuis operators and Kronecker Poisson pencils . 70K. PAWA×OWSKI, A proof of the Laitinen Conjecture. (Joint work with Ronald

Solomon and Toshio Sumi) . . . . . . . . . . . . . . . . . . . . . . . . . . . 71P. POPESCU, M. POPESCU, On the high order geometry on osculator spaces

and anchored vector bundles. (Joint work with Marcela Popescu) . . . . . . 71P. POPESCU, M. POPESCU, On the Lie pseudoalgebra generated by an an-

chored module. (Joint work with Marcela Popescu) . . . . . . . . . . . . . 72A. PRYKARPATSKY, Ergodic measures of Boole type dynamical systems on

axis: analytical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73D. ROYTENBERG, Higher Lie algebras in Poisson geometry and elsewhere . . 73T. RYBICKI, Commutators of equivariant homeomorphisms on G-manifolds with

one orbit type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74M. SADOWSKI, Complete �at manifolds . . . . . . . . . . . . . . . . . . . . . 75P. SEVERA, Dirac structures and deformation quantization . . . . . . . . . . . 75

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V. SHARKO, The L2-invariants and non-singular Morse-Smale �ows on manifolds 76Z. �KODA, Equivariant sheaves and torsors beyond groupoids . . . . . . . . . . 77D. SZEGHY, Conjugate and focal points in semi-Riemann geometry . . . . . . 77J. SZENTHE, Invariant Lagrangians on Homogeneous . . . . . . . . . . . . . . 78C. VIZMAN, A geometric construction of abelian Lie group extensions . . . . . 78W. ZHANG, Bergman kernel and symplectic reduction . . . . . . . . . . . . . . 79N. ZHUKOVA, Singular Foliations with Ehresmann Connections . . . . . . . . 79

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FOREWORD

This is a conference of the cycle initiated in 1998 with a meeting in Konopnica(http://im0.p.lodz.pl/konferencje/) and is organized in B¾edlewo near Poznan from8.05.2005 to 15.05.2005 (Poland) in the Conference Center of the Stefan Banach InternationalMathematical Center. The last meetings were organized in Krynica-Zdrój in Hotel "Baska"of the University of the AGH University of Science and Technology, Kraków. Therefore, thepresent meeting has been named by somebody as "Krynica in B¾edlewo".During this Conference we celebrate the 100th anniversary of Charles Ehresmann�s birthday.

The special sessions dedicated to Charles Ehresmann�s life and work and the in�uence of hisideas on the modern mathematics are organized.

The main aim of the conference series is to present and discuss new results on geometryand topology of manifolds with the particular attention being paid to applications of algebraicmethods. The topics usually discussed include:

� Lie groups (including in�nite dimensional), Lie algebroids and their generalizations, Liegroupoids,

� Characteristic classes, index theory, K-theory , Fredholm operators,

� Singular foliations, cohomology theories for foliated manifolds and their quotients,

� Symplectic, Poisson, Jacobi and special Riemannian manifolds,

� Applications to mathematical physics.

All these topics are very closely related and most interesting results are obtained whenvarious non-standard methods are applied.

ORGANIZERS AND SCIENTIFIC COMMITTEE

VII Conference on

GEOMETRY AND TOPOLOGY OF MANIFOLDSis organized by

� Institute of Mathematics of the Polish Academy of Sciences, Warszawa

� Institute of Mathematics of the Technical University of ×ódz, ×ódz

- Stefan Banach International Mathematical Center, Warszawa

� Université Paul Sabatier, Toulouse

� Institute of Mathematics of the Jagiellonian University, Kraków

� Faculty of Applied Mathematics at the AGHUniversity of Science and Technology, Kraków

Organizing Committee� Jan Kubarski, Chairman, ×ódz, Poland, email: [email protected]

�Institute of Mathematics of the Polish Academy of Sciences, Warszawa,

� Institute of Mathematics of the Technical University of ×ódz, ×ódz.

� Jean Pradines, Toulouse, France, email: [email protected]

�Université Paul Sabatier, Toulouse, France,

� Tomasz Rybicki, Kraków, Poland, email: [email protected]

� Faculty of Applied Mathematics at the AGH University of Science and Technology,Kraków

� Robert Wolak, Kraków, Poland, email: [email protected]

�Institute of Mathematics of the Jagiellonian University, Kraków

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Scienti�c Committee�Dmitri Alekseevsky (UK) �Bogdan Bojarski (Poland)�Ronald Brown (UK) �Pierre Cartier (France)�Andrée Charles Ehresmann (France) �Janusz Grabowski (Poland)�Andre Hae�iger (Switzerland) �André Joyal (Canada)�Mikhail Karasev (Russia) �Anders Kock (Denmark)�Ivan Koláµr (Czech Republic) �Paulette Libermann (France)�Charles-Michel Marle (France) �Kirill Mackenzie (UK)�Peter Michor (Austria) �Alexander Mishchenko (Russia)�Ieke Moerdijk (The Netherlands) �Pierre Molino (France)�Bertrand Monthubert (France) �Valentin Poenaru (France)�Jean Pradines (France) �Jean Renault (France)�Alexandre Martins Rodrigues (Brazil) �James Stashe¤ (USA)�János Szenthe (Hungary) �Nicola Teleman (Italy)�W÷odzimierz Tulczyjew (Italy) �Alan Weinstein (USA)�Yanlin Yu (China) �Weiping Zhang (China)�Michel Zisman (France)

Sponsors: The organizers of the conference are grateful to the following sponsors:

� Rector of the Technical University of ×ódz

� Dean of the Faculty of Technical Physics, Informatics and Wydzia÷Fizyki Technicznej,Informatyki and Applied Mathematics

� Rector of the Jagiellonian University

� Rector of the AGH University of Science and Technology

� State Committee for Scienti�c Research

� State Committee of Polish Academy of Sciences

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LIST OF PARTICIPANTS

1. Abe, Kojun, Mathematical Sciences, Faculty of Science, Shinshu UniversityAsahi 3-1-1, Matsumoto, Nagano Prefecture, Japane-mail: [email protected], http://math.shinshu-u.ac.jp/�kabe

2. Alekseevsky, Dmitri, Department of Mathematics, Hull University, Cottingham Road,Hull, HU6, 7RX, UK.e-mail: [email protected]

3. Arouche, Abdelouahab, USTHB, Fac. Math. BP 32 El Alia 16111, Alger Algeriae-mail: [email protected]

4. Balcerzak, Bogdan, Institute of Mathematics, Technical University of ×ódz, ×ódz,Poland, e-mail: [email protected]

5. Banyaga, Augustin, The Pennsylvania State Universitye-mail: [email protected]

6. Bobienski, Marcin, Warsaw University, Warszawa, Polande-mail: [email protected]

7. Bogdanovich, Sergey A., Minsk, Belarus,e-mail: [email protected]

8. Bojarski, Bogdan, Mathematical Institute, Polish Academy of Sciences, Warszawa,Poland,e-mail: [email protected]

9. Booss-Bavnbek, Bernhelm, Roskilde University, Roskilde, Denmark,e-mail: [email protected],

10. Borak, Ewa, Bia÷ystok University, Bia÷ystok, Poland,e-mail: [email protected]

11. Borodzik, Maciej, Warsaw University, Warszawa, Polande-mail: [email protected]

12. Botelho, Junia, University of S¼ao Paulo, Brazile-mail: [email protected]

13. Brown, Ronnie, Department of Mathematics, University of Wales, Bangor, UKe-mail: [email protected], http://www.bangor.ac.uk/~mas010

14. Cappelletti Montano, Beniamino, University of Bari, Bari, Italye-mail: [email protected]

15. Cartier, Pierre, Institut Mathematique De Jussieu, 175 Rue Du Chevaleret, 75013 ParisFrance, (virtual participant),e-mail: [email protected]

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16. Cicortas, Gratiela, University of Oradea, Oradea, Romania,e-mail: [email protected]

17. Beno Eckmann, ETH-Zentrum, 8092 Zurich, Switzerland, (virtual participant)

18. Ehresmann, Andrée Charles, (the widow), Université de Picardie Jules Verne, Faculté deMathématique et Informatique, Amiens (virtual participant)e-mail: [email protected]

19. Eichhorn, Jürgen, Greifswald University, Greifswald, Germanye-mail: [email protected]

20. Ermolitski, Alexander, Belarus, Minske-mail: [email protected]

21. Ewert-Krzemieniewski, Stanis÷aw, Technical University of Szczecin, Polande-mail: [email protected]

22. Fomenko, Anatoly, Moscow State University, Moscow, Russiae-mail: [email protected]

23. Gao, Ji, Community College of Philadelphia, USAe-mail: [email protected], http://faculty.ccp.edu/faculty/jgao

24. Grabowski, Janusz, Institute of Mathematics of the Polish Academy of Sciences, War-saw, Polande-mail: [email protected], http://www.impan.gov.pl/~jagrab/

25. Hansoul, Sarah, University of Liège, Liège, Belgiume-mail: [email protected]

26. Hausmann, Jean-Claude, University of Geneva, Switzerlande-mail: [email protected], http://www.unige.ch/math/folks/hausmann/

27. Huebschmann, Johannes, Université des Sciences et Technologies de Lille, Francee-mail: [email protected]

28. Jany�ka, Josef, Masaryk University, Brno, Czech Republice-mail: [email protected], http://www.math.muni.cz/~janyska

29. Jelonek, W÷odzimierz, Cracow University of Technology, Kraków, Polande-mail: [email protected]

30. Karasev, Mikhail, (virtual participant)e-mail: [email protected]

31. Khimshiashvili, Giorgi, Georgian Academy of Sciences, Tbilisi, Republic of Georgia,e-mail: [email protected], http://www.rmi.acnet.ge/�khimsh

32. Kock, Anders, University of Aarhus, Denmark, Aarhuse-mail: [email protected]

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33. Koláµr, Ivan, Masaryk University, Brno, Czech Republice-mail: [email protected]

34. Konderak, Jerzy, Univ. di Bari, Italye-mail: [email protected]

35. Krot, Ewa, Bia÷ystok University, Bia÷ystok, Polande-mail: [email protected]

36. Kubarski, Jan, Technical University of ×ódz, ×ódz, Polande-mail: [email protected], http://im0.p.lodz/pl/�kubarski

37. Kushnirevitch, Vitaly, Alber-Ludwigs-Universitaet Freiburg, Freiburg, Germanye-mail: [email protected]

38. Kwasniewski, Andrzej Krzysztof, Higher School of Mathematics and Applied Infor-matics, Bia÷ystok, Polande-mail: [email protected], http://ii.uwb.edu.pl/akk/index.html

39. Léandre, Rémi, Université de Bourgogne, Dijon, Francee-mail: [email protected]

40. Lech, Jacek, AGH University of Science and Technology, Cracow, Poland,e-mail: [email protected]

41. Lecomte, Pierre, University of Liège, Liège, Belgiume-mail: [email protected], http://ulg.ac.be/geothalg

42. Libermann, Paulette, Université Denis Diderot, Paris, France

43. Mackenzie, Kirill, The University of She¢ eld, UK (virtual participant)e-mail: K.Mackenzie@she¢ eld.ac.uk, http://www.shef.ac.uk/~pm1kchm/

44. Maksymenko, Sergey, National Academy of Sciences of Ukraine, Kiev, Ukrainee-mail: [email protected], http://www.imath.kiev.ua/~maks

45. Marle, Charles-Michel, Université Pierre et Marie Curie, Paris, Francee-mail: [email protected], http://www.math.jussieu.fr/~marle/

46. Martinez, Eduardo, Universidad de Zaragoza, Zaragoza, Spaine-mail: [email protected]

47. Meigniez, Gaël, Universite de Bretagne Sud, Francee-mail: [email protected], http://www.univ-ubs.fr/lmam/meigniez

48. Minervini, Giulio, Universita�di Bari, Molfetta, Italye-mail: [email protected]

49. Mishchenko, Alexandr, Moscow State University, Moscow, Russiae-mail: [email protected], http://higeom.math.msu.su/asmish/

50. Mishchenko, Tatiana, Russian Academy of Education, Moscow, Russia

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51. Molino, Pierre, (virtual participant),e-mail: [email protected]

52. Monthubert, Bertrand, Toulouse, Francee-mail: [email protected]

53. Mormul, Piotr, Warsaw University, Warsaw, Polande-mail: [email protected]

54. Negreiros, Caio, State University of Campinas, Brazile-mail: [email protected]

55. Nest, Ryszard, Copenhagen University, Copenhagen, Denmarke-mail: [email protected]

56. Ngui¤o Boyom, Michel, Université Montpellier2, Montpellier, Francee-mail: [email protected]

57. Nikonov, Igor, Moscow State University, Moscow, Russiae-mail: [email protected]

58. Panasyuk, Andriy, Warsaw University, Warszawa, Polande-mail: [email protected]

59. Pawa÷owski, Krzysztof, Adam Mickiewicz University, Poznan, Polande-mail: [email protected], http://main.amu.edu.pl/�kpa

60. Poncin, Norbert, University of Luxembourg, Luxembourg City, Grand-Duchy of Luxem-bourge-mail: [email protected]

61. Popelensky, Fedor, Moscow State University, Moscow, Russiae-mail: [email protected]

62. Popescu, Paul, University of Craiova, Craiova, Romaniae-mail: [email protected]

63. Pradines, Jean, Université Paul Sabatier, Toulouse III, Toulouse, France,e-mail: [email protected]

64. Prykarpatsky, Anatoliy, Faculty of Applied Mathematics, AGH University of Scienceand Technology, Kraków, Poland, Dept. of Nonlinear Mathematical Analysis at IAPMMof NAS, Lviv, Ukrainee-mail: [email protected]

65. Renault, Jean (virtual participant)e-mail: [email protected]

66. Rodrigues, Alexandre A. M., University of S¼ao Paulo, Brazile-mail: [email protected]

67. Rogowski, Jacek, Technical University of ×ódz, Lódz, Polande-mail: [email protected]

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68. Roytenberg, Dmitry, TA Utrecht, The Netherlandse-mail: [email protected]

69. Rybicki, Tomasz, AGH University of Science and Technology, Cracow. Polande-mail: [email protected]

70. Sadowski, Micha÷, University of Gdansk, Gdansk, Polande-mail: [email protected]

71. Scardua, Bruno, Universidade Federal do Rio de Janeiro, Brasile-mail: [email protected]

72. Severa, Pavol, Dept. of Theoretical Physics, Bratislava, Slovakiae-mail: [email protected]

73. Sharko, Vladimir, National Academy of Sciences of Ukraine, Kiev, Ukrainee-mail: [email protected]

74. �koda, Zoran, Institute Rudjer Boskovic, Zagreb, Croatiae-mail: [email protected]

75. Stashe¤, Jim (virtual participant),e-mail: [email protected], http://www.math.unc.edu/Faculty/jds

76. Szeghy, David, Eötvös University, Budapest, Hungarye-mail: [email protected]

77. Szenthe, János, Eötvös University, Budapest, Hungarye-mail: [email protected]

78. Teleman, Nicolae, Dipartimento di Scienze Matematiche Università Politecnica delleMarche, Ancona, Italiae-mail: e-mail: [email protected]

79. Tulczyjew, W÷odzimierz, Universita di Camerino, Monte Cavallo, Italye-mail: [email protected]

80. Urbanski, Pawe÷, University of Warsaw, Warszawa, Polande-mail: [email protected]

81. Vassiliou, Efstathios, University of Athens, Athens, Greecee-mail: [email protected]

82. Vizman, Cornelia, West University of Timisoara, Timisoara, Romaniae-mail: [email protected]

83. Walczak, Pawe÷, University of ×ódz, Lódz, Polande-mail: [email protected]

84. Waliszewski, W÷odzimierz, University of Lodz, Lódz, Poland

85. Weinstein, Alan (virtual participant)e-mail: [email protected]

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86. Witkowski, Pawe÷, Warsaw University, Warsaw, Polande-mail: [email protected]

87. Wolak, Robert, Jagiellonian University, Cracow, Polande-mail: [email protected]

88. Zajtz, Andrzej, Institute of Mathematics, Pedagogical Academy of Cracow, Polande-mail: e-mail: [email protected]

89. Zhang, Weiping, Nankai University, Tianjin, P.R.Chinae-mail: [email protected]

90. Zhukova, Nina, Nizhny Novgorod State University, Russiae-mail: [email protected]

91. Ziemianska, Olga, Warsaw University, Warsaw, Polande-mail: [email protected]

92. Zisman, Michel, Université Paris 7 Denis Diderot, Paris, Francee-mail: [email protected]

93. ·Zo÷¾adek, Henryk, University of Warsaw, Warszawa, Polande-mail: [email protected]

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TITLES OF MINICOURSES

Alexandr S.Mishchenko, Moscow State University, RussiaK-theory over C*-algebras, MINICOURSE

1. Some elementary and evident examples

2. Almost �at bundles from the point of view of C*-algebras

3. Twisted K-theory due to M. Atiyah and G. Segal

TITLES OF LECTURES

1. Kojun Abe (joint work with Kazuhiko Fukui), On the �rst homology group of the groupof di¤eomorphisms of a smooth orbifold and its applications

2. Abdelouahab Arouche, Restriction properties of equivariant K-theory rings

3. Bogdan Balcerzak, Secondary characteristic classes for extensions of anchored Leibnizalgebras

4. Marcin Bobienski (joint work with P.Nurowski), On SO(3) geometry in dimension �ve

5. Ronnie Brown, Groupoids, Local-to-global, Higher dimensions: Three themes in thework of Charles Ehresmann

6. Beniamino Cappelletti Montano, On Legendrian foliations on almost S-manifolds

7. Gratiela Cicortas, Categorical sequences and relative categories

8. Andrée Charles Ehresmann (the widow), (will be read by Ronnie Brown), HowCharles Ehresmann�s vision of Geometry developed with time

9. Jürgen Eichhorn, Absolute and relative characteristic numbers for open manifolds, theirapplication to bordism theory and the Novikov conjecture

10. Alexander Ermolitski, On universality of Kählerian manifold

11. Ji Gao, A Parameter related to W � Topology in Banach Spaces

12. Sarah Hansoul, Existence of natural and projectively equivariant quantizations

13. Jean-Claude Hausmann (joint work with Eugenio Rodriguez), Robot arms and Moe-bius transformation (the snake charmer algorithm)

14. Johannes Huebschmann, Strati�ed Kähler structures on adjoint quotients

15. Josef Jany�ka, Higher order Utiama�s theorem

16. Giorgi Khimshiashvili (joint work with Bogdan Bojarski), Riemann-Hilbert problemsin loop spaces.

17. Anders Kock, Pregroupoids and their enveloping groupoids

18. Ivan Koláµr, Functorial prolongations of Lie groupoids

19. Andrzej Krzysztof Kwasniewski, On a new subcategory of the so called cobwebprefabs

20. Rémi Léandre, Stochastic Poisson-Sigma model

21. Jacek Lech (joint work withTomasz Rybicki), Perfectness at in�nity of di¤eomorphismgroups on open manifolds

22. Pierre Lecomte, From a Euclidian space to Cartan geometries: how to rebuild di¤er-ential operator from their principal symbol?

23. Paulette Libermann, C. Ehresmann concepts in Di¤erential Geometry

24. Sergey Maksymenko, Homotopy types of stabilizers and orbits of Morse mappings ofsurfaces

25. Charles-Michel Marle, The works of Charles Ehresmann on connections: from Cartanconnections to connections on �bre bundles, and some modern applications

26. Gaël Meigniez, The nature of �brations

27. Alexandr Mishchenko, MINICOURSE: "K-theory over C*-algebras"

1. Some elementary and evident examples

2.Almost �at bundles from the point of view of C*-algebras

3. Twisted K-theory due to M.Atiyah and G.Segal

28. Bertrand Monthubert, Groupoids and Index Theory in the singular manifolds setting

29. Piotr Mormul, Nilpotent algebras hidden in special multi-�ags

30. Ryszard Nest, The geometry of the calculus of Fourier integral operators

31. Igor Nikonov (joint work with G.I.Sharygin), On Hopf-type cyclic cohomology withcoe¢ cients

32. Andriy Panasyuk, Algebraic Nijenhuis operators and Kronecker Poisson pencils

33. Krzysztof Pawa÷owski (joint work with Ronald Solomon and Toshio Sumi), A proofof the Laitinen Conjecture

34. Norbert Poncin (joint work with Janusz Grabowski), On the Chevalley-Eilenbergcohomology of some in�nite-dimensional algebras of geometric origin

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35. Fedor Popelensky, TBA

36. Paul Popescu (joint work with Marcela Popescu), On the high order geometry onosculator spaces and anchored vector bundles

37. Jean Pradines, In Charles Ehresmann�s footsteps: from Group Geometries to GroupoidGeometries

38. Anatoliy Prykarpatsky, Ergodic measures of Boole type dynamical systems on axis:analytical aspects

39. Alexandre A. M. Rodrigues, Contact and equivalence of submanifolds of homogeneousspaces

40. Dmitry Roytenberg, Higher Lie algebras in Poisson geometry and elsewhere

41. Tomasz Rybicki, Commutators of equivariant homeomorphisms on G-manifolds withone orbit type

42. Micha÷Sadowski, Complete �at manifolds

43. Pavol Severa, Dirac structures and deformation quantization

44. Vladimir Sharko, The L2-invariants and non-singular Morse-Smale �ows on manifolds

45. Zoran �koda, Equivariant sheaves and torsors beyond groupoids

46. David Szeghy, Conjugate and focal points in semi-Riemann geometry

47. János Szenthe, Invariant Lagrangians on Homogeneous Manifolds

48. Nicolae Teleman, TBA

49. W÷odzimierz Tulczyjew, Modernization of Ehresmann jet theory

50. Cornelia Vizman, A geometric construction of abelian Lie group extensions

51. Pawe÷Walczak, Compact foliations with �nite LS-category

52. Weiping Zhang, Bergman kernel and symplectic reduction

53. Nina Zhukova, Singular Foliations with Ehresmann Connections

54. Michel Zisman, Quasi-commutative cochains in algebraic topology (After Max Karoubi)

55. Henryk ·Zo÷¾adek, (Joint work with Maciej Borodzik), Classi�cation of complex planea¢ ne algebraic curves with zero Euler Characteristic

15

THE MATHEMATICAL LEGACY OFCHARLES EHRESMANN,

ABSTRACTS

Charles Ehresmannwas born in Strasbourg on the 19th April 1905. From 1924 to 1927 hestudied at the prestigious l�Ecole Normale Supérieure in Paris.

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The years 1932-34, Ehresmann spent in Paris and Goettingen where heworked with Elie Cartan and Hermann Weyl, respectively. Then he contin-ued his research in Princeton. In 1934 he received the degree of Docteures Sciences Mathématiques for a thesis prepared under the supervision ofElie Cartan. From 1934 to 1939 he was a researcher at C.N.R.S. In 1939Ehresmann was named an assistant professor and then a professor at theFaculté des Sciences de Strasbourg.

From 1955 to 1975, he was professor at Sorbonne and then l�Université ParisVII.

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In 1975 he retired, but continued to lecture at l�Université de Picardie inAmiens. He died in 1979.

During his active university career he was a visiting professor, among otherplaces, at Universidad do Brasil à Rio de Janeiro, Princeton University,Tata Institute à Bombay, Université de Mexico, Université de Buenos-Aires,Université de Sâo Paulo, Université de Montréal and Kansas University atLawrence.In 1957 he founded the international journalCahiers de Topologie et GéométrieDi¤érentielle, whose editor he remained till his death.

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From 1970 he was the editor of the publication series Esquisses Mathéma-tiques.In 1965, Ch. Ehresmann was the President of the Société Mathématique deFrance .In 1967, he received the honorary doctor�s degree from the Bologna Univer-sity.Ehresmann received the following prizes of l�Académie des Sciences : PrixFrancoeur, Prix Bordin, Prix Petit d�Ormoy (in 1965).He was the supervisor of 29 thèses de Doctorat d�Etat, 4 thèses de Doctoratd�Université, and 47 thèses de Doctorat de troisième cycle.

Compiled by Robert Wolak from information and photographs provided byMme Andrée C. Ehresmann.

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Andrée Charles Ehresmann, (the widow)Faculté de Mathématique et Informatique 33 rue Saint-Leu, 80039 Amiens.E-mail address: [email protected]

How Charles Ehresmann�s vision of Geometrydeveloped with time(Read by Ronnie Brown)

First of all, I would like to thank the organizers of this Conference for having dedicatedit to the memory of Charles. During all his life, my husband has thought about Geometryas the foundation of Mathematics. However his idea of Geometry has evolved with time,and in the second part of his work, from the late �fties on, it came to include very abstractnotions in category theory. Initially they were suggested by geometrical situations butlater developed in a general setting, as he hoped they would allow for the uni�cation ofseveral domains into a simple and harmonious framework. As he said in a address givenin Lawrence in 1966 [O III, p. 759]*1 :� Mathematics is very akin to Art; a mathematical theory not only must be rigorous,

but it must also satisfy our mind in quest of simplicity, of harmony, of beauty... For thePlatonists among the mathematicians, the motivation of their work lies in this search forthe true structure in a given situation and in the study of such an abstract structure foritself... Mathematics is a never �nished creation, which has not to justify its existenceby the importance and the expanding number of its applications... It is the key for theunderstanding of the whole Universe �.This citation reveals Charles�vision of Mathematics and explains why all his life has

been devoted to them. He knew how to communicate his enthusiasm in long discussionsand (not very formal) lectures in which he always tried to convey his ideas by drawing�gures, for he thought that geometrical insights are most illuminating. So it might be ofsome interest to point out how his vision of Geometry progressively changed.

What is Geometry for Charles?In the "Notice" Charles wrote for his candidature to the Paris University in 1955 [O I,

p. 471], he describes the genesis of his research and its development up to this date, whichgives some clues to the turning-point from Geometry toward category theory he beganaround this time. At this end, he proposes a "large" de�nition of Geometry as �the theoryof more or less rich structures, in which are generally intertwined algebraic and topologicalstructures�, obtained by generalization of elementary geometry seen as the study of atopological space (R3) equipped with the action of a group of transformations. First thegroup is replaced by a pseudogroup of transformations which act only on some sub-spacesof the topological space; these sub-spaces form the open sets of a topological structure,so that the pseudogroup has an underlying topological structure. Whence the notion ofa local structure, which is still generalized, and enriched by in�nitesimal notions. It isthese successive generalizations, which abut to the important notions of category theoryto which Charles devoted his life from 1957 to his death in 1979 that give the thread ofCharles�works.I have no time here to recall the main results of Charles, even if I restricted myself to

his papers on Geometry taken in its usual meaning; anyway they are well-known and still

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easily accessible, for instance in the 7 volumes of his complete works [O]. My aim is onlyto indicate the development of his ideas up to his "conversion" to category theory (thatwas not always well understood by his colleagues!).

Homogeneous and locally homogeneous spacesCharles became interested in Geometry under its di¤erent aspects during his years at

the "Ecole Normale Supérieure", when he read the works of Sophus Lie on the advice ofErnest Vessiot, and followed Elie Cartan�s lectures on Riemannian spaces which gave hima glimpse on new directions in di¤erential geometry.His Thesis, obtained in 1934 [O I, p. 3] was supervised by Elie Cartan for whom he

kept a great admiration during all his life. Written during a two years stay in Princeton,it is devoted to the topology of some homogeneous spaces, and it still remains a referenceon Grassmann manifolds. Their homology and Poincaré groups are determined thanksto a powerful and then original method using cellular decompositions, similar to thoseconsidered later on in the theory of CW-complexes.The same method is also applied to more general manifolds in a paper following his

thesis [O 1, P. 55] which in fact formed the second part of his thesis.His following important paper [O I, p. 87] is devoted to Lie locally homogeneous spaces.

He was prompted to this notion by the locally Euclidean spaces studied by Elie Cartan in"Lectures sur la théorie des espaces de Riemann" and by ideas exposed in Veblen�s book"The foundation of Di¤erential Geometry".The Klein program de�ned a geometry by the data of a group operating on a space.

To de�ne a locally homogeneous spaces, the group is replaced by a local continuous group(or germ of group), and the space is obtained by gluing together more elementary spaceson which this local group acts, the gluing respecting the structure. Charles proved thatsuch a Lie locally homogeneous space which is compact and locally connected is equivalentto a Lie homogeneous space. In particular, he studies locally projective spaces and locallya¢ ne spaces. This paper is important by its later developments, for it conducted Charlesto a general de�nition of a pseudogroup of transformations and of the associated "localstructures", the avenue through which he later came to category theory, as we�ll explainlater on.

Fiber bundles and foliationsIn the early forties, Charles introduced the notion of a �ber bundle independently

from Steenrod, while the war had broken communications between France and the UnitedStates. He develops a general theory in a series of Notes from 1941 to 1944 [O I, p. 310-321],de�ning locally trivial principal bundles and their associated �bre bundles.Among the main results, let us note Lifting of Homotopy Theorems and the exact

sequence associated to a �bration [O I, p. 105], generalizing theorems proved in specialcases in his former papers.The problem of restricting the structural group is raised and solved in several instances,

for instance, for the tangent bundle on a di¤erentiable manifold [O I, p. 133]; as a by-product, he proved that, if the Universe of Relativity is compact, its Euler-Poincaré char-acteristic must he zero (he was proud that this result [O I, p. 319] brought him his �rstinvitation to Rio de Janeiro in 1952). Other examples he considers are the almost complex,quaternionian or hermitian manifolds, studied in the Theses of P. Libermann and Wu WenTsun.

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To study manifolds equipped with a completely integrable �eld of contact elements [OI, p. 322], Charles introduced the theory of a foliated manifold [O I, p. 155], which wasdeveloped by Reeb whose Thesis is a reference in this domain. Later on, Charles de�nedmore general foliations and adapted the notion of holonomy and the stability theoremsto locally simple foliations [O I, p. 370]. These results are re�ned in a substantial paperwritten in 1961 [O II, p. 563], unfortunately ignored by most specialists.

Pseudogroups of transformations and local structuresLie locally homogeneous spaces, topological manifolds, di¤erentiable or analytic man-

ifolds, but also �ber bundles and foliated manifolds, are structures constructed by gluingtogether more elementary structures of the same speci�c type in a well-de�ned sense. Inhis quest for simplicity and harmony, Charles has searched for a unifying theory.At this end, making more precise ideas of Veblen-Whitehead, he gives the de�nition

of a pseudogroup of transformations and of the associated structures. This notion is �rstde�ned in his 1947 paper on �ber bundles [O I, p. 133], and is generalized in followingpapers. For instance a �ber bundle becomes a structure associated to an appropriatepseudogroup of transformations on the product of its basis by the �ber.Roughly a pseudogroup of transformations G on a set E is formed by a set of 1-1

mappings between sub-sets of E closed by inversion, composition, and such that g belongsto G if it glues elements of G. Then the domains of the transformations in G generate atopology on E. A structure associated to G on a set V is obtained by gluing together theimages of charts of a complete atlas compatible with G (meaning that the change of chartspertains to G).In particular given a group of automorphisms of a topological space E, their restrictions

to the open sub-spaces form a pseudogroup of transformations on E, and the associatedstructures are the structures locally equivalent to E (equipped with the structure de�nedby G).In 1952, Charles proposes a more general theory of "local structures" in the frame

of Bourbaki�s species of structures [O I, 352, 411]. The local automorphisms of such alocal structure S form a pseudogroup of transformations. Conversely any pseudogroup oftransformations G gives rise to local structures, namely the structures associated to it,which are de�ned by complete atlases having their changes of charts in G.

Foundation of Di¤erential GeometryBundle theory has important applications in Di¤erential Geometry. In particular, the

di¤erentiable bundles are an appropriate setting for the study of in�nitesirnal connections,as it is shown in [O I, p. 179, 233].But Charles was unsatis�ed with notations on di¤erentials, and this prompted him

to introduce a coordinate-free representation, namely the in�nitesimal jets. He de�nedbundles of jets, in the holonomic, semi-holonomic and non-holonomic cases [O I, p. 343-369] which led him to give a modern foundation to Di¤erential Geometry (now folklore).In particular he developed a beautiful theory of prolongations of di¤erentiable manifoldsin the 3 cases, Holonomic and semi-holonomic jets (not yet fully exploited) simplify severalproblems involving di¤erential systems (to be compared with sprays).In�nitesimal structures (which generalize geometrical objects) and their covariants,

G-structures and their associated pseudogroups are introduced, and he raises the localequivalence problem for G-structures, studied in the thesis of P. Libermann.

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Let us note that the coordinate-free handling of di¤erential geometry initiated byCharles has been fundamental in the more recent development of Synthetic Di¤erentialGeometry in the categorical frame of topos theory. For instance Kock recognizes it hasstrongly in�uenced him.

The �rst step toward categories: the groupoidsIn his notes on Di¤erential Geometry of the early �fties, Charles makes an extensive

use of the notion of a groupoid (in the sense introduced by Brandt in 1926); as such agroupoid can be de�ned as a category in which every morphism is invertible, it is his �rststep toward category theory (though he did not realize it at this time).In fact he had already used the term groupoid in his 1950 paper on connections [O I,

p. 179], where he says that the isomorphisms from �ber to �ber of a �ber bundle form agroupoid HH�1. In 1952 [O I, 355], he de�nes the actions of this principal groupoid onthe bundle and uses it, for instance to de�ne covariant maps.Groupoids of in�nitesimal jets, with their di¤erentiable structure, as well as their op-

erations on manifolds, become an essential tool for the study of prolongations of manifoldsand of �ber bundles. In his view at this time, Di¤erential Geometry consists in the studyof the spaces on which groupoids of jets act and of their prolongations.Though he also describes the general composition law on jets and indicates its proper-

ties, that amounts to de�ne the category of jets, he does not use the term (he said to methat the link with categories was later pointed to him by Constantino de Barros).In fact, the turning point came in the late �fties, when Charles wrote 2 important

papers which initiate his work in category theory and from which many of his papers ofthe following ten years are more or less directly issued. They provide a unifying setting inthe category framework to improve and generalize his former results in several ways.

Species of local structuresIn 1951 [O 1, p. 153], Charles had already mentioned that a pseudogroup of transfor-

mations is a particular groupoid, and that only the open sets of the associated topologyare used, not the points, whence the idea of replacing the pseudogroup of transformationsby a groupoid and the topology by a paratopology (i.e., a complete distributive lattice).Let us remark that "topologies without points" have later taken a great importance; sinceparatopologies are extensively used, under the name of locales, in connection with Topostheory.This idea is formalized in the 1957 seminal paper "Gattungen von lokalen Strukturen"

[O II, p. 126], the �rst one written in a categorical framework.The pseudogroup of transformations is replaced by a local groupoid and even, more

generally, by a local category, that is a category equipped with a "local" order compatiblewith its structure (in modern term it is a category internal to a category of locales).He describes how such a local category can act on a space, determining a species of local

structures on it. This species is complete if it satis�es a gluing property (sheaf axiom).The main result of the paper is the Complete Enlargement Theorem (or Associated

Sheaf Theorem) which constructs the complete species of local structures associated toa given species, by a process generalizing the gluing process used to construct the localstructures associated to a pseudogroup of transformations. Various applications are given.

Locally trivial groupoids and associated bundles

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The following paper in 1959 [O I, p. 237] transposes the theory of �ber bundles in thecategorical setting.In former papers [O I, 355, 367], Charles had noted that the groupoid HH�1 of isomor-

phisms between �bers of a �ber bundle has a di¤erentiable structure compatible with itscomposition, and that it acts on the associated bundles. And he had de�ned its prolonga-tions, used for example to de�ne higher order connection elements [O I, p. 233]. Now hecharacterizes the properties of such a principal groupoid.First he de�nes the notions of a topological or di¤erentiable category and of its actions,

which he proposes to consider as de�ning generalized �ber spaces.To recover the principal groupoids, a topological groupoid must be locally trivial; it

means that each unit x has an open neighborhood on which there exists a local section sof the codomain map such that dom�s is the constant map on x.Charles proves that the theory of locally trivial groupoids is equivalent to that of prin-

cipal bundles, and that the spaces on which such a groupoid K acts correspond to theassociated �ber bundles.In a 1959 paper [O I, p. 421], he generalizes to di¤erentiable groupoids the theorem he

had proved in a Note in 1958 [O I, p. 374], namely that the largest group of transformationsincluded in a Lie pseudogroup of �nite type is a Lie group.

ConclusionIn the sixties, Charles wrote only 4 brief papers on Di¤erential Geometry, all in the

categorical frame, in which he developed a theory ot pro,dongations of di¤erentiable cat-

egories and of their actions (generalizing that of manifolds and bundles). His last viewson Di¤erential Geometry are summarized in the abstract [O I, p. 271] of the lecture hegave during a Conference on categories we organized in Amiens in 1973: it is the study ofdi¤erentiable categories, of their actions on manifolds and of their prolongations.The topological and di¤erentiable categories led Charles in 1963 to the general notion

of internal categories in a category, and of internal actions, and in 1968 to the theoryof sketched structures and of completions of categories and functors. In all these moreabstract works (reprinted in Parts II and IV of his "Oeuvres"), the initial motivation wasoften of a geometric nature, even if it is not always apparent in the paper.Thus the geometric part of Charles� work has been fundamental, not only for the

important results he obtained in it, particularly the study of �ber bundles and foliatedspaces and the foundations of di¤erential geometry on jets, but also for it gave a support forthe more general notions he developed in category theory and for their many applicationsboth in Physics, in Computer Science, and, more recently, in Systemics and Biology. Tocover all these aspects, the Bedlewo geometrically centered conference will be followed inOctober by the international conference "Charles Ehresmann : 100 ans" we are organizingin Amiens, in which we�ll insist more particularly on Category Theory and multidisciplinaryapplications.

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Ronnie BrownDepartment of Mathematics, University of Wales, Bangor, UK.E-mail address: [email protected]

Groupoids, Local-to-global, Higher dimensions: Three themes inthe work�of Charles Ehresmann

Charles Ehresmann�s work on category theory is unusual in its emphasis on the relationsbetween local and global methods and results, often with the notion of di¤erentiable (orLie) groupoids, and for early ventures into higher order categories, through notions ofstructured categories.This talk will show two of his strong in�uences on my work.

One, obtained via Jean Pradines, is the use of admissible local sections of a locally Liegroupoid. The background work of Ehresmann on Lie groupoids and germs allowed for aninteresting algebraic encapsulation of the intuition of �iteration of local procedures�, andso to obtain a holonomy groupoid, and then a monodromy groupoid, in situations moregeneral than foliations.

Second, Ehresmann�s initial work on double categories was published when I wastrying to express a putative higher order van Kampen theorem in terms of conjecturedhigher homotopy groupoids. These were found with Philip Higgins in the late 1970s. Theyyielded nonabelian methods for certain local-to-global problems in dimensions more than1, and a new version of the interface between homology and homotopy.

Some work has been done on the interaction of these two themes, in which local ad-missible sections are generalised to certain local homotopies of an identity morphism to anautomorphism.

Much more work is needed to apply these methods in the theory and applications ofmanifolds.

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Jean-Claude Hausmann

University of Geneva, Switzerland.E-mail address: [email protected], http://www.unige.ch/math/folks/hausmann/

Robot arms and Moebius transformation(the snake charmer algorithm)(Joint work with Eugenio Rodriguez)

Consider a polygonal robot arm in Rd, starting from the origin and made ofm segmentsof length 1. We would like to move this arm so that its end follows a prescribed curve in Rd.The Snake charmer algorithm presented here performs this task via the diagonal actionof the Moebius group M(d � 1) on each segment. Computer animations will be shown.The notion of genarized connection in the sense of Ehresmann plays a central role and theholonomy arising from following closed curves is quite interesting.

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Anders KockUniversity of Aarhus, Denmark.E-mail address: [email protected]

Pregroupoids and their enveloping groupoids

Abstract:The notion of pregroupoid (or a¢ noid space, Weinstein) is a many-object gen-eralization of the notion of principal �bre bundle, with the structure encoded in terms of oneternary operation. To a principal �bre bundle P , Ehresmann constructed a groupoid PP�1

which in some sense is �equivalent�to P . This construction also works for pregroupoids.A pregroupoid does not embed naturally in PP�1. But there exists a somewhat biggercanonical �enveloping�groupoid P+ (roughly, P+ is four times the size of PP�1), in whichP embeds naturally. In fact the embedding P ,! P+ is the unit for a pair of adjointfunctors between the categories of pregroupoids and groupoids.

A special case of pregroupoids are pregroups, which have been studied (under variousnames, e.g. Schar), by Prüfer, Baer, Certaine, Vagner, and others.

We shall consider in particular the di¤erentiable case.

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Ivan KoláµrMasaryk University, Brno, Czech Republic, Masaryk University, Brno, Czech Republic.E-mail address: [email protected]

Functorial prolongations of Lie groupoids

Our starting point are the results by C. Ehresmann on the jet prolongations of Liegroupoids and their actions. We extend his construction to the case of an arbitrary �berproduct preserving bundle functor F on the category of all �bered manifolds with m-dimensional bases and �ber preserving maps with local di¤eomorphisms as base maps.We deduce that every Lie groupoid � over M induces a Lie groupoid F� over M andevery action of � on a �bered manifold Y ! M induces an action of F� on FY !M . The principal bundle version of this result was deduced by M. Doupovec and theauthor in Monatsh. Math. 134(2001), 39�50. Our procedure is essentially based on thecharacterization of F in terms of Weil algebras by W. Mikulski and the author, Di¤erentialGeometry and its Applications 11(1999), 105�115. We also clarify that for every Weilfunctor TA on the category of all manifolds, TA� is a Lie groupoid over TAM and everyaction of � on Y is canonically extended into an action of TA� on TAY ! TAM ..

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Paulette LibermannUniversité Denis Diderot, Paris, France

C. Ehresmann concepts in Di¤erential Geometry

We �rst give a survey of C. Ehresmann career, pointing out the great in�uence of hisSeminars on the Mathematical Community.

Then we outline some of the tools that C. Ehresmann has introduced in Di¤erentialGeometry (which are often used in Mechanics and Physics).

1. Pseudogroups of transformations. Use of an atlas compatible with a pseudogroup tode�ne manifolds, �ber bundles, foliations and more generally local structures.

2. Reduction of the structure group of a principal bundle. In particular G-structures(among them almost complex and almost symplectic structures).

3. Connections on a principal bundle; Cartan connections.

4. Groupoids.

5. Jets (holonomic, semi-holonomic, non-holonomic). Prolongations.

6. Lie groupoids and Lie psendogroups.

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Charles-Michel MarleUniversité Pierre et Marie Curie, Paris, France.E-mail address: [email protected], http://www.math.jussieu.fr/~marle/

The works of Charles Ehresmann on connections: from Cartanconnections to connections on �bre bundles, and some modern

applications

In a series of three papers published in Ann. Ec. Norm. (vol. 40, 1923, pp. 325�412,vol. 41, 1924, pp. 1�25 and vol. 42, 1925, pp. 17�88), Élie Cartan introduced a¢ ne connec-tions on manifolds and de�ned the main related concepts: torsion, curvature, holonomygroups, : : : He discussed applications of these concepts in Classical and Relativistic Me-chanics; in particular he explained how parallel transport with respect to a connectioncan be related to the principle of inertia in Galilean Mechanics and, more generally, canbe used to model the motion of a particle in a gravitational �eld. In subsequent papers,Élie Cartan extended these concepts for other types of connections on a manifold, calledconformal, Euclidean and projective connections.

Around 1950, Charles Ehresmann introduced connections on a �bre bundle and, whenthe bundle has a Lie group as structure group, connection forms on the associated principalbundle, with values in the Lie algebra of the structure group. He called Cartan connectionsthe various types of connections on a manifold previously introduced by É. Cartan, andexplained how they can be considered as special cases of connections on a �bre bundle witha Lie group G as structure group: the standard �bre of the bundle is then an homogeneousspace G=G0; its dimension is equal to that of the base manifold; a Cartan connectiondetermines an isomorphism of the vector bundle tangent to the the base manifold onto thevector bundle of vertical vectors tangent to the �bres of the bundle along a global section.

These works will be reviewed, and some applications of the theory of connections inmodern physics will be sketched.

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Gaël MeigniezUniversite de Bretagne Sud, France.E-mail address: [email protected]

The nature of �brations

The notion of �bration, as a map that veri�es the homotopy lifting property � say,for polytopes �was pointed out by Ehresmann in the early 40�s. In this talk, I shall �rstpresent a characterization, in terms of three properties.

� the �rst one (homotopical submersion) is local at the source, and thus veri�ed bysubmersions, as well as by �brations;

� the two other ones (triviality of emerging cycles and of vanishing cycles of everydimension dimension) express the isomorphism of the homotopy groups of the neigh-boring �bres.

If we turn to the category of open manifolds of �nite dimension and smooth maps, thisallows a caracterization of bundles (maps which are trivial over an open covering of thebase) among submersions (maps whose di¤erential at each point is onto), in terms of thehomotopy type of the space of embeddings of (large) compact domains in the �bres.In particular, consider a submersion E �! B where E; B are open manifolds; and

assume that it is a �bration, i.e. it veri�es the homotopy lifting property. In general, itmay not be a bundle. For example, there is a submersion-�bration over B = R with all�bres R3 but one which is the Whitehead manifold, a contractible open 3-manifold nothomeomorphic to R3.We show that every submersion-�bration is necessarily a bundle under any of the fol-

lowing hypothesis on the topology of the �bres:

� The dimension of the �bres is 2.

� Each �bre is di¤eomorphic to Rp.

� Each �bre is topologically �nite, of dimension at least 5, and its boundary at in�nityis simply connected.

Also, there is a stabilization result: every submersion-�bration becomes a bundle if wemultiply the �bres by some RN .Finally, we give applications to foliations.

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Pierre MolinoMontpellier, FranceE-mail address: [email protected]

(Speech read by Jean Pradines)

L�oeuvre originale de Charles Ehresmann mérite d�autant plus d�etre commémorée quece grand visionaire a été, dans une certaine mesure, méconnu. Peu de mathématiciensont innové comme lui, a la maniere d�un artiste qui découvre de nouveaux paysages, dansune reverie féconde ou apparaissent des perspectives que d�autres découvriront plus tard.Ainsi de la théorie la plus générale des feuilletages singuliers. Souvent, on a eu peine a lesuivre dans des anticipations dont, seul peut-etre, il voyait d�emblée le contenu géométrique(ainsi de ses travaux sur les catégories). Il a devancé bien des développements a venir,comme le héros Balzacien du "Chef d�oeuvre inconnu", ce peintre génial dont les toilesrestent indéchi¤rables dans leur modernité abstraite, laissant pourtant deviner, par undétail admirable, une vision nouvelle et profonde.

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Bertrand Monthubert

Toulouse, France.E-mail address: [email protected]

Groupoids and Index Theory in the singular manifolds setting

The introduction of groupoids in Index Theory has been realized by A. Connes in thecontext of foliations. It was later developped and we will present an overview of the waygroupoids intervene in index theory. We will also give examples where groupoids techniquesinduce index results in the context of singular manifolds.

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Jean PradinesUniversité Paul Sabatier, Toulouse III, Toulouse, France.E-mail address: [email protected]

In Charles Ehresmann�s footsteps: from Group Geometries toGroupoid Geometries

In this lecture we intend to focus on one of the turning points in Charles Ehresmann�swork: we mean, when dealing with the theory of principal bundles, the introduction,parallel to that of the structural group, of the structural groupoid. (Note that nowadaysthese are often known too as gauge group and gauge groupoid, rather conveniently thoughambiguously, since these terms possess also quite di¤erent meanings, according to theauthors). The actions, on the bundle, of the group and the groupoid are commuting.Together with the study of the category of jets (and more speci�cally the groupoid of

invertible jets), this led him to the basic idea (rediscovered much later by various authors)of considering (small) internal groupoids (more generally categories) in various (large) cat-egories, originally and basically the category of (morphisms between) manifolds. In par-ticular this leads to an inductive de�nition of multiple (smooth) groupoids. This unifyingconcept involves a very far-reaching intertwining between algebra and geometry.He also stressed the interest of choosing the source and target maps and/or the anchor

map (or transitor) in suitable subclasses.We shall show it is convenient to impose to thesesubclasses some suitable stabibity properties, and propose various choices unifying varioustheories. We point out that the stabiblity properties we need are very easily satis�ed inthe topos setting, but one gets much wider ranging theories (especially when aiming atapplications to Di¤erential Geometry) when not demanding the structuring category to bea topos.We note that the concepts of gauge group and groupoid have a strong intuitive (geo-

metrical or physical) interpretation. When thinking the elements of the principal bundleP as �events�or �observations�, the �bres, which are also the elements of the base or theobjects of the groupoid, may be viewed as �observers�.In that respect, the unique object of the group plays the role of a �universal observer�,

and the group itself of an �absolute� gauge reference. On the other hand the groupoidallows direct comparisons between the various observers, without using the medium ofthe absolute observer. The comparison between these two �points of view�(absolute andrelative) is realized by means of the projections of the bundle P onto the bases of thegroup and of the groupoid (the former being a singleton), and by the basic fact that thegroup and the groupoid induce on P the �same� groupoid (more precisely isomorphicgroupoids). From a purely algebraic point of view, this describes a Morita equivalencebetween the structural group and groupoid. But in Ehresmann�s internal setting, thisacquires a lot of di¤erent meanings depending on the above-mentioned various choices.This groupoid (which is the core of the structure) inherits from the two projections a

very rich extra structure, which may be recognized as a particular instance of a very specialand interesting structure of smooth double groupoid. In the purely algebraic context, thisstructure has been described in the literature under various names and equivalent ways:rule of three, a¢ noid (A. Weinstein), pregroupoid (A. Kock).

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Now it turns out that all the essential features of this description are still valid whenreplacing the structural group by a groupoid, and the situation becomes perfectly symmet-rical (up to isomorphism). In the purely algebraic (hence also topos) setting, this symmetricsituation has been described by A. Kock, in the language of torsors and bitorsors.In the �physical�interpretation, we can say that we have now two classes of observers,

and a comparison between two �conjugate points of view�. We also note that it is no longerdemanded the groupoids to be transitive (then certain pairs of observers cannot comparetheir observations), and also the rank of the anchor map to be constant (then the isotropygroups are allowed to vary, and this may be thought as symmetry breakings or changes ofphase).We shall give various examples of this unifying (purely diagrammatic) situation (and

more general ones), which encompasses the construction of associated bundles as well, therealization of a non abelian cocycle, including Hae�iger cocycles, the construction of theholonomy groupoids of foliations, and the Palais globalization of a local action law.By lack of time, we shall not describe the corresponding in�nitesimal situation, which

of course was Ehresmann�s main motivation for introducing the structural groupoid, inorder to understand the meaning of �in�nitesimally connecting� the �bres. This will betackled by other lecturers at the present Conference.

Our conclusion will be that the transition, from the Kleinian conception of Geometryas the study of group actions, to the Ehresmannian enlarged point of view, consisting inconsidering groupoid actions, involves a conceptual revolution which parallels the physicalrevolution from the classical concept of an absolute universe to the modern visions aboutour physical space.

It might well be that this revolution is not enough well understood presently, and isliable to come out into unexpected developments.

36

Alexandre A. M. Rodrigues

University of S¼ao Paulo, Brazil.E-mail address: [email protected]

Contact and equivalence of submanifolds of homogeneous spaces

All manifolds and maps are assumed to be of class C1 unless otherwise stated.Let G be a Lie group acting on the manifold N . Two submanifolds S, S of N , S,

S � N , are G-equivalent if there exists g 2 G such that g:S = S. S and S have G-contact of order k � 0 at points a 2 S and a 2 S if there exists g 2 G such that g:a = aand g:S and S have contact of order k at the point a. S and S are locally G-equivalentat points a and a if there are open neighborhoods of a and a which are G-equivalent.We shall consider the following problems:

(1) Let ' : S �! S be a di¤eomorphism such that S and S have G-contact of order k atcorresponding points a 2 S and '(x) 2 S. Can we always choose k su¢ ciently highto ensure that S and S are G-equivalent?

(2) Under which conditions there exists g 2 G such that ' is the restriction to S of themap Lg : x 2M �! g:x 2M?

(3) Assume that G sets transitively on M . Give conditions on the contact elements of Sfor S to be a homogeneous space of a Lie subgroup of G.

Assuming regularity conditions on S and S, the integer asked for in problem 1) alwaysexists.Let p be the dimension of S. G acts in a natural way in the manifold Ck;pM of contact

elements of order k and dimension p of M [ ]. For a point a 2 S, let CkaS, Gka and dkra berespectively the contact element of order k of S ate the point a, the isotropy subgroup ofG at the point CkaS 2 Gk;pM and the dimension of Gk. Let X = CkaS and denote by O

kx

and hk(x) respectively the orbit of CkxS in Gk;pM and the dimension of the vector space

TXOkx \ TXCkS, where CkS is the submanifold of Gk;pM of all contact elements of order

k of S and TXOkx and TXCkS are the tangent spaces of Okx and C

kS at the point x.For k0 � k, dk(a) � dk0(a) and hk(a) � hk0(a). Hence, there exists an integer k � 1,

such that dk(a) = dk�1(a) and hk(a) = hk�1(a). We say that a is a G-regular point of S ifthere exists k � 1 such that 1) dk(a) = dk�1(a) and hk(a) = hk�1(a), 2) dk(x) and ha(x)are constant for x in a neighborhood of a in S. If these conditions are satis�ed we say thata is a k-regular point of S under the action of G. If a is a k-regular point of S thus g:a isa k-regular point of g:S. The order of a is the least integer k satisfying the two conditionsabove.

Theorem 1 Let S; S � M be two submanifolds of M of same dimension p. Let a 2 Sand a 2 S be two points. Assume that a is a k-regular point of S and that there exists acontinuous map ' : V �! G, de�ned in a neighborhood V of a in S such that '(a):a = a,

37

'(x):x 2 S and '(x):CkxS = Ck'(x)Sa for all x 2 V . Then, there exist open neighborhoodsW and W of a and a in S and S which are G-equivalent.

The proof of theorem 1 rests on the theorem of uniqueness of solutions of partial dif-ferential systems of �nite type [ 1 ], [ 2 ].

In theorems 2 and 3 we assume that G is a compact Lie group and H is a closedsubgroup of G. Let L be the union of all G-orbits of Ck;pM of type H that is, orbits whoseisotropy subgroups are conjugate to H. Denote by N the quotient space of L by the orbitsand by � : L �! N the natural projection. It is known [ 3 ] that (L;N; �;G=H;G) is adi¤erentiable �ber bundle with structural group G.Let f : S �! S be a di¤eomorphism and such that S and S have G-contact of order

k � 1 at corresponding points x 2 S and x = f(x) 2 S and let a 2 S and a 2 f(a) 2 S betwo points. Considering suitable cross sections of the �ber bundle (L;N; �;G=H;G) onecan prove the existence of a neighborhood V of a in S and of a di¤erentiable map ' : V�! G such that '(x):x = f(x) and '(x):CkxS = C

kxS. Hence, theorem 1 can be restated as

follows:

Theorem 2 Assume that there exists k � 1 and that

Theorem 3 Assume that S and S are connected and that there exists an integer k � 1such that:

Consider again the �ber bundle (L;N; �;G=H;G). There exists a �nite number of realvalued di¤erentiable functions ~�i, 1 � i � r, de�ned in L, such that two contact elementsX;X 2 L are in the same orbit of G if and only if ~�i(X) = ~�i(X). Given a submanifoldS � M of dimension p, if the orbits of CkxS are of type H for all x 2 S one can pull backthe functions ~�i by the map �k : x 2 S ! CkxS 2 L. The functions �i = ~�i � �k are calleda complete set of G-invariants of order k of the submanifold S of M . Often the invariantscan be de�ned in a natural way and have deep geometrical meaning as for instance, thecurvature and torsion of curves in R3.Assuming that the isotropy subgroups of CkxS and C

kxS are of type H for all x 2 S and

x 2 S, the invariants of order k, �i and �i are de�ned in S and S. One can then restatetheorems 2 and 3 replacing condition (3) by the following condition:The condition hk(x) = 0 in theorem 3 is equivalent to the statement that the rank of

the di¤erentials d�i, 1 � i � r, is p at every point x 2 S.Assume now that G acts transitively on M and let S be the orbit of a Lie subgroup

K of G. Then, hk(x) = p and Ckx , Ckx0 are conjugate subgroups for all x; x

0 2 S and everyk � 0. Hence, there exists k � 1 such that every x 2 S is a k-regular point of S.

Theorem 4 A necessary and su¢ cient condition for a submanifold S �M to be an openset of an orbit of a connected Lie subgroup K of G is that there exists k � 1 such that x isa k-regular point of S and hk(x) = p for all x 2 S.

If we assume moreover that G is compact, then a complete set �i of invariants of orderk is de�ned on S. Clearly hk(x) = p for every p 2 S if and only if the functions �i areconstant on S. Hence, the following corollary to theorem 4 holds:

38

Theorem 5 Let S �M be a connected submanifold and assume G compact. Assume alsothat there exists k � 1 such that every point of S is k-regular and that the isotropy groupsGkx are conjugate in G. Then, S is an open set of an orbit of a connected Lie subgroup ofG if and only if the invariants of order k are constant on S.

Bibliographie

[1] M. Kuranishi, Lectures on involutive systems of partial di¤erential equations, Publica-coes da Sociedade de Matematica de Sao Paulo, 1967.

[2] J. A. Verderesi, Contact et conguence de sous varietes , Duke Math. J., 49, 1982,513�515.

[3] Bredon, G. E., Introduction to compact transformation groups, Academic Press, NewYork, 1972.

39

W÷odzimierz Tulczyjew

Universita di Camerino, Monte Cavallo, Italy.E-mail address: [email protected]

Modernization of Ehresmann jet theory

Extentions of Ehresmann jet theory are presented. These extentions were introducedto form a basis for modern Analytical Mechanics and Field Theory. A presentation ofthe theory in a form less dependent on coordinate systems is introduced using ideals ofthe algebra of di¤erentiable functions. This presentation is somewhat related to A. Weil�stheory of points proches.

40

Wu Wen-Tsun

Beijing, People�s Republic of China

(Speech read by Jean Pradines)

In the winter of 1947 several Chinese students including myself arrived at Strasbourgto study Mathematics there. We were ardently accepted by Ch. EHRESMANN andbegan our mathematics researches since that time. During my stay about two years inStrasbourg, I learned the topics about complex manifolds and Grassmannians. For bothsubjects Professor EHRESMANN was doubtless one of the main creators. In particular theGrassmannians becomes henceforth one of the main tools in my further research works.Even in quite recent years I have discovered by means of Grassmannians a method ofde�ning the Chern classes and Chern numbers for an algebraic variety with arbitrarysingularities. These newly de�ned classes and numbers may be computed explicitely in aneasy way through the well known structure of Grassmannians mainly due to EhresmannAll these were achieved owing to my acquaintance with the Grassmannians during my stayin Strasbourg. I shall explain in some details how this is done in a mathematical paperdedicated to my former teacher EHRESMANN in near future.

41

Michel Zisman

Université Paris 7 Denis Diderot, Paris, France.E-mail address:[email protected]

Quasi-commutative cochains in algebraic topology (After MaxKaroubi)

We introduce a new type of �di¤erential forms�on a simplicial set (= a space) over abasic commutative coherent ring k, for instance Z , Fp;Q;R;C : : : They de�ne a di¤erentialgraded algebra, D�(X), called quasi-commutative, which is in fact quasi-isomorphic tothe usual algebra of cochains C�(X). The algebra D�(X) determines (assuming some�niteness conditions) the homotopy type of X when k = Z, a result linked with the workof Quillen and Sullivan in rational homotopy, in what case the di¤erential algebras aregenuine commutative algebras and not only quasi-commutative, as in our case.Given two spaces X and Y , the cup-product is de�ned thanks to a map

mX;Y : D�(X)D�(Y )! D�(X � Y )

We shall construct functorially a subcomplex D�(X) �D�(Y ) � D�(X)D�(Y ) called thereduced tensor product, and verifying mainly the two following conditions :�the preceding inclusion is a quasi-isomorphism (i.e. induces an isomorphism in coho-

mology)�the following diagram (where the vertical arrows are induced by switching round X

and Y ) is commutative :

D�(X) �D�(Y ) mX;Y��! D�(X � Y )??y ??yD�(Y ) �D�(X) mY;X��! D�(Y �X):

The main aim of the talk will be to present the construction of the functor D providedwith its additional structure given by mX;Y and the reduced tensor product �.

42

ABSTRACTS OF MINICOURSES

Alexandr S. Mishchenko

Department of Mathematics, Moscow State University, Leninskije Gory, 119992Moscow,Russia.E-mail address: [email protected], http://higeom.math.msu.su/asmish/

K-theory over C*-algebras

1 Some elementary and evident examples

The most general picture for the Hirzebruch formula for oriented smooth manifolds can berepresented as follows. Let M be a closed oriented non simply connected manifold withfundamental group �. Let B� be the classifying space for the group � and let

fM :M�!B�;

be a map inducing the isomorphism of fundamental groups. Let �(B�) denote the bordismgroup of pairs (M; fM). Recall that �(B�) is a module over the ring � = �( pt ).In [1] a homomorphism was constructed

sign : �(B�)�!L�(C�) (1)

which for every manifold (M; fM) assigns the element sign (M) 2 L�(C�), where L�(C�)is the Wall group for the group ring C�. The homomorphism sign satis�es the followingconditions:(a) sign (M) does depend only on the homotopy equivalence class of the manifold M .(b) if N is a simply connected manifold and �(N) is its signature then

sign (M �N) = sign (M)�(N) 2 L�(C�):

We shall be interested only in the groups after tensor multiplication with the �eld ofrational numbers Q, in other words in the homomorphism

sign : �(B�)Q�!L�(C�)Q:

Since �(B�)Q � H�(B�;Q) � the homomorphism sign can be considered as

sign : H�(B�;Q)�!L�(C�)Q:

Therefore sign represents the cohomology class � 2 H�(B�;L�(C�)Q):The key idea is that for any manifold (M; fM) the signature can be presented by a

version of the general Hirzebruch formula

sign (M; fM) = hL(M)f �M(�); [M ]i 2 L�(C�)Q: (2)

44

If � : L�(C�)Q�!Q is an additive functional and �(��) = x 2 H�(B�;Q) then the lefthand side of

sign x(M; fM) = hL(M)f �M(x); [M ]i 2 Q

is called the higher signature due to S.P.Novikov. This gives a description of the family ofall homotopy-invariant higher signatures.The general Hirzebruch formula (2) has di¤erent settings: smooth, combinatorial, func-

tional, algebraic. All of them have useful applications in non commutative geometry andtopology.

2 Almost �at bundles from the point of view of C*-algebras

Connes, Gromov and Moscovici [2] showed that for any almost �at bundle � over themanifoldM; the index of the signature operator with values in � is a homotopy equivalenceinvariant ofM: From here it follows that a certain integer multiple n of the bundle � comesfrom the classifying space B�1(M): The geometric arguments allow to show that the bundle� itself, and not necessarily a certain multiple of it, comes from an arbitrarily large compactsubspace Y � B�1(M) trough the classifying mapping.Using a natural construction of [3], one can present a simple description of such bundles

as a bundle over a C��algebra. For this we modify the notion of almost �at structure onbundles over smooth manifolds and extend this notion to bundles over arbitrary CW -spacesusing quasi-connections [4]. More of that it is possible to construct so called classifyingspace for almost �at bundles.

3 Twisted K-theory due to M.Atiyah and G.Segal

In the paper [5] M. Atiyah and G. Segal have considered families of Fredholm operatorsparametrized by points of a compact space K which are continuous in a topology weakerthan the uniform topology, i.e. the norm topology in the space of bounded operators B(H)in a Banach space H.Therefore, it is interesting to ascertain whether the conditions, characterized families of

Fredholm operators, from the paper [5] precisely describe the families of Fredholm operatorswhich forms a Fredholm operator over the C��algebraA = C(K) of all continuous functionson K.It is not supposed by the authors of the paper [5] that an operator over algebra A

admits the adjoint one or in their terms continuity of the adjoint family.Here we aim to clarify the question of description of the class of Fredholm operators

which in general case do not admit the adjoint operator. For the �rst time, operators whichplay the role of Fredholm operators and may not have the adjoint ones were considered in

45

the paper [6]. Since the main class of operators considered in the paper [6] is the class ofpseudodi¤erential operators, for any element of which the adjoint operator automaticallyis a bounded one, then existence of the adjoint operator was not the actual question forthe main goals of this paper.However, in their paper [5] authors have considered operators, which may not have the

adjoint one, in the form of families of operators continuous in the compact-open topologythe adjoint families of which, in general case, may not be a continuous one. We can showthat the class of Fredholm operators over arbitrary C��algebra, which may not admit theadjoint ones, can be extended in a such way that the class of compact operators used inthe de�nition of the class of Fredholm operators contains compact operators both with andwithout existence the adjoint ones.In the case when the C��algebra is a commutative algebra of continuous functions on

a compact space appropriate topologies in the classic spaces of Fredholm and compactoperators in the Hilbert space can be constructed which fully describe the sets of Fredholmand compact operators over the C��algebra without assumption of existence boundedadjoint operators over the algebra.

References

[1] A. S. Mishchenko. Homotopy invariant of non simply connected manifolds. Ra-tional invariants. I. Izv.Akad.Nauk.SSSR, 34(3):501�514, 1970.

[2] A. Connes, M. Gromov, and H. Moscovici. Conjecture de Novikov et �brés presqueplats. C. R. Acad. Sci. Paris, série I, 310:273 �277, 1990.

[3] B. Hanke and T. Schick. Enlargeability and index theory. preprint, March 17,2004.

[4] Nicolae Teleman. Distance function, linear quasi�connections and chern character.IHES, /M/04/27:11, 2004.

[5] M. Atiyah and G. Segal. Twisted k-theory. arXiv:math.KT/0407054, pages 1�49,3 October 2003. Eltctronic archive: arXiv:math.KT/0407054 v1 5 Jul 2004.

[6] Index of elliptic operators over C��algebras. Izvrstia AN SSSR, ser. matem.,43(4):831�859, 1979.

46

ABSTRACTS OF LECTURES

Kojun Abe

Department of Mathematical Sciences, Shinshu University, Matsumoto, Japan.E-mail address: [email protected], http://math.shinshu-u.ac.jp/�kabe

On the structure of the groups of di¤eomorphisms of manifoldswith boundary and its applications

(Joint work with Kazuhiko Fukui)

In this talk we shall consider the �rst homology group D(M) of the group of di¤eomor-phisms of a smooth manifold M which are isotopic to the identity through isotopies withcompact support. In case M a smooth manifold without boundary, Thurston proved thatthe group D(M) is perfect. For the caseM = [0; 1] Fukui proved H1(D([0; 1]) �= R�R. IfM is a smooth manifold with boundary of dimM � 2, then we shall prove that D(M) isperfect. The key point of the proof is combining the result by Tsuboi of the group of leafpreserving di¤eomorphisms of a foliated manifold with the method by Takens normalizingthe singularities of smooth vector �elds.We apply those results to calculate the �rst homology group of the group DG(M) of G-

equivariant di¤eomorphisms of a smooth G-manifold M which are isotopic to the identitythrough equivariant isotopies with compact support. If M has one orbit type, then thegroup DG(M) is perfect. In the case of M with codimension one orbit we can calculateH1(DG(M)) by analyzing the behavior of G-equivariant di¤eomorphisms of M around thesingular orbits using the functional smooth structure on the orbit space M=G and Baker-Campbell-Hausdor¤ formula for the Lie group. WhenM=G is a smooth orbifold with sometype of isolated singularities, we can calculate H1(D(M=G)) using the method by Takensand apply to the case of a S1-action on S3. If M is the O(n)-manifold investigated byHirzebruch-Mayer, we shall prove that DO(n)(M) is perfect by analyzing the structure ofequivariant di¤eomorphisms around the singular orbits applying the method by Bierstoneand Schwarz.

��

Abdelouahab Arouche

USTHB, Fac. Math. BP 32 El Alia 16111, Alger AlgeriaE-mail address: [email protected]

48

Restriction properties of equivariant K-theory rings

An important ingredient in the completion theorem of equivariant K-theory given byS. Jackowski is that the representation ring R(G) of a compact Lie group satis�es tworestriction properties. We give in this note su¢ cient conditions on a compact G-space Xsuch that these properties hold with K�(X) instead of R(G).

��

Bogdan Balcerzak

Institute of Mathematics, Technical University of ×ódz, ×ódz, Poland.E-mail address: [email protected]

Secondary characteristic classes foranchored extensions Leibniz algebras

A construction of a secondary characteristic homomorphism for some pairs of anchoredLeibniz algebras is proposed. The generalization of secondary characteristic classes for Liealgebroids is obtained.

��

Marcin Bobienski

Warsaw University, Warszawa, Poland.E-mail address: [email protected]

On SO(3) geometry in dimension �ve(Joint work with P. Nurowski)

49

A nonstandard inclusion SO(3) � SO(5) associated with the irreducible representation� of SO(3) inR5 is considered. The tensor t reducing the groupO(5) to this SO(3) is found.It is given by a ternary symmetric form with some special properties. A 5-dimensionalmanifold (M; g; t) with the Riemannian metric g and the ternary form t generated bysuch tensor de�nes an SO(3) structure on M . The Gray-Hervella-type classi�cation ofsuch structures is established using the so(3)-valued connections with torsion. Structureswith antisymmetric torsions are studied in detail. In particular, it is shown that theintegrable models (those with vanishing torsion) are isometric to the symmetric spacesM+ = SU(3)=SO(3), M� = SL(3;R)=SO(3), M0 = (SO(3) �� R5)=SO(3). We also�nd all SO(3) structures with transitive symmetry groups. Given an SO(3) structure(M; g; t) we de�ne its twistor space Z to be an S2-bundle of those 2-forms on M whichspan a 3-dimensional irreducible representation of SO(3) and which have unit length. The7-dimensional twistor manifold Z is then naturally equipped with several CR-structuresand G2-structures. The integrability conditions for these structures are discussed andinterpreted in terms of the Gray-Hervella-type classi�cation of (M; g; t).

��

Bogdan Bojarski[1] and Giorgi Khimshiashvili[2]

[1]Mathematical Institute, Polish Academy of Sciences, Warszawa, Poland.E-mail address: [email protected]

[2]Georgian Academy of Sciences, Tbilisi, Republic of Georgia.E-mail address: [email protected], http://www.rmi.acnet.ge/�khimsh

Riemann-Hilbert problems in loop spaces

Two recent generalizations of the classical Riemann-Hilbert transmission problem inthe context of loop spaces will be discussed. The �rst one is concerned with piecewiseholomorphic vector-functions with values in a complex representation space of a compactLie group G. Such problems are naturally described in terms of the grassmannian modelof the loop group LG. In particular, a version of Fredholm theory for such problems canbe formulated in terms of Fredholm pairs of subspaces and Kato grassmannians, whichdevelops previous results of the authors concerning the geometry of Fredholm pairs andKato grassmannians.The second generalization is concerned with piecewise pseudoholomorphic functions

taking values in a certain space of immersed loops in a three-dimensional riemannian

50

manifold introduced by J.-L.Brylinski and L.Lempert. It will be shown that solutions tosuch problems can be constructed by solving Cauchy problem for a certain nonlinear partialdi¤erential equation on the underlying manifold. A geometric method for solving latterproblem will be described and a few explicit examples of solutions will be given.

��

Maciej Borodzik[1] and [2]Henryk ·Zo÷¾adek

Warsaw University, Warszawa, Poland.[1]E-mail address: [email protected]

[2]E-mail address: [email protected]

Classi�cation of complex plane a¢ ne algebraic curves with zeroEuler Characteristic

The topological classi�cation of complex algebraic curves is well known; they are sur-faces of genus g deprived of n points and with some self- intersections. But, when we �xthe topology, the classi�cation with respect to automorphisms of the a¢ ne space is notthat simple. Abhyankar, Moh, Suzuki, Zaidenberg and Lin have classi�ed all plane curvesof the type (0,1) and without self-intersections (the contractible ones).We present complete classi�cation of curves of the type (0,1) with one self-intersection

and of the type (0,2) (with zero Euler characteristic). The classi�cation contains 19 casesfor the �rst type and 23 cases for the second type.We present also new tools used in the proofs. These are: estimation of invariants of

a singularity (like the Milnor number) via its order and codimension, the Poincaré-Hopfformula and a bound for the sum of codimensions of singularities.

��

Beniamino Cappelletti Montano

University of Bari, Bari, Italy.E-mail address: [email protected]

51

On Legendrian foliations on almost S-manifolds

Given an almost S-manifold (M2n+r; ��; ��; �; g), � = 1; : : : ; r, a Legendrian folia-tion on M is a n-dimensional integrable distribution of the 2n-dimensional distribution

H =rT

�=1

ker (��). We attach to these foliations a connection which we call be-Legendrian.

Using this connection we are able to �nd some results on characteristic classes, Ehresmannconnections and projections of Legendrian foliations onto Lagrangian foliations.

��

Gratiela Cicortas

University of Oradea, Oradea, Romania.E-mail address: [email protected]

Categorical sequences and relative categories

MSC (2000): 55M30Key words and phrases: Lusternik- Schnirelmann category, relative category, cate-

gorical sequence, relative categorical sequence

R. H. Fox [2] characterized the Lusternik- Schnirelmann category using categoricalsequences. In [1], E. Fadell de�ned the relative category. I de�ne relative categoricalsequences and I prove that, in suitable hypothesis, the relative category is the minimumlength of relative categorical sequences. Possible applications are the product inequalities.

References

[1] E. Fadell, Lectures in cohomological index theories of G- spaces with appli-cations to critical point theory, Universita della Calabria, Cosenza, Italy, 1985

[2] R. H. Fox, On the Lusternik- Schnirelmann category, Ann.Math. 42 (1941),pp. 333�370

52

��

Jürgen Eichhorn

Greifswald University, Greifswald, Germany.E-mail address: [email protected]

Absolute and relative characteristic numbers for open manifolds,their application to bordism theory and the Novikov conjecture

We develop a general approach to de�ne characteristic numbers for open manifolds, anabsolute and a relative version, prove certain invariance properties, apply them to bordismtheory and discuss several versions of the Novikov conjecture for open manifolds.

��

Alexander Ermolitski

Minsk, Belarus.E-mail address: [email protected]

On universality of Kählerian manifold

Let (M; g) be a smooth n-dimensional Riemannian manifold. We have considered thealmost Hermitian structure (J; bg) on TM constructed with help of a Riemannian metricg and the second fundamental tensor �eld h of (J; bg), see [1]. So, the following injectivemapping has been obtained

E : R(M)! AH(M) : g 7! (J; bg);where R(M) is the set of all Riemannian metrics onM and AH(M) is the set of all almostHermitian structures on TM .For any point p 2 M we can consider such a set f�(p)g of positive numbers that the

mapping expjU(�(p)) is de�ned and injective on U(�(p)) � TpM . Let "(p) = supf�(p)g.

53

Lemma, [2]. The mapping M ! R+ : p 7! "(p) is continuous on M .Proposition. There exists a normal tubular neighborhood Tb(M; "(p)) =

Sp2M D(p; "(p))

in TM with respect to bg = E(g) on TM induced by some g 2 R(M), whereM is consideredas the null section OM in TM(M 3 p$ Op 2 OM � TM) and "(p) is continuous functionon M .All the constructions announced in [3] and considered in [4] for the case " = const can

be generalized for the case of positive continuous function "(p) on M .Theorem. Let M be a smooth real manifold and Tb(M; "(p)) be the corresponding

normal tubular neighborhood in TM with respect to some bg = E(g) on TM .ThenM(OM) is a totally geodesic submanifold of the Kählerian manifold (Tb(M;

"(p)2; J; g)),

where the almost Hermitian structure (J; g) is the deformation of the structure (J; bg), see[3],[4].So, any smooth real manifold of dimension n can be embedded in a Kählerian manifold

of dimension 2n as a totally geodesic submanifold.References

1. S. A. Bogdanovich, A. A. Ermolitski: "On almost hyperHermitian structures onRiemannian manifolds and tangent bundles" CEJM 2(5) 2004, 615-623.2. D. Gromol, W. Klingenberg, W. Meyer: Riemannsche geometrie im groben, Springer,

Berlin, 1968 (in German).3. A. A. Ermolitski: "Deformations of structures on a tubular neighborhood of a

submanifold", 9th Belarusian mathematical conference. Abstracts., Grodno (2004), part2, pp. 74-75.4. A. A. Ermolitski: "Deformation of structures, embedding of a Riemannian manifold

in a Kählerian one and geometric antigravitation" (to appear).

��

Ji Gao

Community College of Philadelphia, USA.E-mail address: [email protected], http://faculty.ccp.edu/faculty/jgao

A Parameter related to W �Topology in Banach Spaces

Let X be a Banach space, S(X) = fx 2 X : kxk = 1g be the unit sphere of X. Theparameter, modulus of W �-convexity, W �(�) = inff< x�y

2; fx >: x; y 2 S(X); kx � yk �

�; fx 2 rxg, where 0 � � � 2 and rx � S(X�) be the set of norm 1 supporting functionals

54

of S(X) at x, is investigated. The relationship among uniform nonsquareness, uniformnormal structure and the parameter W �(�) are studied, and a known result is improved.The main result is that for a Banach space X, if there is �, where 0 < � < 1

2such that

W ��(1 + �) >

�2where W �

�(1 + �) = lim�!��W�(1 + �), then X has normal structure.

��

Janusz Grabowski [1] and [2]Norbert Poncin

[1]Institute of Mathematics of the Polish Academy of Sciences, Warsaw, Poland.E-mail address: [email protected], http://www.impan.gov.pl/~jagrab/

[2]University of Luxembourg, Luxembourg City, Grand-Duchy of Luxembourg.E-mail address: [email protected]

On the Chevalley-Eilenberg cohomology of somein�nite-dimensional algebras of geometric origin

We give a complete and explicit description of the derivations of the Lie algebraD(M) ofall linear di¤erential operators of a smooth manifoldM , of its Lie subalgebra D1(M) of alllinear �rst-order di¤erential operators ofM , and of the Poisson algebra S(M) = Pol(T �M)of all polynomial functions on T �M , the symbols of the operators in D(M). The problemof distinguishing those derivations that generate one-parameter groups of automorphismsand describing these one-parameter groups will also be solved.

��

Sarah Hansoul

University of Liège, Liège, Belgium.E-mail address: [email protected]

Existence of natural and projectively equivariant quantizations

55

In this talk, what we call a quantization is a linear bijection

Qr : S(M)! D(M);

where D(M) is a space of di¤erential operators and S(M) its associated space of symbols.This map depends on a linear connection r on a di¤erential manifold M .A few years ago, P. Lecomte conjectured the existence of a natural and projectively

equivariant quantization, namely a quantization which is natural and takes the same valueson two connections projectively equivalent. The existence of such a quantization is ageneralization on an arbitrary manifold of the existence of the so-called slm+1-invariantquantization previously obtained on Rm.We obtain a su¢ cient condition for the existence of such quantizations, when the dif-

ferential operators considered act between sections of vector bundles associated to the �berbundle P 1M of linear frames of M . In this proof, we use both approaches of projectivestructures due to T.Y. Thomas and J.H.C. Whitehead in the 1920�s. In particular, we usethe theory of Cartan connexions, and construct a Casimir operator depending on a Cartanconnexion. We show the existence of a quantization when the eigenvalues of this operatorare of multiplicity one.

��

Johannes Huebschmann

Université des Sciences et Technologies de Lille, France.E-mail address: [email protected]

Strati�ed Kähler structures on adjoint quotients

The complexi�cation of a compact Lie group, endowed with a biinvariant Riemannmetric, inherits a Kähler structure having twice the kinetic energy of the metric as itspotential, and Kähler reduction with reference to the adjoint action yields a strati�edKähler structure on the adjoint quotient. The resulting singular Poisson-Kähler geometryof the adjoint quotient and the corresponding singular Kähler quantization on the reducedlevel exhibits nice features which will be explained in the talk, including the quantizationof the geodesic �ow on the reduced level. The ultimate goal is to extend this procedureto certain models for lattice gauge theory, with applications to molecular mechanics andmoduli spaces.

56

��

Josef Jany�ka

Masaryk University, Brno, Czech Republic.E-mail address: [email protected], http://www.math.muni.cz/~janyska

Higher order Utiama�s theorem

We prove higher order version of the Utiyama�s theorem. To prove the Utiyama�stheorem in order r � 2 we have to use auxiliary classical connections on base manifolds.We prove that any natural (invariant) operator of order r for principal connections onprincipal G-bundles and for classical connections on base manifolds with values in a (1; 0)-order G-gauge-natural bundle factorizes through curvature tensors of both connectionsand their covariant di¤erentials, where the covariant di¤erential of curvature tensors ofprincipal connections is considered with respect to both connections.

��

Rémi Léandre

Université de Bourgogne, Dijon, France.E-mail address: [email protected]

Stochastic Poisson-Sigma model

Cattaneo-Felder have given a formal path integral representation of Kontsevitch�s for-mula of *-product on a Poisson manifold. We introduced a stochastic regulator in order tode�ne rigorously some-thing similar of Cattaneo-Felder formula, and we perform the quasi-classical limit, in order to de�ne a stochastic *-product. This procedure was done alreadyby Klauder in order to de�ne some path integral in quantummechanic, by introducing someGaussian regulator. In order to study the e¤ect of the regulator, we de�ne the Lebesguemeasure in in�nite dimension as a distribution. The hope is to get someting analoguous of

57

stochastic quantization of Parisi-Wu, the role of the in�nite dimensional Langevin equationbeing replaced by the in�nite dimensional Brownian motion of Airault-Malliavin. We startby doing a small survey on random surfaces got by stochastic analysis.

��

Jacek Lech[1] and Tomasz Rybicki[2]

AGH University of Science and Technology, Cracow, Poland.[1]E-mail address: [email protected]

[2]E-mail address: [email protected]

Perfectness at in�nity of di¤eomorphism groups on openmanifolds

Let us recall that a group G is called perfect if G = [G;G], where the commutatorsubgroup is generated by all commutators [g1; g2] = g1g2g

�11 g

�12 , g1; g2 2 G. In terms of

homology of groups this means that H1(G) = G=[G;G] = 0.It is well known that the identity component of the group of all compactly supported

Cr-di¤eomorphisms of a manifold is perfect and simple provided 1 � r � 1, r 6= n + 1,and n is the dimension of the manifold (theorems of Herman, Thurston and Mather).Several generalizations for the automorphism groups of geometric structures are known.The problem of the perfectness of analogous groups with no restriction on support isstudied by making use of results of Segal. It is only very loosely related to the problem ofthe perfectness of compactly supported di¤eomorphism groups. A clue role in the problemplays the notion of the perfectness at in�nity. We show that the identity component ofthe group of all Cr-di¤eomorphisms on Rn is perfect at in�nity. Also we formulate someconditions which ensure together with the perfectness at in�nity that the di¤eomorphismgroup in question is perfect.

��

Pierre Lecomte

University of Liège, Liège, Belgium.E-mail address: [email protected], http://ulg.ac.be/geothalg

58

From a Euclidian space to Cartan geometries: how to rebuilddi¤erential operator from their principal symbol?

The space of di¤erential operators acting between spaces of sections of natural bundles isa �ltered representation of the Lie algebra of vector �elds in a natural way. Since about tenyears, the relations between that module and the associated graded modules are studied.The purpose of the talk is to give a account of the obtained results, of the methods usedto get them and of the perspective left open in that �eld.

��

Sergey Maksymenko

National Academy of Sciences of Ukraine, Kiev, Ukraine.E-mail address: [email protected], http://www.imath.kiev.ua/~maks

Homotopy types of stabilizers and orbits of Morse mappings ofsurfaces

LetM be a smooth compact surface (oriented or not, with boundary or without it) andP either R or S1. The group D(M) of di¤eomorphisms of M naturally acts on C1(M;P )by the following rule: if h 2 D(M) and f 2 C1(M;P ), then h�f = f�h: For f 2 C1(M;P )let S(f) = fh 2 D(M) j f � h = fg be the stabilizer and O(f) = ff � h jh 2 D(M)g bethe orbit of f under this action. Let �f be the set of critical points of f and D(M;�f )the group of di¤eomorphisms h of M such that h(�f ) = �f . Then the stabilizer S(f;�f )and the orbit O(f;�f ) of f under the restriction of the above action to D(M;�f ) arewell de�ned. Evidently, S(f) � D(M;�f ), whence S(f;�f ) = S(f). Let Sid(f) be theidentity path-component of S(f) and Of (f) and Of (f;�f ) the connected components ofO(f) and O(f;�f ) (resp.) in the corresponding compact-open topologies. We endowSid(f), Of (f), and Of (f;�f ) with C1-topologies. A function f : M �! P will be calledMorse if �f � IntM , all critical points of f are non-degenerated, and f is constant onthe connected components of @M . A Morse mapping f is generic if every level-set of fcontains at most one critical point; f is simple if every critical component of a level-set off contains precisely one critical point. Evidently, every generic Morse mapping is simple.Let f : M �! P be a Morse mapping. We �x some orientation of P . Then the index of anon-degenerated critical point of f is well-de�ned. Denote by ci, (i = 0; 1; 2) the numberof critical points of f of index i.

59

Theorem 1 If either c1 > 0 orM is non-orientable, then Sid(f) is contractible. Otherwise,Sid(f) is homotopy equivalent to S1.

Theorem 2 Suppose that c1 > 0. Then (1) Of (f;�f ) is contractible; (2) �iOf (f) � �iMfor i � 3 and �2Of (f) = 0. In particular, Of (f) is aspherical provided so is M . Moreover,�1Of (f) is included in the following exact sequence

0 �! �1D(M)� Zk �! �1Of (f) �! G �! 0; (1)

where G is a �nite group and k � 0. If f is simple, then for the surfaces presented inTable 1, the number k is determined only by the number of critical points of f .

Table 1:M k

M = S2, D2, S1 � I, T 2, P 2 with holes c1 � 1M is orientable and di¤ers from the surfaces above c0 + c2

(3) Suppose that f is generic. Then the group G in Eq. (1) is trivial, whence �1O(f) ��1D(M) � Zk. In particular, �1O(f) is abelian. The homotopy type of Of (f) is given inTable 2.

Table 2:Surface M Homotopy type of Of (f)S2, P 2 SO(3)� (S1)c1�1

D2, S1 � I, Möbius band Mo (S1)c1

T 2 (S1)c1+1

Klein bottle K (S1)k+1

other cases (S1)k

Theorem 3 If c1 = 0, then f can be represented in the following form

f = p � ef :M ef�! eP p�! P;

where ef is one of the mappings shown in Table 3, eP is either R or S1, and p is either acovering map or a di¤eomorphism. In this case the homotopy types of Of (f) and Of (f;�f )depend only on ef and are given in Table 3. HereK is represented as the factor space of the2-torus T 2 � S1�S1 by the involution (x; y) 7! (x+ �=2;�y) and � means contractibility.

60

Table 3:Type ef :M �! eP c0 c1 c2 Of (f) Of (f;�f )(A) S2 �! R ef(x; y; z) = z 1 0 1 S2 �(B) D2 �! R ef(x; y) = x2 + y2 1=0 0 0=1 �(C) S1 � I �! R ef(�; t) = t 0 0 0 �(D) T 2 �! S1 ef(x; y) = x, 0 0 0 S1

(E) K �! S1 ef(fxg; fyg) = f2xg 0 0 0 S1

��

Piotr Mormul

Warsaw University, Warsaw, Poland.E-mail address: [email protected]

Nilpotent algebras hidden in special multi-�ags

Special �ags locally induce nilpotent algebras.

Throughout the present text we use the same terminology and notation that have beenintroduced, and used, in [M3]. The natural parameter k is the width, and r is the lengthof a special multi-�ag under consideration.

Let us de�ne a linear automorphism �j of Rk+1 associated to any �xed operation j 2

f1; : : : ; k+1g, by its values on a ��xed once for all �basis e1; : : : ; ek+1 of Rk+1. Namely,let �j send: e1 7! e1 + e2 + � � � + ek+1, e2 7! e1; : : : , ej 7! ej�1, ej+1 7! ej+1; : : : ,

ek+1 7! ek+1 (so that, naturally, �1 sends e2 7! e2; : : : ; ek+1 7! ek+1, and

��k+1 sends

e2 7! e1; : : : ; ek+1 7! ek). With these notations,

Theorem 1. Every rank-(k + 1) distribution D generating a special k��ag of a lengthr � 1 is locally nilpotentizable (i. e., is locally weakly nilpotent in the sense of [M1]) and alocal nilpotent basis around a point p is fZ1; : : : ; Zk+1g, where (Z1; : : : ; Zk+1) 2 j1: j2 : : : jrissuing from [M3] is an EKR form for D around p satisfying the least upward jumps rule.The nilpotency order of the real Lie algebra L( j1: j2 : : : jr)

def= LR

�Z1; : : : ; Zk+1

�is equal

to the component of ek+1 (i. e., the last one) in the vector �j1 �j2 � � �

�jr (e1 + � � �+ ek+1).

Moreover, if a germ of D at a certain point admits a local EKR form with no non-zeroconstants, then that germ is also strongly nilpotent in the sense of [AGau] and [M1].

61

Remark 1. The algebra L( j1: j2 : : : jr) appearing in this theorem is well de�ned. Indeed,for two di¤erent EKRs (Z1; : : : ; Zk+1) and ( eZ1; : : : ; eZk+1) belonging in the same EKR classj1: j2 : : : jr, it is easy to write down a simple translation in the space R(r+1)k+1 inducing an(inner) isomorphism of LR

�Z1; : : : ; Zk+1

�and LR

� eZ1; : : : ; eZk+1�, sending every generatorZl to eZl, l = 1; : : : ; k+1. The translation vector has as components the di¤erences betweenthe respective constants sitting in the �elds Z1 and eZ1 (and many other components 0).Thus LR

�Z1; : : : ; Zk+1

�and LR

� eZ1; : : : ; eZk+1� are two identical copies of one underlyingLie algebra [that turns out to be nilpotent by this very theorem].In the light of recent constructions [M2], all this is geometric and to any distribution

germ D generating a special k-�ag there is associated such an algebra A. We conjecturehere, extending the partial results and conjectures of [M4], that A is always minimal for D,in the sense of having the minimal nilpotency order among the Lie algebras induced overR by all possible nilpotent bases for D.

It is also to be noted that whenm = max(j1; : : : ; jr) < k, then not only the last component,but precisely k+1�m last components in

�j1 �j2 � � �

�jr (e1+ � � �+ek+1) are mutually equal

and equal to the nilpotency order of L( j1: j2 : : : jr). This will be visible in the proof.

Corollary 1. For distribution germs D as in the �Moreover� part of Thm. 1 (for k =1 they are called �tangential� in [M1]), one is able to e¤ectively compute the degree ofnonholonomy of D, too. It is equal to the nilpotency order of the nilpotent Lie algebragiven by the nilpotent approximation of D, and in the discussed case that algebra is justLR�Z1; : : : ; Zk+1

�. Hence the nonholonomy degree of such D equals the last component in

the vector �j1 �j2 � � �

�jr (e1 + � � �+ ek+1), too.

Still before proving Thm. 1, let us demonstrate how it works.

Example 1. Fix k = 2. The nilpotency order of the algebra L(1:1:2) is equal to 6,because, computing in that �xed for ever basis e1; e2; e3,

�1 �1 �2

0@ 111

1A =

0@ 1 0 01 1 01 0 1

1A0@ 1 0 01 1 01 0 1

1A0@ 1 1 01 0 01 0 1

1A0@ 111

1A =

0@ 256

1A :

Likewise, the nilpotency order of the algebra L(1:2:2) is equal to 7, because

�1 �2 �2

0@ 111

1A =

0@ 1 0 01 1 01 0 1

1A0@ 1 1 01 0 01 0 1

1A0@ 1 1 01 0 01 0 1

1A0@ 111

1A =

0@ 357

1A :

Remark 2. For k = 1, the description of nilpotency orders of the EKRs given in Thm.1allows to re-prove a part of Jean�s results [J] on the nonholonomy degrees of Goursat germs.A bridge to those results rests on certain simple algebraic identities satis�ed (for, we repeat,k = 1) by the operators

�1 and

�2 when applied to e1 + e2. The details are given in [M4].

62

PROOF of Theorem 1 uses re�nements of methods of [M1, M4]. By Rem. 1, one canchoose as handy an EKR in the class j1: j2 : : : jr as possible. Let us choose, then, theuniquely de�ned EKR (Z1; : : : ; Zk+1) 2 j1: j2 : : : jr having all constants zero. (Withsuch a choice, also minimized will be the work on the �Moreover�part in the theorem.)

We are going to associate weights w(�) to all coordinates t; x01; : : : ; xrk, and to basicvector �elds (versors) w

�@@x

�= �w(x) in such a way that �under the classical de�nition

of the weight of a monomial vector �eld that goes back to the 1970s,

w�yi1yi2 � � � yis

@

@x

�= w(yi1) + w(yi2) + � � �+ w(yis) + w

� @@x

�(1)

�all Z1; Z2; : : : ; Zk+1 will be homogeneous of degree �1, and all v. f. appearing in thestepwise construction of the involved �eld Z1 will be homogeneous, of growing degrees, aswe go backwards in that construction. The manner of growth will highly depend on theEKR class, and the nilpotency order will be a function of that growth.

In our situation without constants,

Z1 = xr1Z

r�11 + xr2

@

@xr�11

+ � � �+ xrjr�1@

@xr�1jr�2+

@

@xr�1jr�1+ xrjr

@

@xr�1jr

+ � � �+ xrk@

@xr�1k

; (2)

where the involved v. f. from the one before last step is denoted by Zr�11 (and there is noconstants in it, too). Let us start by declaring w(Z1) = w(Z2) = � � � = w(Zk+1) = �1,hence also w(xr1) = � � � = w(xrk) = 1. By writing w(Z1) = �1, we stipulate that allsummands in (2) are of weight �1. In particular, �1 = w(xr1 Zr�11 ) = w(xr1) + w(Z

r�11 ) =

1 + w(Zr�11 ), which allows to compute the weight of Zr�11 , and to proceed further. Theparticulars of the entire de�nition can be formulated inductively as follows, understandingby Zr1 the starting v. f. Z1.

Fix an arbitrary s such that 1 � s � r and assume that Zs1 is a homogeneous v. f.of known weight w(Zs1), and that the weights of the coordinates x

s1; : : : ; x

sk are already

de�ned. Recall that

Zs1 = xs1Zs�11 + xs2

@

@xs�11

+ � � �+ xsjs�1@

@xs�1js�2+

@

@xs�1js�1+ xsjs

@

@xs�1js

+ � � �+ xsk@

@xs�1k

:

Analyzing the consecutive summands on the RHS, the homogeneity of Zs1 implies that

w(Zs�11 ) = w(Zs1) + w� @

@xs1

�(3)

and thatw� @

@xs�1l

�= w(Zs1) + w

� @

@xsl+1

�for l = 1; : : : ; js � 2 ; (4)

w� @

@xs�1js�1

�= w(Zs1) ; (5)

w� @

@xs�1l

�= w(Zs1) + w

� @

@xsl

�for l = js; : : : ; k : (6)

63

In this way inductively de�ned are the negative weights of all (r+1)k+1 versors, including@@t= Z01 , and the positive (opposite to the former) weights of all coordinates. Now denote

by d the biggest weight of a coordinate, say x (there can be, sometimes, several coordinatesof the weight d).The key property of weights de�ned by (1) is the additivity under Lie multiplication. Sothe Lie products of any s factors from among fZ1; : : : ; Zk+1g are homogeneous v. f.�s ofweight �s. In particular, products of more than d factors automatically vanish (our vectormonomials have only weights � �d). On the other hand, D, generating a special �ag, iscompletely nonholonomic. In particular @

@xat 0 2 R(r+1)k+1 should be a combination of

products of factors from among fZ1; : : : ; Zk+1g evaluated at 0.In such a combination, let us reiterate, only products of at most d factors count, whileproducts of any e < d factors have the @

@x�components of the form P @

@x, P �a homogeneous

polynomial of degree d� e > 0. Those P�s are, naturally, polynomials in shifted variables(denoted by big letters) of the EKR in question. But in the present EKR the variables areactually not shifted! �there is no constants shifting them. In consequence, all such P�svanish at 0. In this way only products of exactly d factors contribute in the production(s) of@@xat 0. This implies a [modest if cardinal] information that certain product(s) of exactly

d factors from among fZ1; : : : ; Zk+1g is (are) non-zero.At this point we know, therefore, that LR

�Z1; : : : ; Zk+1

�is a nilpotent Lie algebra of

nilpotency order d and also, as a byproduct, that the small �ag of D, D = V1 � V2 � V3 �� ends (locally, as a germ) on its term Vd,

Vd = TR(r+1)k+1: (7)

How to better visualise that value d ?

In what follows we are going to use the matrix notation for the operators �j1 ; : : : ;

�jr being

written in the mentioned �xed basis e1; : : : ; ek+1 of Rk+1. Observe that the formulas (3)�(6) can be most compactly expressed as

�w(Zs�11 ); w

� @

@xs�1l

�l=1;:::; k

�Tis the value of

�js at

�w(Zs1); w

� @

@xsl

�l=1;:::; k

�T: (8)

Hence, by making the composition over s = r; r � 1; : : : ; 1,�w� @@t

�; w� @

@x0l

�l=1;:::; k

�Tis the value of

�j1 �j2 � � �

�jr at [�1; �1; : : : ; �1 ] T

l=1;:::; k :

(9)Each Zs�11 (s � 2) has as a component the bare versor @

@xs�2js�1�1, and Z01 simply is

@@t. Thus

w(Zs�11 ) is a weight of a versor, and in (8) one set of versors�weights is transformed intoanother set of versors�weights.

Fact. The lowest negative versor weight � d occurs among the components of the vectoron the LHS of (9).

64

Pf. Passing (for the reader�s convenience) to the positive weights of variables, one startsfrom the vector [1; 1; : : : ; 1]T and notes that each

�j evaluated at any vector [a1; : : : ; ak+1]T

with all positive components,

�j [a1; a2; : : : ; ak+1]

T = [a1 + a2; : : : ; a1 + aj; a1; a1 + aj+1; : : : ; a1 + ak+1]T ;

has the maximum of components bigger than a1; a2; : : : ; aj; aj+1; : : : ; ak+1, hence biggerthan max(a1; : : : ; ak+1). �It remains to explain why the last component, and possibly not only it, on the LHS of

(9) is the biggest in modulus. Not surprisingly, the least upward jumps rule satis�ed by ourEKR is responsible for that. In fact, form = max(j1; : : : ; jr), the last k+1�m componentsof �j1 �j2 � � �

�jr [1; 1; : : : ; 1]

T are pairwise equal (we work in the sequel with positive weightsof variables only).

If m appears for the �rst time in the sequence j1; : : : ; jr as jl = m, then the m-thcomponent of

�jl �jl+1 � � �

�jr [1; 1; : : : ; 1]

T

is smaller than the components No m+ 1; : : : ; k + 1, and this relation keeps holding afterapplying each of the subsequent operators

�jl�1; : : : ;

�j1 .

Now either m = 1 (in which case we are already done) or, by the least upward jumpsrule, the value m � 1 does appear in the sequence j1; : : : ; jr, and so let it appear for the�rst time as js = m� 1, s < l. Then the (m� 1)-th component in

�js � � �

�jl � � �

�jr [1; 1; : : : ; 1]

T

is smaller than the components No m; m + 1; : : : ; k + 1, and this relation keeps holdingafter applying each of the subsequent operators (if there remains any)

��js�1; : : : ;

�j1 .

Thus at this point we know that, in the eventual outcome �j1 � � �

�jr [1; 1; : : : ; 1]

T , the(m�1)-th component is smaller than them-th component which in turn is smaller than thek + 1�m last components that are pairwise equal. Then we proceed likewise downwards,constantly using the least upward jumps property of j1: j2 � � � jr, obtaining in the end that,in the vector

�j1 � � �

�jr [1; 1; : : : ; 1]

T , the 1st component is smaller than the 2nd which issmaller than : : : which is smaller than the m-th which is smaller than the k + 1�m last,mutually equal, components.

The �Moreover�part in Thm. 1. Thanks to the choice of an EKR in the beginning, wecan work all the time with the same fZ1; : : : ; Zk+1g. Divide the constructed weights ofcoordinates t; x01; : : : ; x

rk into groups of equal values, the highest value being d: w1 = w2 =

� � � = wn1 = 1, n1 = k + 1 (it is clear that, on top of 1 = w(xr1) = � � � = w(xrk), preciselyone more variable xr�1jr�1 has weight 1), wn1+1 = � � � = wn2 = 2; : : : , wnl�1+1 = � � � = wnl =l; : : : , wnd�1+1 = � � � = wnd = d, where nd = (r + 1)k + 1.Attention. In this de�nition it may often happen that a given integer l is not the value ofa weight, and then simply nl�1 = nl.

65

Proposition 1. [n1; n2; : : : ; nd] is the small growth vector of D at 0 2 R(r+1)k+1. Moreover,all members Vj of the small �ag of D have at 0 the description

Vj(0) =� @

@x; x : w(x) � j

�: (10)

That is to say, the coordinates t; x01; : : : ; xrk are �for D �linearly adapted at 0.

Pf. Vj is spanned by the products of � j factors from among fZ1; Z2; : : : ; Zk+1g. Suchproducts are, as we already know, homogeneous of weights � �j and, evaluated at 0, arecombinations of bare versors (because there is no constants in (Z1; : : : ; Zk+1)). Those are,therefore, the versors of coordinates x s. t. w(x) � j. This means the inclusions : � : in(10). Clearly, for j = 1, with rkV1 = k + 1, there is an equality in (10). Suppose that j isthe smallest positive integer s. t. there holds in (10). (In view of (7), assuredly j < d.)That is, that there exists a combinationX

x: w(x)=j

ax@

@x=2 Vj(0) : (11)

Then

Vj+s(0) = span�Vj(0); certain combinations of

@

@x; j < w(x) � j + s

�for = 1; 2; : : : In particular, for s = d� j,

Vd(0) = span�Vj(0); certain combinations of

@

@x; j < w(x) � d

�:

This together with (11) implyP

x: w(x)=j ax@@x=2 Vd(0) which contradicts (7).

The description (10), now shown to hold, allows to compute the s.gr.v. of D at 0. Forj = 2; : : : ; d, dimVj(0) = n1 + (n2 � n1) + � � �+ (nj � nj�1) = nj. Prop. 1 is proved.Summarizing now, our weights w(�) are modelled on the pattern established by the

s.gr.v. of D at 0. With this knowledge, Prop. 1 says precisely that the coordinates inquestion are linearly adapted for D at 0 (cf., for inst., [AGamS, B, M4]). In fact, they areeven adapted.This (last) part is standard, because, on the one side, �linear adaptedness� implies thatthe nonholonomic orders of these variables t; x01; : : : ; x

rk do not exceed their respective

weights wi ([B], p. 35), that is �do not exceed our weights w(�). And, on the other side,any nonholonomic derivative of a variable x from this set, wrt fZ1; : : : ; Zk+1g and oforder l < w(x), is �by our weights�construction �a homogeneous polynomial of positiveweight w(x) � l in the variables that are not shifted by constants, and vanishes as such(cf. the earlier argument in this long proof that certain products of d factors from amongfZ1; : : : ; Zk+1g do not vanish). Thus the nonholonomic orders and the weights wi coincide,and the coordinates are indeed adapted.

66

Once found (any) set of adapted coordinates for a distribution around a point, one cancompute the nilpotent approximation of that distribution at that point. In the present caseall terms in Z1, and in Z2; : : : ; Zk+1, too, are of degree (weight) �1, so that the nilpotentapproximation of D at 0 simply coincides with D. This exactly means ([AGau]; see also[M1],Def. 5) that the germ D is strongly nilpotent. �

Algorithmic issues

Do there exist more algorithmic ways of computing the nilpotency orders, not viabringing in the entire matrices of operators

�j required in Thm. 1 ? For k = 1 this task

has been carried out in [M1, M4]. We will give here an answer for k = 2. It is less compactthan in the Goursat case. For k = 3 the answer is still more involved than for k = 2, andso on when k grows further. It is important that the computation keeps being recursive,although the recurrence patterns become more and more involved.

So, from now on, we set k = 2 and use the letter C for codes of di¤erent EKRs.Moreover, we denote by f(C) the nilpotency order of the Lie algebra L(C) issuing fromThm. 1. A string of l same ciphers, like for inst. a string of l 1�s, going in row in a codewill be written shortly as 1l. Thus 12:23 is the EKR 1:1:2:2:2, and so on. Here are therecipes for computing the nilpotency orders of L(C) from Thm. 1.

Theorem 2. In the situation k = 2, for any EKR C satisfying the least upward jumpsrule (C = ; not excluded), the nilpotency order of the Lie algebra L(C) equals f(C), wherethe function f(�) is de�ned recursively below. In this de�nition an � stands for any cipherfrom among f1; 2; 3g, while a � �it is important �stands only for 2 or 3.

(i) f(;) = 1; f(1) = 2 ,

(ii) f(C: � : 1) = 2f(C: �)� f(C) ,

(iii) f(1l: 2) = 2l + 2, l = 1; 2; 3; : : : ,

(iv) f(C: � : � : 1l: 2) = (2l + 2)f(C: � : �)� 2l f(C: �)� f(C) for anyl = 0; 1; 2; : : : ,

(v) f(C: � : � : 1l: 3) = (2l + 1)f(C: � : �)� (2l � 1)f(C: �) + f(C) for anyl = 0; 1; 2; : : : .

It can be useful to compare right now the above recurrencies with those governing the casek = 1 (Thm. 1 in [M1]). In the Goursat case the nilpotency orders g(�) satisfy the relations

g(;) = 1 , g(1) = 2 ,

g(C: � :1) = 2 g(C: �)� g(C) ,

g(C: � :2) = g(C: �) + g(C) ,

where an � is for 1 or 2, and C starts on the left from 1.1. (This particular start forGoursat�s normal forms is due to the famous Engel 1889 theorem that holds only for

67

1-�ags.)

Example 2. In order to illustrate the algorithm, let us compute in two ways thenilpotency order of L(13:2:3:1:2:12:3:2:14:3). Firstly, according to Thm. 1,

� �1�3 �

2 �3 �1 �2� �1�2 �

3 �2� �1�4 �

3

0@ 111

1A =

0@ 32811281193

1A ;

which implies that the order in question is 1193. Secondly, according to Thm. 2, oneproceeds stepwise, expanding progressively the code to the right:

f(13:2) = 2 � 3 + 2 = 8 by (iii),

f(13:2:3) = f(13:2) + f(13) + f(12) = 8 + 4 + 3 = 15 by (v), l = 0,

f(13:2:3:1:2) = 4f(13:2:3) � 2f(13:2) � f(13) = 4 � 15 � 2 � 8 � 4 = 40 by (iv),l = 1. Additionally, f(13:2:3:1) = 2 � 15� 8 = 22 by (ii), and sof(13:2:3:1:2:12:3) = 5f(13:2:3:1:2)� 3f(13:2:3:1) + f(13:2:3)= 5 � 40� 3 � 22 + 15 = 149 by (v), l = 2.

Additionally, f(13:2:3:1:2:1) = 2 � 40� 22 = 58 as well as f(13:2:3:1:2:12)= 2 � 58� 40 = 76 by (ii) again, and sof(13:2:3:1:2:12:3:2) = 2f(13:2:3:1:2:12:3)� f(13:2:3:1:2:1)= 2 � 149� 58 = 240 by (iv), l = 0,

f(13:2:3:1:2:12:3:2:14:3)

= 9f(13:2:3:1:2:12:3:2)� 7f(13:2:3:1:2:12:3) + f(13:2:3:1:2:12)= 9 � 240� 7 � 149 + 76 = 1193 by (v), l = 4,

as in the �rst way.

References

[AGamS] A.A.Agrachev, R.V.Gamkrelidze, A.V. Sarychev; Local invariants ofsmooth control systems, Acta Appl. Math. 14 (1989), 191�237.

[AGau] � � , J�P.Gauthier; On subanalyticity of Carnot�Carathéodory distances,Ann. Inst. H. Poincaré �AN 18 (2001), 359�382.

[B] A.Bellaïche; The tangent space in sub-Riemannian geometry, in: A.Bellaïcheand J�J.Risler (eds.), Sub-Riemannian Geometry, Birkhäuser, Basel 1996,1�78.

[HLuSul] H.Hermes, A. Lundell, D. Sullivan; Nilpotent bases for distributions and con-trol systems, J.Di¤.Eqns 55 (1984), 385�400.

68

[J] F. Jean; Car with N trailers: characterisation of the singu-lar con�gurations, ESAIM: Control Optimisation Calc. Var.(http://www.edpsciences.org/cocv/) 1 (1996), 241�266.

[KRub] A.Kumpera, J. L.Rubin; Multi��ag systems and ordinary di¤erential equa-tions, Nagoya Math. J. 166 (2002), 1�27.

[LaSus] G.La¤erriere, H. J. Sussmann; A di¤erential geometric approach to motionplanning, in: F.Canny and Z.Li (eds.), Nonholonomic Motion Planning,Kluwer, Dordrecht 1993, 235�270.

[M1] P.Mormul; Goursat distributions not strongly nilpotent in dimensions notexceeding seven, in: A. Zinober and D.Owens (eds.), Nonlinear and AdaptiveControl NCN4 2001, Lecture Notes in Control and Information Sciences 281,Springer, Berlin 2003, 249�261.

[M2] � � ; Geometric singularity classes for special k-�ags, k � 2, of arbitrarylength. Preprint in: Singularity Theory Seminar, S. Janeczko (ed.), WarsawUniv. of Technology, Vol. 8 (2003) pp. 87�100.

[M3] � � ;Multi-dimensional Cartan prolongation and special k-�ags, H.Hironakaet al (eds.), Geometric Singularity Theory, Banach Center Publications 65,Inst. of Math., Polish Acad. Sci., Warszawa 2004, 157�178.

[M4] � � ; Minimal nilpotent bases for Goursat distributions of coranks not ex-ceeding six, Univ. Iagel. Acta Mathematica 42 (2004), 15�29.

[Mu] R.M.Murray; Nilpotent bases for a class of nonintegrable distributionswith applications to trajectory generation for nonholonomic systems,Math. Control Signals Syst. 7 (1994), 58�75.

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Ryszard Nest

Copenhagen University, Copenhagen, Denmark.E-mail address: [email protected]

The geometry of the calculus of Fourier integral operators

69

Let be a Fourier Integral Operator and let and denote the algebras of pseudodi¤erentialoperators on X and Y . A natural object to consider is the bimodule, which carries thegeometric information about the original operator. Since the algebras of pseudodi¤erentialoperators admit microlocal description via symbol calculus, the same kind of study canbe applied to the bimodule. We will construct the microlocal bimodules associated tothis situation and explain the kind of geometry involved in their study. As examplesof applications we�ll give homological interpretation of the composition of Fourier IntegralOperators and of their traces. An important case where this kind of operators appear is theconstruction of Guillemin and Sternberg of Integral projections associated to coisotropicsubmanifolds of cotangent bundle of a closed manifold. We will apply the above methodsto give a formula for the index of the operators of this type.

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Igor Nikonov

Moscow State University, Moscow, Russia.E-mail address: [email protected]

On Hopf-type cyclic cohomology with coe¢ cients(Joint work with G.I.Sharygin)

Hopf-type cyclic cohomology with coe¢ cients were introduced recently by P.M.Hajac,M.Khalkali, B.Rangipour, M.Sammerhauser. We present explicit calculations of such co-homology for a couple of interesting examples. The second part of the talk is devoted toa general construction of coupling between algebraic and coalgebraic version of Hopf-typecohomology, taking values in the usual cyclic cohomology of an algebra.

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Andriy Panasyuk

Warsaw University, Warszawa, Poland.E-mail address: [email protected]

70

Algebraic Nijenhuis operators and Kronecker Poisson pencils

We give a criterion of (micro-)kroneckerity of the linear Poisson pencil on g� related toan algebraic Nijenhuis operator N : g ! g on a �nite-dimensional Lie algebra g. As anapplication we get a series of examples of completely integrable systems on semisimple Liealgebras related to Borel subalgebras and a new proof of the complete integrability of thefree rigid body system on gln.

��

Krzysztof Pawa÷owski

Adam Mickiewicz University, Poznan, Poland.E-mail address: [email protected], http://main.amu.edu.pl/�kpa

A proof of the Laitinen Conjecture(Joint work with Ronald Solomon and Toshio Sumi)

For a �nite group G, consider the subset Sm(G) of the real representation ring RO(G)consisting of the di¤erences U �V of real G-modules U and V which are Smith equivalent,i.e., there exists a smooth action ofG on a sphere S such that the �xed point set SG = fx; ygand as G-modules, Tx(S) �= U �W and Ty(S) �= V �W for a real G-module W . In 1960,P.A. Smith asked is it true that Sm(G) = 0? Now, consider the subset LSm(G) of Sm(G)by imposing the restriction that the action of G on S is such that the H-�xed point setSH is connected for every cyclic subgroup H of G of order 2n with n � 3. In 1996, E.Laitinen posed a conjecture that for a �nite Oliver group G, LSm(G) 6= 0 if and only ifaG � 2, where aG is the number of the real conjugacy classes of G represented by elementsof G not of prime power order. By using equivariant surgery and character theory of �nitegroups, we present a proof of the Laitinen Conjecture and as an application of the result,we answer the original Smith question about Sm(G) to the e¤ect that Sm(G) 6= 0 for any�nite Oliver group G with aG � 2. By using a classi�cation of �nite Oliver groups G withaG = 0 or 1, we obtain that if G is simple, Sm(G) 6= 0 if and only if aG � 2.

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71

Paul Popescu

University of Craiova, Craiova, Romania.E-mail address: [email protected]

On the high order geometry on osculator spaces and anchoredvector bundles

(Joint work with Marcela Popescu)

Higher order geometry was studied by M. de Leon and R. Miron using the bundles ofaccelerations. R. Miron studies Finsler and Lagrange spaces of higher order, constructinga dual theory of higher order Hamilton spaces by means of some Legendre transformationsthat relates a lagrangian and a hamiltonian of the same order. These Legendre trans-formations use essentially an a¢ ne section and they are not intrinsic associated with thelagrangian or the hamiltonian. In order to remove this inconvenient, we de�ne an a¢ nehamiltonian, which can be related by Legendre transformations with a lagrangian of thesame order. These Legendre transformations are intrinsic associated with the lagrangian orthe a¢ ne hamiltonian and give a bijective correspondence between the regular lagrangiansand a¢ ne hamiltonians of the same order.A theory of Hamilton submanifolds was considered by R. Miron also in the case of

higher order geometry. He de�nes an induced hamiltonian on the submanifold, which isnot intrinsic (the induction procedure is not uniquely de�ned by the hamiltonian and thesubmanifold). We study here some intrinsic ways to induce a hamiltonian and an a¢ nehamiltonian of higher order on a submanifold.All the constructions considered above are also possible for anchored vector bundles.

One obtain a general theory of higher order geometry of anchored vector bundles, usingalmost Lie structures, a generalization of Lie algebroids.

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Paul Popescu

University of Craiova, Craiova, Romania.E-mail address: [email protected]

72

On the Lie pseudoalgebra generated by an anchored module(Joint work with Marcela Popescu)

A Lie pseudoalgebra is a natural generalization of a Lie algebra. Any module gener-ates a Lie algebra, called by Bourbaki the free Lie algebra of the module. We presenta construction of the free Lie pseudoalgebra of an anchored module that allows a linearconnection. In the case of theLie algebra of a module, the construction is di¤erent fromthat considered by Bourbaki.

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Anatoliy Prykarpatsky

Faculty of Applied Mathematics, AGH University of Science and Technology, Kraków,Poland.Dept. of Nonlinear Mathematical Analysis at IAPMM of NAS, Lviv, Ukraine.E-mail address: [email protected]

Ergodic measures of Boole type dynamical systems on axis:analytical aspects

We study invariant nonatomic measures realated with Boole type discrete mappings onaxis. Based on the basic properties of the Perron-Frobenius operator we study analyticalapproximations of its solutions and construct the invariant measures, which appear to beergodic.

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Dmitry Roytenberg

TA Utrecht, The Netherlands.E-mail address: [email protected]

Higher Lie algebras in Poisson geometry and elsewhere

73

We discuss the appearance of both strict and weak Lie 2- and 3-algebras in the contextof Poisson manifolds and Courant algebroids.

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Tomasz Rybicki

AGH University of Science and Technology, Cracow, Poland.E-mail address: [email protected]

Commutators of equivariant homeomorphisms on G-manifoldswith one orbit type

Let M be a manifold and let a compact and connected Lie group G act on M with oneorbit type. The long-standing problem of the perfectness of the identity component of thecompactly supported equivariant homeomorphism group of M is solved in the a¢ rmativeunder the assumption that the orbit codimension is greater than one. The result makes useof a well-known theorem of J.N. Mather [4] on the simplicity of homeomorphism groupson manifolds.In other categories analogous results are known. Namely, a theorem for equivariant Cr-

di¤eomorphisms, where r = 1; : : : ;1 and r�1 is di¤erent than the orbit space dimension,is due to A. Banyaga [3] for G = T n , and is due to K. Abe and K. Fukui [1] for anarbitrary compact Lie group G. Recently the latter authors showed in [2] such a theoremfor Lipschitz homeomorphisms. However, the proofs of these analogs were much easier thanthe present one as they made use of a stability property for di¤eomorphisms or Lipschitzhomeomorphisms. This property is no longer true in the topological category. On theother hand, a clue part of the proof of our result is no longer true even in the C1-case. Thetheorem seems to be a starting point for computing the �rst homology of the equivarianthomeomorphism group for more complicated G-manifolds.

References

[1] K. Abe, K. Fukui, On commutators of equivariant di¤eomorphisms, Proc.Japan Acad. 54, Ser. A (1978), 53�54.

[2] K. Abe, K. Fukui, On the structure of the group of Lipschitz homeomor-phisms and its subgroups, J. Math. Soc. Japan 53 (2001), 501�511.

74

[3] A. Banyaga, On the structure of the group of equivariant di¤eomorphisms,Topology 16 (1977), 279�283.

[4] J. N. Mather, The vanishing of the homology of certain groups of homeo-morphisms, Topology 10 (1971), 297�298.

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Micha÷Sadowski

University of Gdansk, Gdansk, Poland.E-mail address: [email protected]

Complete �at manifolds

The aim of the talk is to discuss some properties of complete �at manifolds (cf-manifolds). Each such a manifoldM is a �at vector bundle over a closed �at manifold XM ,in particular, the homotopy type of M is determined by �1(M): The following topics willbe discussed during the talk:

1) the results concerning �niteness of the set of di¤eomorphism classes of cf-manifolds ofa �xed dimension m;

2) an algebraic criterion of an a¢ ne equivalence,

3) topological structure of cf-manifolds with cyclic holonomy groups,

4) topological and a¢ ne classi�cation of m-dimensional cf-manifolds (m � 4),5) immersions of cf-manifolds into Euclidean spaces.

For the formulation of most of the results we refer to M. Sadowski, Topological and a¢ nestructure of complete �at manifolds, preprint, arXiv: math /DG/ 0502449

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Pavol Severa

Dept. of Theoretical Physics, Bratislava, Slovakia.E-mail address: [email protected]

75

Dirac structures and deformation quantization

Dirac structures were introduced (by Courant and Weinstein) as a common generaliza-tion of both Poisson structures and closed 2-forms. In the talk we shall notice that they alsodescribe Hamiltonian families of Poisson structures (i.e. families that solve the Maurer-Cartan equation). These families are easily quantized via Kontsevich formality map. Asan application, we�ll get a quantization of twisted Poisson structures to non-commutativedeformations of gerbes.

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Vladimir Sharko

National Academy of Sciences of Ukraine, Kiev, Ukraine.E-mail address: [email protected]

The L2-invariants and non-singular Morse-Smale �ows onmanifolds

A smooth �ow 't on smooth closed manifoldMn is called non-singular Morse-Smale if:the chain-recurrent set of 't consist of �nite number hyperbolic closed orbit; the unstablemanifold of any closed orbit has transversal intersection with the stable manifold of anyclosed orbit.

De�nition 1 The i-th Morse S1-number of manifoldMn is the minimum number of closedorbits of index i taken over all non-singular Morse-Smale vector �elds with untwisted closedorbits on manifold Mn. Denote the i-th Morse S1-number of manifold Mn byMS1

i (Mn).

Let Nn�1 � Mn be a closed submanifold of the closed manifold Mn. We shall saythat (Mn; Nn�1) form admissible par of the level i if : MnnNn�1 consist from twocomponents, closing which we denote by Mun and Mup ; Mun (Mup) have homotopy typeof k-dimensional (l-dimensional) CW-complex, where k � i; l � n� i� 1.

De�nition 2 The admissible par (Mn; Nn�1) of level i we call �-minimal if the number(�1)i�(Mun) is minimal between all admissible pars of the level i of the manifold Mn. Thevalue of number (�1)i�(Mun) for minimal admissible par of the level i of the manifold Mn

we shall call i-th Euler characteristic of manifold Mn and shall denote by �i(Mn).

76

Lemma 1 Let Mn be a closed smooth manifold (n � 9).On Mn there is ordered Morsefunction f : Mn �! [0; n] such that: f have one critical point of index 0 (1) ; all criticalpoints of index � of the function f lie on f�1(�); f have minimal number critical pointsof �xed index i.If 4 � i � n� 5, then the par (Mn; f�1(i + 1=2)) is �-minimal admissiblepar of the level i .

Lemma 2 Let (Mn; Nn�1) �-minimal admissible par of the level i. Ifn � 9 and 4 � i � n � 5, then �i(Mn) is invariant of homotopy type of Mn and equal�i(M

n) = (�1)iPi

j=0 dimN(G)(Hi(2)(M

n)) + bSi+1(2) (Mn).

Theorem 1. Let Mn, n � 9, be arbitrary closed manifold with zero Euler characteristic,then i-th Morse S1-number of manifold Mn for 4 � i � n�5 is the invariant of homotopytype of Mn and equal MS1

i (Mn) = �(�i(M

n)), where � : Z �! N such that �(x) =1=2(x+ j x j).

��

Zoran �koda

Institute Rudjer Boskovic, Zagreb, Croatia.E-mail address: [email protected]

Equivariant sheaves and torsors beyond groupoids

In my earlier work in noncommutative geometry I introduced analogues of torsors forcoactions of Hopf algebras on nona¢ ne noncommutative spaces. A puzzling class of �big�examples called for a non�at descent, which should be replaced for geometric purposesby (derived) descent up to higher homotopies. Combinatorially, similar higher cocyclesappear also in the study of descent for weak actions of categorical groups (in ordinary"commutative" geometry) and bigroupoids. These objects are relevant for study of "di¤er-ential geometry of gerbes" now popular in geometry and physics. The notion of equivariantsheaves in this setup is far more involved than in 1-categorical situation. I will outline ourwork in progress concerning these ideas.

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77

David Szeghy

Eötvös University, Budapest, Hungary.E-mail address: [email protected]

Conjugate and focal points in semi-Riemann geometry

The focal locus F (p) of a smooth submanifold P of a semi-Riemannian manifold isconsidered in the normal bundel NP of P and also the image of F (P ) in M under theexponential map. The possible types of focal points are studied.

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János Szenthe

Eötvös University, Budapest, Hungary.E-mail address: [email protected]

Invariant Lagrangians on Homogeneous

Let M = G=H be a homogeneous manifold, a Lagrangian L : TM ! R is said tobe invariant if for the canonical left action �g : M ! M; g 2 G of the homogeneousmanifold, L � T�g = L; g 2 G holds. Since thus a natural generalization of the concept ofhomogeneous Riemannian manifolds is obtained, the extension of their classical theory toinvariant Lagrangians is interesting and will be discussed in the lecture.

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Cornelia Vizman

West University of Timisoara, Timisoara, Romania.E-mail address: [email protected]

78

A geometric construction of abelian Lie group extensions

We extend Kostant�s central extension of a group of di¤eomorphisms preserving a vectorvalued closed 2-form ! (the kernel of a �ux homomorphism), to an abelian extension ofa group of di¤eomorphisms preserving ! up to a linear isomorphism (the kernel of a �ux1-cocycle). We use this abelian extension to build abelian Lie group extensions.

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Weiping Zhang

Nankai University, Tianjin, P.R.China.E-mail address: [email protected]

Bergman kernel and symplectic reduction

We describe joint results with Xiaonan Ma on the asymptotic behaviors of Bergmankernels on symplectic manifolds with Hamiltonian group actions.

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Nina Zhukova

Nizhny Novgorod State University, Russia.E-mail address: [email protected]

Singular Foliations with Ehresmann Connections

The concept of Ehresmann connection for a foliation was introduced by Blumenthal andHebda [1] as a natural generalization of Ehresmann connection for submersions. We haveextended this concept on foliations (M;F) with singularities in sense of Stefan and Sus-mann. Complete singular Riemannian and totally geodesic foliations have natural Ehres-mann connections.

79

We have de�ned an Ehresmann connection Q of singular foliation (M;F) as a gener-alized distribution Q on M; which is transverse to F ; and a vertical-horizontal propertyis satis�ed. This property allows to transfer an integral curve � of Q (called horizontal)along admissible curves (called vertical) in the leaf L = L(�(0)) of F . Unlike regular casethis transfer is not unique in general. For singular Riemannian foliation this transfer keepslengths of horizontal curves. We used this property and proved a criterion of local stabilityof leaves of singular Riemannian foliations.By singular Ehresmann foliation (M;F ; Q) we mean a singular foliation F with an

Ehresmann connection Q: We have introduced a concept of �Q-holonomy group of thesingular Ehresmann foliation (M;F ; Q) [3].This group has a global character.On some natural assumptions we have proved that the existence of a compact regular

leaf L with a �nite holonomy group �HQ(L) implies compactness of each leaf L� of thefoliation (M;F ; Q) and �niteness of the holonomy group �HQ(L�):Using �Q-holonomy groups we have introduced a topological groupoid �GQ(F ) of a

singular Ehresmann foliation (M;F ;Q): An important advantage of the groupoid �GQ(F )is that its topological space is always Hausdor¤. In the case of regular foliation thisgroupoid is equal to the graph GQ(F ) [2].

References

1. R.A.Blumenthal, J.J.Hebda. Ehresmann connections for foliations// Indiana Univ.Math. J. 33 (1984), 597-611.2. N.I.Zhukova. The graph of a foliation with Ehresmann connection and stability of

leaves// Russian Math.(Iz.VUZ) 38, ü2 (1994), 78-81.3. N.I.Zhukova. Ehresmann connections for foliations with singularities// Russian

Math.(Iz.VUZ) 48, ü10 (2004), 45-56.Singular Foliations with Ehresmann Connections.

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VII International Conference GEOMETRY AND TOPOLOGY OF …im0.p.lodz.pl/.../07_BedlewoEhresmann2005/folder05abcde.pdf · 2008-09-18 · Singular foliations, cohomology theories for