are continuous. From this it is clear that one should seek a topological structure ~ on G such that the maps

are continuous. Here we do not need to consider the whole group structure on G and it is not even necessary to know that such a structure exists on G. Generalizing the situation, we

can even assume that an arbitrary set X is given and two maps {:X---~• and ~:X--~% of it,

that these maps commute, f has a unique fixed point ~o, ~ is an involution with fixed point

~o. It is required to find a topological structure ~ on X such that ~: IX,T)---~C~,T) and

~: i~,~) - ~X,~) are continuous.

In the second version when the continuity of one map p':iG~G,~) ~(~,T),~ =~IX is

required, the map p':i~i~)--~(~) imposes no conditions on T since p'(~,~)=l for any ~ ~ ~.

BIBLIOGRAPHY ON BITOPOLOGICAL SPACES

A. A. Ivanov UDC 515.145

A bibliography on bitopological spaces.

In compiling this bibliography materials from the reviewing journals Matematika and Mathematical Reviews were mainly used. The references in papers were only partly used since not all the papers relating to the theme were considered by the compiler. For the compiler, there is not complete certainty that he has included all the literature on the theory of bitopological spaces. In the next collection "Studies in Topology 7" which will appear in 1991, a continuation of this bibliography will be given. In it, references omitted here and references to newly published papers will be included. In this connection the compiler asks that corresponding information and reprints of papers be sent to him at the following ad- dress: SSSR, 191011, Leningrad, Nab. r. Fontanki, 27, LOMI, Ivanov A. A.

I. D. Adnadjevid, "Ordered spaces and bitopology," Glasnik Mat., Ser. 3, i0 (30), No. 2, 337-340 (1975).

2. D. Adnadzhevich, "Separation axioms and bitopological quotient spaces," Math. Balkan., !, 1-6 (1977).

3. D. Adnadzhevich, "Separation axioms and convergence in bitopological ordered spaces," Soobshch. Akad. Nauk Gruz. SSSR, 94, No. 2, 285-288 (1979).

4. D. Adnadjevid, "Converging in B-spaces," Mat. Veer., 36, No. 4, 259-265 (1984). 5 D. Adnadzhevich, "Bicompactness of bitopological spaces," Zap. Nauchn. Seminar. Leningr.

Otdel. Mat. Inst., 143, 19-25 (1985). 6 D. Adnadjevid, "Some weak forms of continuity in bitopological spaces," Mat. Vesn., 38,

No. i, 17-23 (1986). 7 C. Amihaesei, "Sur lee hi-spaces extr~ment discontinue," An. sti. Univ. lasi, Sect.

I a, 19, No. i, 19-25 (1973). 8 Hovard Anton, "Measures on bitopological spaces," Ann. Soc. Sci. Bruxelles, Ser. i, 91,

No. i, 3-11 (1977). 9 S.P. Arya, "A note on pairwise ~i spaces," Glee. Mat., Ser. 3, 14, No. i, 147-150

(1979). I0 S.P. Arya, "Separation axioms in bitopological ordered spaces," Glas. Mat., Ser. 3, 15,

No. i, 169-178 (1980).

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 167, pp. 63-78, 1988.

2790 0090-4104/90/5201-2790512.50 �9 1990 Plenum Publishing Corporation

Ii. S. P. Arya and M. P. Bhamini, "Some generalization of Tp-spaces," Mat. Vesn., 6, Noo 3, 221-230 (1982).

12. S. P. Arya and M. P. Bhamini, "A note on semi-US-spaces," Ranchi Univ. Math. J., i__33, 60-68 (1982).

13. John August and Charles Byrn, "Completion of lattices of semicontinuous functions," J. Austral. Math. Soc., Ser. A, 26, No. 4, 453-464 (1978).

14. T. Birsan, "Sur les espaces bitopologiqus connexes," An. sti. Univ. Iasi, Sec. la, 14, No. 2, 293-296 (1968).

15. T. Birsan, "Une extension des bitopologiques," An. sti. Univ. Iasi, Sec. la, 15, No. i, 21-27 (1969).

16o T. Birsan, "Compacit~ dans les espaces bitopologiques," An. sti. Univo lasi, See. la, 15, No. 2, 317-328 (1969).

17. T. Birsan, "Sur les espaces bitopologiques completement reguliers," An. stio Univ. Iasi, Sec. la, 16, No. I, 29-34 (1970).

18. T. Birsan, "Contribution g l'etude des groups bitopologiques," An. sti. Univ. lasi, Sec. la, 19, No. 2, 297-310 (1973).

19. T. Birsan, "Transitive quasi-uniformities and zero dimensional bitopological spaces," An. sti. Univ. Iasi, Sec. la, No. 2, 317-322 (1974).

20. I. Bittner, "Erweiterungsrgume innerhalb der Kategorie der Biadhgrenzriume," Math. Nachro, iI__~5, 249-263 (1984).

21. I. Bittner, "Kompactifizierungen yon Biadhirensri~en. - Beispiele," Math. Nachro, 124, 183-198 (1985).

22. D. Borsan, "Bitopologii Generate de o G-quasi-Metrica," Stud. Univ. Babes-Bolyai Math., 22, No. 2, 72-76 (1977).

23. Shantha Bose, "Weak Hausdorff axiom in bitopological spaces," Bull. Calcutta Math. Soc., 72, No. 2, 95-106 (1980).

24. Shantha Bose, "Semi-open sets, semicontinuity and semi-open mappings in bitopological spaces," Bull. Calcutta Math. Soc., 73, No. 4, 237-246 (1981).

25. Shantha Bose and Dipti Sinha, "Almost open, almost closed, Q-continuous and almost quasicompact mappings in bitopological spaces," Bullo Calcutta Math. Soc., 73, Noo 6, 345-354 (1981).

26. Shantha Bose and Dipti Sinha, "Pairwise almost continuous map in bitopological spaces," Bull. Calcutta Math. Soc., 74, No. 4, 195-206 (i982).

27. L. M. Brown, "On extensions of bitopological spaces," in: Topology. 4th Colloq. Buda- pest, 1978, Vol. i, Amsterdam (1980), pp. 181-213.

28. Lawrance M. Brown, "Sequentially normal bitopological spaces," J. Fac. Sci. Karadeniz Techn. Univ., ~, 18-22 (1981).

29. Lawrance M. Brown, "Para-quasi-uniformities," Hacettepe Bullo Natur. Sci. Engo, i~2, 267-278 (1983).

30. G. C. L. Briimmer and S. Salbani, "On the notion of real compactness for bitopological spaces," Math. Colloq. Univ. Cape Town, ii, 89-99 (1977).

31. G. C. L. Briimmer, "Two procedures in bitopology," Lect. Notes Math., 71__9, 35-43. (1979)o 32. G. C. L. Brthmmer, "On some bitopological induced monads in topology," in: Structure

of Topological Categories, Proc. Conf. Univ. Bremen, 1978, Math. Arbeitspapier Univ. Bremen, Bremen (1979), pp. 13-30ao

33. G. C. L. Briimmer, "On the nonunique extension of topological to bitopological proper- ties," Lect. Notes Math., 91__~5, 50-67 (1982).

34. Charles Byrne, "On compactness of bitopological spaces," Kyungpook Math. J., 15, No. 2, 159-162 (1975).

35. M. J. Canfell, "Semi-algebras and rings of continuous functions," Thesis, Univ. Edin- burgh (1968).

36. Gyu lhn Chae and Kul Pyo Hong, "On the continuity in bitopological space," Ulsan Inst. Tech. Rep., 12, No. i, 147-150 (1981).

37. Gyu lhn Chae, Yoon Je, and Lee Ii Young, "On the continuity in bitopological spaces," Ulsan Inst. Tech. Rep., 13, No. i, 191-!93 (1982).

38. J. Chvalina, "On certain topological state spaces of X-automata," in: Topology. 4-th Colloq. Budapest, 1978, Vol. i, Amsterdam e.a. (1980), pp. 287-299.

39 J. Chvalina, "Separation properties of topologies associated with diagraphs," Scripta Fac. Sci. Natur. UJEP Brunensis-Math., I0, No. 8, 399-410 (1980).

49. Dusan M. Cirid, "Aksiome separacije u bitoploskim prostorima," Mat. Vesn., 11 (26), No. I, 10-21 (1974).

41. M. Dusan Cirid, "Dimension of bitopological spaces," Math~ Balkan., No. 4, 99-1135 (1974).

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43.

44.

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73 Peter Fletcher, Hughes B�9 Hoyle, and C. W. Patty, "The comparison of topologies, '~ Duke Math�9 J., 36, No. 2, 325-331 (1969)�9

74 All A. Fora and Hasan Z�9 Hdeib, "On pairwise Lindel6f spaces," Rev. Colomb. Mat., 17, No. 2, 31-57 (1983).

75 Ali A. Fora, "Strongly zero-dimensional bitopological spaces," J. Univ. Kuwait Sci., ii, No. 2, 181-190 (1984).

76 Takayoshi Fukutake, "On generalized closed sets in bitopological spaces," Bull. Fukuoka Univ. Nat. Sci., 35, 19-28 (1985).

77 J. C. Ganguli and D. Sinha, "Mixed topology for a bitopological space," Bullo Calcutta Math. Soc., 76, No. 5, 304-314 (1984)o

78 S. Ganguly and M. N. Mukherjee, "A note on bitopological spaces which are not pairwise Hausdorff," Math. Today, ~, 41-48 (1984)�9

79 George C�9 Gastl, "Bitopological spaces from quasi-proximities," Port. Math., 32_ i, Noso 3-4, 213-218 (1974).

80 C�9 K. Goel and A. Singhal, "Weaker form of compactness in bitopological hyperspaces," Indian J. Math�9 21, No. 3, 169-176 (1979).

81 Gary Gruenhage, "Convering properties on X 2 \ A, W-sets and compact subsets of ~-product, m~ Top. Appl., 17, No. 3, 287-304 (1984).

82 A. Gutierrez, "A note on a paper by S. Lal," Jo Austral~ Math. Soc., A35, Noo 2, 197-199 (1983)�9

83. Angel Gutierrez and Salva Romaguera, "Sobre espacios pairwise estratificable," Revo Roum. Math�9 Pures Appl., 31, No. 2, 141-150 (1986).

84. Hao-Xuan-Zhon, "On the small diagonals," Top. Appl., 13, No. 3, 283-293 (1982). 85. Hasan Z. Hdeib and Ali A. Fora, "On pairwise paracompact spaces," Dirasat Res~ J. Natur.

Sci., 2, No. 2, 21-29 (1982)�9 86. Taqdir Husain, "Semi-topological groups and linear spaces," Math. Ann~ 160, Noo 2,

146-160 (1965). 87. M. Husek, "Topological spaces without x-accessible diagonal, '~ Comment~ Math. Univo

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�9 . " Proc. Japan Acad. 44, 1009-1012 (1968). 102 Y W. Kim, "Pseudo-quasi-metric spaces, , __ 103. Yong Woom Kim, "Partial order in bitopological spaces," Notices Amer. Math. Soc., 16,

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Indian J�9 Pure Appl. Math., 12, No. 7, 799-803 (1981)~ 106. S. S. Lakshmi, "An alternative definition of local connectedness in bitopological spaces,"

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108. E. P. Lane, "Concerning extensions of semicontinuous functions in bitopological spaces," Notices Amer. Math. Soc., 12, No. I, 128 (1965).

109. Ernest Paul Lane, "Bitopological spaces," Doct. Diss. Purdue Univ., 1965; Diss. Ab- stracts, B27, No. 2, 543-544 (1966).

ii0. E. P. Lane, "Bitopological spaces and quasi-uniform spaces," Proc. London Math. Soc., 1-2, No. 2, 241-256 (1967).

III. E. P. Lane, "Quasi-proximities and bitopological spaces," Port. Math., 288, Nos. 3-4, 151-159 (1969).

112. Ii Yong Lee and Yoon Lee, "Quasisemiopen sets and quasisemicontinuity in bitopological spaces," Ulsan Inst. Techn. Rep., i~3, No. i, 171-173 (1982).

113. Tadeusz Lipski, "Multivalued maps of countably paracompact bitopological spaces," Demonstr. Math., I_88, No. 4, 1143-1151 (1985).

114. Manuel Lopez-Pellicer and Angel Gutierrez, "A generalization of Tong's theorem and properties of pairwise perfectly normal spaces," a. Austral. Math. Soc., A39, No. 3, 353-359 (1985).

115. S. N. Maheshwari and R. Prasad, "On pairwise s-normal spaces," Kyungpook Math. J., No. I, 37-40 (1975).

116. S. N. Maheshwari and R. Prasad, "Some new separation axioms in bitopological spaces," Mat. Vesn., ~, No. 2, 159-162 (1975).

117. S. N. Maheshwari and R. Prasad, "On pairwise irresolute functions," Mathematica (Cluj), 18 (41), No. 2, 169-172 (1976).

118. S. N. Maheshwari and R. Prasad, "Semi open sets and semi continuous functions in bitopo- logical spaces," Math. Notae, 26, 29-37 (1977-1978).

119. S. N. Maheshwari and R. Prasad, "On pairwise s-regular spaces," Riv. Math. Univ. Parma, i , 45-58 (1978).

120. S. N. Maheshwari and U. Tapi, "Sur les espaces bitopologiques s-connexes," An. sti. Univ. lasi, Sec. la, 255, No. i, 57-59 (1979).

121 S. N. Maheshwari, P. C. Jain, and Gyu lhn Chae, "On quasiopen sets," Ulsan Inst. Tech. Rep., l!l, No. 2, 291-292 (1980).

122 S. N. Maheshwari, Gyu lhn Chae, and S. S. Thakur, "Some new mappings in bitopological spaces," Ulsan Inst. Tech. Rep., i__22, No. 2, 301-304 (1981).

123 Z. P. Mamusic, "Some further aspects of G. Preuss' theory of E-connected spaces," Mat. Vesn., 388, No. I, 65-74 (1986).

124 A. S. Mashhour, F. H. Khedr, and S. N. EI-Deeb, "S-separation axioms in bitopological spaces," Bull. Fak. Sci. Assiut Univ., All, No. i, 53-67 (1982).

125 A. S. Mashhour, F. H. Khedr, I. A. Hasanein, and A. A. Allam, "S-closedness in bitopo- logical spaces," Kyungpook Math. J., 2_~4, No. i, 93-99 (1984).

126 A. S. Mashhour, F. H. Khedr, I. A. Hasanein, and A. A. Allam, "S-closedness in bito- pological spaces," Ann. Soc. Sci. Bruxelles, Set. I, 9-6, No. 2, 69-76 (1982).

127 K. N. Meenakshi, "Completion of bitopological spaces," J. Madras Univ., B35-B36, 27-31 (1965-1966 (1968)).

128. D. N. Mishra and K. K. Dube, "Pairwise R-space," Ann. Soc. Sci. Bruxelles, Ser. I, 8__7, No. I, 3-15 (1973).

129. S. N. Mishra, "Remarks on some fixed point theorems in bimetric spaces," Indian J. Pure Appl. Math., 2, No. 2, 1271-1274 (1978).

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139. Mila Mrsevid, "Local compactness in bitopological hyperspaces," Mat. Vesn., ~, No. I, 41-52 (1979).

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142. Mila Mrsevid, "On quasiproximity spaces," Indian J. Pure Appl. Math., 14, Noo 4, 511- 514 (1983).

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144. Mila Mrsevid, "Some properties of bitopological hyperspaces," Kyungpook Math. I,, 25, No. i, 61-70 (1985).

145. Mila Mrsevid, "Some characterisations of bitopological connectivity properties," Mat. Vesn., 38, No. i, 75-87 (1986).

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148. M. N. Mukherjee, "Pairwise category properties of some pairwise minimal bitopological spaces," Ann. Soc. Sci. Bruxelles, Ser. i, 96, No. 3, 117-127 (1982(1983)).

149. M. N. Mukherjee, "Pairwise semi-connectedness in bitopologica! spaces," Indian J. Pure Appl. Math., I-4, No. 9, 1166-1173 (1983).

150 M. N. Mukherjee, "On pairwise S-closed bitopological spaces," International J. Math. Sci., 8, No. 4, 729-745 (1985).

151 M. N. Mukherjee, "Biotopologicai space with mixed topology," Bull. Calcutta Math. Soc., 77, No. 5, 274-280 (1985).

152 M. N. Mukher3ee and G. K. Banerjee, "Pairwise setconnected mappings in bitopological spaces," Indian J. Pure Appl. Math., i-7, No. 9, 1106-1113 (1986)o

153 M. G. Murdeshwar and S. A. Naimpally, Quasi-Uniform Topological Spaces, Noordhoff, Groningen (1966).

154 L. Nachbin, "Sur les espaces uniformes ordonnes," Compt. Rend. Paris, 22___66, 774-.775 (1948).

155 S. I. Nedev, "O-metrizable spaces," Trudy Mosk. Mat. Obshch., 24, 201-236 (1971). 156. Takashi Noiri, "On pairwise s-regular spaces,"Atti Accad. Naz. Lincei Rend. CI. Sci.

Fis. Mat. Natur., (8), 62, No. 6, 787-790 (1977). 157. Takashi Noiri, "A note on pairwise s-normal spaces," Kyungpook Math. J., 22, No. i,

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Supplement I

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25. R. Vasudevan and C. K. Goel, "A note on pairwise C-compact bitopological spaces~" Mat. Vesn., 1 (14) (29), No. 2, 179-187 (1977).

CONWAY AND KAUFFMAN MODULES OF A SOLID TORUS

V. G. Turaev UDC 515.162.8

The Conway and Kauffman modules of a solid torus are calculated.

I. Formulation of Results

i.i. Conway Module. Let M be an oriented three-dimensional manifold. One says that 3

oriented links L+~L_, L 0 c M form a Conway triple if h,,h_,h 0 are isotopic to links in M

which coincide (considering orientations) outside a ball ~ cM and in this ball look as shown

in Fig. i. In this figure the ball B is identified with a ball in ~ by means of a homeo-

morphism of degree I; here B is endowed with the orientation induced by the given orienta-

tion in M and ~ is endowed with the right orientation. For example, if L is an arbitrarily

oriented link in M and ~ is a trivial knot in M \ L then ( L~ L ~ h ~ ~) is a Conway triple

(see Fig. 2).

We shall consider that the empty set ~ is the unique "empty" oriented knot in M up to

isotopy. The triple Q~ ~), where ~ is a trivial knot in M will be considered a Conway

triple in what follows.

We denote by g the set of isotropy types of oriented links in M. We denote by A the

ring of Laurent polynomials in 2 variables ~ [~, -~ ~! ~i] o We consider the free A-module

/I[~ ] whose elements are formal linear combinations of the elements of the set ~ with co-

efficients in A. Let C be the submodule of this module generated by the elements m L+ - -i h- -~ho corresponding to all Conway triples L+~L-TL 0CM, The quotient module 7~[~]/O

is denoted by ~'~QM) and is called the Conway module of the manifold M The module Co~(M)

is also called the module of skein equivalence of links in M. For an oriented link L in M

we shall denote by [L] the class ~Q~O~O)ECo~(M). In particular~ [~]=Q~_~-i)~i[~].

Interest in the Conway module is due, in particular, to the fact that the elements of

the dual module QC0~QM~--H0~L~C0~QM),]I) are in bijective correspondence with the "gen-

eralized Jones-Conway invariants of links in M," i.e., with isotopy invariants P of oriented

links in M which assume values in A and have the property that ~QL+)-~-I~Q~_)= ~i~0) for

any Conway triple h+,h_, Lo c M.

_ ' Fig. 2 ~+ L0

Fig. 1

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 167, pp. 79-89, 1988.

0090-4104/90/5201-2799512.50 �9 1990 Plen~m Publishing Corporation 2799