Manifolds: Where Do We Come From?WhatAre We?Where Are We GoingMishaGromovSeptember13,2010Contents1 IdeasandDenitions. 22 HomotopiesandObstructions. 43 GenericPullbacks. 94 DualityandtheSignature. 125 TheSignatureandBordisms. 256 ExoticSpheres. 367 IsotopiesandIntersections. 398 Handlesandh-Cobordisms. 469 ManifoldsunderSurgery. 49110 EllipticWingsandParabolicFlows. 5311 Crystals,LiposomesandDrosophila. 5812 Acknowledgments. 6313 Bibliography. 63AbstractDescendantsofalgebraickingdomsofhighdimensions, enchantedbythemagicofThurstonandDonaldson,lostinthewhirlpoolsoftheRicciow,topologistsdreamofanideallandofmanifoldsperfectcrystalsofmathematicalstructurewhichwouldcaptureourvaguementalimagesofgeometricspaces. Webrowsethroughtheideasinheritedfromthepasthopingtopenetratethroughthefogwhichconcealsthefuture.1 IdeasandDenitions.Wearefascinatedbyknotsandlinks. Wheredoesthisfeelingofbeautyandmystery come from?To get a glimpse at the answer let us move by 25 millionyears in time.25 106is, roughly, what separates us from orangutans: 12 million years toour common ancestor on the phylogenetic tree and then 12 million years backby another branch of the tree to the present day orangutans.But are there topologists among orangutans?Yes, there denitely are: many orangutans are good at proving the triv-ialityofelaborateknots, e.g. theyfastmastertheartofuntyingboatsfromtheir mooring when they fancy taking rides downstream in a river, much to theannoyance of people making these knots with a dierent purpose in mind.A more amazing observation was made by a zoo-psychologist Anne Russonin mid 90s at Wanariset Orangutan Reintroduction Project (see p. 114 in [68]).... Kinoi [ajuvenilemaleorangutan], whenhewasinapossessionof ahose, invested every second in making giant hoops, carefully inserting one endof his hose into the other and jamming it in tight. Once hed made his hoop,he passed various parts of himself back and forth through it an arm, his head,his feet, his whole torso as if completely fascinated with idea of going throughthe hole.Playing with hoops and knots, where there is no visible goal or any practicalgainbeitanapeora3D-topologistappearsfullynon-intelligenttoapractically minded observer. But we, geometers, feel thrilled at seeing an animalwhose space perception is so similar to ours.2Itisunlikely, however, thatKinoi wouldformulatehisideasthewaywedo and that, unlike our students, he could be easily intimidated into acceptingequivalenceclassesofatlasesandringedspacesasappropriatedenitionsof his topological playground. (Despite such display of disobedience, we wouldenjoy the company of young orangutans; they are charmingly playful creatures,unlike the aggressive and reckless chimpanzees our nearest evolutionary neigh-bors.)Apart from topology, orangutans do not rush to accept another human def-inition, namely that of tools, as ofexternal detachedobjects(toexcludeabranchusedforclimbingatree)employed for reaching specic goals.(The use of tools is often taken by zoo-psychologists for a measure of intel-ligence of an animal.)Being imaginative arboreal creatures, orangutans prefer a broader denition:For example (see [68]): they bunch up leaves to make wipers to clean their bodies without detach-ing the leaves from a tree;theyoftenbreakbranchesbutdeliberatelyleavethemattachedtotreeswhen it suits their purposes these could not have been achieved if orangutanswere bound by the detached denition.Morale. Ourbestdenitions, e.g. thatofamanifold, towerasprominentlandmarks over our former insights. Yet, we should not be hypnotized by de-nitions. After all, they are remnants of the past and tend to misguide us whenwe try to probe the future.Remark.There is a non-trivial similarity between the neurological structuresunderlying the behaviour of playful animals and that of working mathematicians(see [31]).32 HomotopiesandObstructions.Formorethanhalf acentury, startingfromPoincare, topologistshavebeenlaboriously stripping their beloved science of its geometric garments.Nakedtopology, reinforcedbyhomological algebra, reachedits to-daybreathtakingly high plateau with the followingSerre |Sn+NSN|-Finiteness Theorem. (1951) Thereareat mostnitelymanyhomotopyclassesof mapsbetweenspheres Sn+NSNbut forthe two exceptions: equivi-dimensional case wheren = 0N(SN) = Z; the homotopy class of amapSN SNinthiscaseisdeterminedbyanintegerthatisthedegreeofamap.(Brouwer 1912, Hopf 1926. We dene degree in section 4.)This is expressedin the standard notation by writingN(SN) = Z. Hopf case, whereNis even andn = 2N 1. In this case2N1(SN) containsa subgroup of nite index isomorphic to Z.It follows thatthe homotopy groupsn+N(SN) are nite forN >> n,where, by the Freudenthal suspension theorem of 1928 (this is easy),the groupsn+N(SN) forN n do not depend onN.These are called the stable homotopy groups of spheres and are denotedstn .H. Hopf provedin1931thatthemapf S3S2=S3]T, forthegroupT C of the complex numbers with norm one which act on S3 C2by (z1, z2) (tz1, tz2), is non-contractible.In general, the unit tangent bundle X = UT(S2k) S2khas nite homologyHi(X) for 0 < i < 4k 1. By Serres theorem, there exists a mapS4k1 Xofpositive degree and the composed mapS4k1 X S2kgenerates an innitecyclic group of nite index in4k1(S2k).The proof by Serre a geometers nightmare consists in tracking a multi-tude of linear-algebraic relations between the homology and homotopy groupsof innite dimensional spaces of maps between spheres and it tells you next tonothingaboutthegeometryof thesemaps. (See[58] forasemi-geometricproof of the niteness of the stable homotopy groups of spheres and section 54of thisarticleforarelateddiscussion. Also, theconstructionin[23] mayberelevant.)Recall that theset of thehomotopyclasses of maps of asphere SMtoaconnectedspaceXmakesagroupdenotedM(X), (isforPoincarewhodened thefundamental group1) where the denition of the group structuredepends on distinguished pointsx0 Xands0 SM. (The groupsMdenedwithdierent x0aremutuallyisomorphic, andif Xissimplyconnected, i.e.1(X) = 1, then they are canonically isomorphic.)This point inSMmay be chosen with the representation ofSMas the onepoint compacticationof theEuclideanspaceRM, denotedRM, wherethisinnitypoint istakenfors0. Itisconvenient, insteadofmapsSm= Rm (X, x0), todealwithmapsf RMXwithcompactsupports, wherethesupportofanfistheclosureofthe(open)subsetsupp(f) =suppx0(f) Rmwhich consists of the pointss Rmsuch thatf(s) x0.Apairofmapsf1, f2 RM Xwithdisjointcompactsupportsobviouslydenes the joint map f RMX, where the homotopy class of f(obviously)dependsonlyonthoseof f1, f2, providedsupp(f1)liesinthelefthalf spaces1 0 RM, wheres1isanon-zerolinearfunction (coordinate) on RM.Thecompositionofthehomotopyclassesoftwomaps,denoted |f1| |f2|,is dened as the homotopy class of the joint off1 moved far to the left withf2moved far to the right.Geometryissacricedhereforthesakeofalgebraicconvenience: rst, webreakthesymmetryofthesphereSMbychoosingabasepoint,andthenwedestroythesymmetryof RMbythechoiceof s1. If M =1, thenthereareessentially two choices: s1and s1, which correspond to interchangingf1withf2 nothing wrong with this as the composition is, in general, non-commutative.Ingeneral M 2, theses1 0are, homotopicallyspeaking, parametrizedbytheunit sphere SM1RM. Since SM1is connectedfor M 2, thecomposition is commutative and, accordingly, the composition inifori 2 isdenoted |f1| + |f2|. Good for algebra, but theO(M + 1)-ambiguity seems toogreat a price for this. (An algebraist would respond to this by pointing out thattheambiguityisresolvedinthelanguageofoperadsorsomethingelseofthiskind.)Butthisis, probably, unavoidable. Forexample, thebestyoucandoformaps SMSMin a given non-trivial homotopy class is to make them symmet-ric (i.e. equivariant) under the action of the maximal torus Tkin the orthogonalgroupO(M + 1), wherek = M]2 for evenMandk = (M + 1)]2 forModd.And ifn 1, then, with a few exceptions, there are no apparent symmetricrepresentatives in the homotopy classes of maps Sn+NSN; yet Serres theoremdoes carry a geometric message.Ifn 0, N 1, then every continuous mapf0 Sn+N SNis homotopic toa mapf1 Sn+NSNof dilation bounded by a constant,dil(f1) =defsups1s2Sn+Ndist(f(s1), f(s2))dist(s1, s2) const(n, N).DilationQuestions. (1)Whatistheasymptoticbehaviourof const(n, N)forn, N ?5For all we know the SerredilationconstantconstS(n, N) may be boundedforn and, say, for 1 N n 2, but a bound one can see ohand is thatbyanexponential tower (1 + c)(1+c)(1+c)..., of height N, sinceeachgeometricimplementation of the homotopy lifting property in a Serre brations may bringalong an exponential dilation. Probably, the (questionably) geometric approachtotheSerretheoremviasingularbordisms(see[75], [23],[1]andsection5)delivers a better estimate.(2) Let f Sn+NSNbe a contractible map of dilation d, e.g. fequals them-multiple of another map wherem is divisible by the order ofn+N(SN).What is, roughly, the minimum Dmin = D(d, n, N) of dilations of maps Fofthe unit ballBn+N+1SNwhich are equal tofon(Bn+N+1) = Sn+N?Of course, this dilation is the most naive invariant measuring the geometricsize of a map. Possibly, an interesting answer to these questions needs a moreimaginative denition of geometric size/shape of a map, e.g. in the spirit ofthe minimal degrees of polynomials representing such a map.Serrestheoremanditsdescendantsunderlymostof thetopologyof thehigh dimensional manifolds. Below are frequently used corollaries which relatehomotopy problems concerning general spaces X to the homology groups Hi(X)(see section 4 for denitions) which are much easier to handle.|Sn+NX|-Theorems. LetXbeacompactconnectedtriangulatedorcellular space, (dened below) or, more generally, a connected space with nitelygeneratedhomologygroupsHi(X), i = 1, 2, ... . IfthespaceXissimplycon-nected, i.e. 1(X) = 1, then its homotopy groups have the following properties.(1) Finite Generation. The groups m(X) are (Abelian!)nitely generatedfor allm = 2, 3, ....(2) Sphericity. Ifi(X) = 0 fori = 1, 2, N 1, then the (obvious) HurewiczhomomorphismN(X) HN(X),which assigns, to a mapSN X, theN-cycle represented by thisN-sphere inX, is an isomorphism. (This is elementary, Hurewicz 1935.)(3) Q-Sphericity. If the groups i(X) are nite for i = 2, N1 (recall that weassume1(X) = 1), thentheHurewiczhomomorphismtensoredwithrationalnumbers,N+n(X) Q HN+n(X) Q,is an isomorphism forn = 1, ..., N 2.Because of the nite generation property, The Q-sphericity is equivalent to(3)Serrem-SphericityTheorem. Let the groupsi(X) be nite (e.g.trivial) fori = 1, 2, ..., N 1 andn N 2. Thenanm-multiple of every (N +n)-cycle inXfor somem 0 is homologous toan (N + n)-sphere continuously mapped toX;every two homologous spheresSN+nXbecome homotopic when composedwithanon-contractiblei.e. of degreem 0, self-mappingSn+NSn+N. Inmore algebraic terms, the elements s1, s2 n+N(X) represented by these spheressatisfyms1 ms2 = 0.The following is the dual of them-Sphericity.6Serre |SN|Q- Theorem. LetXbeacompacttriangulatedspaceofdimensionn + N, where eitherNis odd orn < N 1.Thenanon-zeromultipleofeveryhomomorphismHN(X) HN(SN)canbe realized by a continuous mapX SN.If twocontinuousmapsaref, g X SNarehomologous, i.e. if theho-mologyhomomorphismsf, g HN(X) HN(SN) = Zareequal, thenthereexists a continuous self-mapping SN SNof non-zero degree such that thecomposed maps fand f X SNare homotopic.These Q-theorems follow from the Serre niteness theorem for maps betweenspheres by an elementary argument of induction by skeletons and rudimentaryobstruction theory which run, roughly, as follows.Cellular andTriangulatedSpaces. Recall that acellular spaceis atopological space Xwithanascending(niteorinnite)sequenceof closedsubspaces X0 X1 ... Xi ... called he i-skeleta of X, such that i(Xi) = Xand such thatX0is a discrete nite or countable subset.every Xi, i > 0 is obtained by attaching a countably (or nitely) many i-ballsBito Xi1 by continuous maps of the boundaries Si1= (Bi) of these balls toXi1.For example, ifXis a triangulated space then it comes with homeomorphicembeddings of the i-simplices iXi extending their boundary maps, (i) Xi1 Xiwhere one additionally requires (here the word simplex, which is,topologically speaking, is indistinguishable from Bi, becomes relevant) that theintersection of two such simplices iand jimbedded intoXis a simplex kwhich is a face simplex in i kand in j k.If X is a non-simplicial cellular space, we also have continuous maps BiXibut they are, in general, embeddings only on the interiorsBi(Bi), since theattachingmaps(Bi) Xi1arenotnecessarilyinjective. Nevertheless, theimages of Biin Xare called closed cells, and denoted Bi Xi, where the unionof all thesei-cells equalsXi.Observethatthehomotopyequivalenceclassof Xiisdeterminedbythatof Xi1andbythehomotopyclassesofmapsfromthespheresSi1=(Bi)toXi1. WearefreetotakeanymapsSi1Xi1wewishinassemblingacellular X which make cells more ecient building blocks of general spaces thansimplices.For example, the sphereSncan be made of a 0-cell and a singlen-cell.If Xi1 = Slfor some l i1 (one has l < i1 if there is no cells of dimensionsbetweenl andi 1) then the homotopy equivalence classes ofXiwith a singlei-cell one-to-one correspond to the homotopy groupi1(Sl).On the other hand, every cellular space can be approximated by a homotopyequivalentsimplicialone,whichisdonebyinductiononskeletonsXiwithanapproximationofcontinuousattachingmapsbysimplicial mapsfrom (i 1)-spheres toXi1.RecallthatahomotopyequivalencebetweenX1andX2isgivenbyapairof maps f12 X1 X2andf21 X2 X1, suchthatbothcomposedmapsf12 f21 X1X1andf21 f12 X2X2are homotopic to the identity.ObstructionsandCohomology. LetY be a connected space such thati(Y ) = 0fori = 1, ..., n 1 1, letf X Y beacontinuousmapandlet7usconstruct, byinductiononi = 0, 1, ..., n 1, amapfnew X Y whichishomotopic tofand which sendsXn1to a pointy0 Yas follows.Assumef(Xi1) =y0. ThentheresultingmapBifY , foreachi-cell BifromXi,makes ani-sphere inY ,because the boundaryBi Xi1goes to asingle point our toy0inY .Sincei(Y ) = 0, thisBiinYcan be contracted toy0 without disturbing itsboundary. We do it alli-cells fromXiand, thus, contractXitoy0. (One cannot, in general, extend a continuous map from a closed subset X X to X, butone always can extend a continuous homotopyft X Y ,t |0, 1|, of a givenmapf0 X Y , f0|X =f0,toahomotopyft X Y forallclosedsubsetsX X, similarly to how one extends R-valued functions fromX XtoX.)Thecontractionof XtoapointinY canbeobstructedonthen-thstep,wheren(Y ) 0, andwhereeachorientedn-cell BnXmappedtoY with(Bn) y0representsanelementc n(Y )whichmaybenon-zero. (Whenwe switch an orientation inBn, thenc c.)We assume at this point, that our space X is a triangulated one, switch fromBnto nand observe that the function c(n) is (obviously) an n-cocycle in Xwith values in the groupn(Y ), which means (this is what is longer to explainfor general cell spaces) that the sum of c(n) over the n+2 face-simplices nn+1equals zero, for all n+1in the triangulation (if we canonically/correctlychoose orientations in all n).The cohomology class |c| Hn(X; n(X)) of this cocycle does not depend(by an easy argument) on how the (n 1)-skeleton was contracted. Moreover,every cocyclecin the class of |c| can be obtained by a homotopy of the maponXnwhichiskeptconstantonXn2. (TwoA-valuedn-cocycles candc,foranabeliangroupA, areinthesamecohomologyclassif thereexistsanA-valuedfunctiond(n1)ontheorientedsimplicesn1Xn1, suchthatn1n d(n1) =c(n) c(n)forall n. Thesetof thecohomologyclasses ofn-cocycles with a natural additive structure is called the cohomologygroup Hn(X; A). It can be shown that Hn(X; A) depends only on X but not ana particular choice of a triangulation of X. See section 4 for a lighter geometricdenitions of homology and cohomology.)Inparticular, if dim(X) =nwe, thus, equatetheset |X Y |oftheho-motopyclassesof maps X Y withthecohomologygroupHn(X; n(X)).Furthermore, thisargumentappliedtoX =Snshowsthat n(X) =Hn(X)and, in general, thatthe set of the homotopy classes of mapsX Y equals the set of homomor-phismsHn(X) Hn(Y ), providedi(Y ) = 0 for 0 < i < dim(X).Finally, whenweusethisconstructionforprovingtheaboveQ-theoremswhereoneof thespacesisasphere, wekeepcomposingourmapswithself-mappings of this sphere of suitable degreem 0 that kills the obstructions bythe Serre niteness theorem.For example, if X is a nite cellular space without 1-cells, one can dene thehomotopy multiple lX, for every integer l, by replacing the attaching maps of all(i+1)-cells, SiXi, by lki-multiples of these maps in i(Xi) for k2 n, every closed, i.e. compact without bound-ary, n-manifold X comes from the generic pullback construction applied to mapsffromSn+N= Rn+Nto the Thom spaceVof the canonical N-vector bundleV X0 = GrN(Rn+N),X = f1(X0) for genericf Sn+NV X0 = GrN(Rn+N).Inaway, Thomhas discoveredthesourceof all manifolds intheworldandrespondedtothequestionWherearemanifoldscomingfrom?withthefollowing1954Answer. All closedsmoothn-manifoldsXcomeaspullbacksoftheGrassmanniansX0 =GrN(Rn+N)intheambientThomspacesV X0undergeneric smooth mapsSn+NV.The manifolds X obtained with the generic pull-back construction come witha grain of salt: generic maps are abundant but it is hard to put your nger onanyoneofthemwecannotsaymuchabouttopologyandgeometryofanindividual X. (Itseems, onecannotputallmanifoldsinonebasketwithoutsome random string attached to it.)But, empowered with Serres theorem, this construction unravels an amazingstructure in the space of all manifolds (Before Serre, Pontryagin and followinghim Rokhlin proceeded in the reverse direction by applying smooth manifoldsto the homotopy theory via the Pontryagin construction.)Selectinganobject X, e.g. asubmanifold, fromagivencollection Xofsimilarobjects,wherethereisnodistinguishedmemberXamongthem,isanotoriously dicult problem which had been known since antiquity and can betraced to De Cael of Aristotle. It reappeared in 14th century as Buridans assproblem and as Zermelos choice problem at the beginning of 20th century.A geometer/analyst tries to select an X by rst nding/constructing a valuefunction on Xand then by taking the optimalX. For example, one may goforn-submanifoldsXof minimalvolumes in an (n + N)-manifoldWendowedwith a Riemannian metric. However, minimal manifolds X are usually singularexcept for hypersurfacesXn Wn+1wheren 6 (Simons, 1968).Picking up a generic or a randomXfrom Xis a geometers last resortwhenall deterministicoptionshavefailed. Thisisaggravatedintopology,since there is no known construction delivering all manifoldsXin a reasonablycontrolled manner besides generic pullbacks and their close relatives;11 on the other hand, geometrically interesting manifolds X are not anybodyspullbacks. Often, theyarecomplicatedquotientsof simplemanifolds, e.g.X = S], whereS is a symmetric space, e.g. the hyperbolicn-space, and is adiscrete isometry group acting onS, possibly, with xed points.(It is obvious that every surface Xis homeomorphic to such a quotient, andthis is also so for compact 3-manifolds by a theorem of Thurston. But ifn 4,one does not know if every closed smooth manifoldXis homeomorphic to suchanS]. It is hard to imagine that there are innitely many non-dieomorphicbut mutually homeomorphicS] for the hyperbolic 4-spaceS, but this may bea problem with our imagination.)Startingfromanotherend, onehasramiedcovers X X0of simplemanifolds X0, where one wants the ramication locus 0 X0 to be a subvarietywithmildsingularitiesandwithaninterestingfundamentalgroupofthecomplement X00, but nding such 0 is dicult (see the discussion following(3) in section 7).And even for simple 0 X0, the description of ramied coveringsX X0where X are manifolds may be hard. For example, this is non-trivial for ramiedcoveringsovertheatn-torusX0 = Tnwhere0istheunionof several at(n2)-subtori in general position where these subtori may intersect one another.4 DualityandtheSignature.Cycles and Homology.If X is a smooth n-manifold X one is inclined to denegeometrici-cyclesCinX, whichrepresenthomologyclasses |C| Hi(X),ascompactorientedi-submanifoldsC Xwithsingularitiesofcodimensiontwo.This, however, istoorestrictive, asitrulesout, forexample, closedself-intersecting curves in surfaces, and/or the double covering mapS1S1.Thus, weallowC Xwhichmayhavesingularitiesof codimensionone,and, besides orientation, a locally constant integer valued function on the non-singular locus ofC.First, we dene dimension on all closed subsets in smooth manifolds with theusual properties of monotonicity, locality and max-additivity, i.e. dim(AB) =max(dim(A), dim(B)).Besides we want our dimension to be monotone under generic smooth mapsof compactsubsetsA, i.e. dim(f(A)) dim(A)andif f Xm+nYnisageneric map, thenf1(A) dim(A) + m.Then we dene the generic dimension as the minimal function with theseproperties whichcoincides withtheordinarydimensiononsmoothcompactsubmanifolds. Thisdepends, of course, onspecifyinggenericateachstep,but this never causes any problem in-so-far as we do not start taking limits ofmaps.An i-cycle C X is a closed subset in X of dimension i with a Z-multiplicityfunction onCdened below, and with the following set decomposition ofC.C = Creg C Csing,such that12Csingis a closed subset of dimension i 2. Cregis an open and dense subset in C and it is a smooth i-submanifold inX.C Csingisaclosedsubsetofdimension i 1. Locally,ateverypoint,x Cthe unionCreg Cis dieomorphic to a collection of smooth copies ofRi+inX, called branches, meeting along their Ri1-boundaries where the basicexample is the union of hypersurfaces in general position.TheZ-multiplicitystructure, is givenbyanorientationof Cregandalocallyconstantmultiplicity/weight Z-functiononCreg, (wherefori = 0thereisonlythisfunctionandnoorientation)suchthatthesumoftheseorientedmultiplicities over the branches ofCat each pointx Cequals zero.EveryCcanbemodiedtoCwithemptyCandif codim(C) 1, i.e.dim(X) > dim(C), also with weights = 1.Forexample, if 2l orientedbranchesof Cregwithmultiplicities1meetatC, dividethemintol pairswiththepartnershavingoppositeorientations,keepthesepartnersattachedastheymeetalongCandseparatethemfromthe other pairs.Nomatterhowsimple, thisseparationof branchesis, saywiththetotalweight 2l, it can be performed inl! dierent ways. PoorC burdened with thisambiguity becomes rather non-ecient.If Xisaclosedorientedn-manifold,thenititselfmakesann-cyclewhichrepresentswhatiscalledthefundamental class |X| Hn(X). Othern-cyclesare integer combinations of the oriented connected components ofX.It is convenient to have singular counterparts to manifolds with boundaries.Sincechainswereappropriatedbyalgebraictopologists, weusethewordplaque, where an (i + 1)-plaqueD with a boundary(D) D is the same asa cycle, except that there is a subset(D) D, where the sums of orientedweights do not cancel, where the closure of(D)equals(D) D and wheredim((D) (D)) i 2.Geometrically, we impose the local conditions on D(D) as on (i+1)-cyclesand add the locali-cycle conditions on (the closed set)(D), where this(D)comes with the canonical weighted orientation induced fromD.(Therearetwooppositecanonical inducedorientations ontheboundaryC = D, e.g. on the circular boundary of the 2-disc, with no apparent rationalfor preferring one of the two. We choose the orientation in (D) dened by theframes of the tangent vectors1, ..., isuch that the orientation given toDbythe (i +1)-frames, 1, ..., i agrees with the original orientation, where is theinward looking normal vector.)Everyplaquecanbesubdividedbyenlargingtheset D(and/or, lessessentially,Dsing). We do not care distinguishing such plaques and, more gen-erally, the equalityD1 = D2 means that the two plaques have a common subdi-vision.We go further and writeD = 0 if the weight function onDregequals zero.We denote by D the plaque with the either minus weight function or withthe opposite orientation.We deneD1 + D2if there is a plaqueD containing bothD1andD2as itssub-plaques with the obvious addition rule of the weight functions.Accordingly, we agree thatD1 = D2ifD1 D2 = 0.13On Genericity. We have not used any genericity so far except for the deni-tion of dimension. But from now on we assume all our object to be generic. Thisis needed, for example, to deneD1+ D2, since the sum of arbitrary plaques isnot a plaque, but the sum of generic plaques, obviously, is.Also if you are used to genericity, it is obvious to you thatIfD Xis ani-plaque (i-cycle) then the imagef(D) Y under a genericmapf X Yis ani-plaque (i-cycle).Notice that fordim(Y ) = i + 1 the self-intersection locus of the imagef(D)becomes apart of f(D)andif dim(Y ) =i + 1, thenthenewpart the -singularity comes fromf((D)).It is even more obvious thatthe pullbackf1(D) of ani-plaqueD Ynunder a generic mapf Xm+nYnis an (i +m)-plaque in Xm+n; if D is a cycle and Xm+nis a closed manifold(or the mapfis proper), thenf1(D) is cycle.Asthelasttechnicality, weextendtheabovedenitionstoarbitrarytri-angulatedspacesX,withsmooth genericsubstitutedbypiecewisesmoothgeneric or by piecewise linear maps.Homology. Twoi-cycles C1andC2inXarecalledhomologous, writtenC1 C2, if there is an (i+1)-plaque D in X|0, 1|, such that (D) = C10C21.For example every contractible cycleC Xis homologous to zero, since theconeover CinY =X |0, 1|correspondingtoasmoothgenerichomotopymakes a plaque with its boundary equal toC.SincesmallsubsetsinXarecontractible,acycleC Xishomologoustozeroifandonlyifitadmitsadecompositionintoasumofarbitrarilysmallcycles, i.e. if, foreverylocallynitecoveringX = iUi, thereexistcyclesCi Ui, such thatC = iCi.The homology groupHi(X) is dened as the Abelian group with generators|C|forall i-cycles CinXandwiththerelations |C1| |C2| =0wheneverC1 C2.SimilarlyonedenesHi(X; Q),fortheeld Qofrationalnumbers,byal-lowingCandD with fractional weights.Examples. Every closed orientablen-manifoldXwithkconnected compo-nents hasHn(X) = Zk, whereHn(X) is generated by the fundamental classesof its components.ThisisobviouswithourdenitionssincetheonlyplaquesDinX |0, 1|with(D) (X |0, 1|) =X 0 X 1arecombinationof theconnectedcomponentsof X |0, 1| andsoHn(X)equalsthegroupof n-cyclesinX.Consequently,every closed orientable manifoldXis non-contractible.Theaboveargumentmaylooksuspiciouslyeasy, sinceitisevenhardtoprovenon-contractibilityof SnandissuingfromthistheBrouwerxedpointtheorem within the world of continuous maps without using generic smooth orcombinatorial ones, except forn = 1 with the covering map R S1and forS2with the Hopf brationS3S2.The catch is that the diculty is hidden in the fact that a generic image ofan (n + 1)-plaque (e.g. a cone overX) inX |0, 1| is again an (n + 1)-plaque.14What is obvious, however without any appeal to genericity is thatH0(X) =Zkfor every manifold or a triangulated space withk components.ThespheresSnhaveHi(Sn) = 0for0 0, -small moves of the vertices of S for 0, where these moves themselvesdepend on the embedding f of X into RMand on , such that the simplices of thecorrespondingly moved triangulation,sayS = S = S(X, f) are-transversalto the-scaledX, i.e. toX = X f RM, wherethe-transversality of an ane simplex RMtoX RMmeans thatthe ane simplices obtained from by arbitrary -moves of the vertices of for some = (S, ) > 0 are transversal to X. In particular, the intersectionangles betweenXand thei-simplices,i = 0, 1, ..., M 1, inSare all .If issucientlylarge(andhence, X RMisnearlyat), thenthe-transversality (obviously) implies that the intersection of X with each simplexand its neighbours in S in the vicinity of each point x X RMhas the samecombinatorial patternastheintersectionofthetangentspaceTx(X) RMwith these simplices. Hence, the (cell) partition = f of Xinduced from Scan be subdivided into a triangulation ofX = X.Almostallofwhatwehavepresentedinthissectionsofarwasessentiallyunderstood by Poincare, who switched at some point from geometric cycles totriangulations, apparently, in order to prove his duality. (See [41] for pursuingfurther the rst Poincare approach to homology.)The language of geometic/generic cycles suggested by Poincare is well suitedfor observing and proving the multitude of obvious little things one comes acrosseverymomentintopology. (Isuspect, geometric, evenworse, somealgebraictopologists think of cycles while they draw commutative diagrams. Rephrasing22J.B.S. Haldanes words: Geometry is like a mistress to a topologist: he cannotlive without her but hes unwilling to be seen with her in public.)But if you are far away from manifolds in the homotopy theory it is easier towork with cohomology and use the cohomology product rather than intersectionproduct.The cohomology product is a bilinear pairing, often denoted HiHj Hi+j,which is the Poincare dual of the intersection productHniHnjHnijinclosed orientedn-manifoldsX.The -product can be dened for all, say triangulated,Xas the dual of theintersectionproductontherelativehomology, HMi(U; ) HMj(U; ) HMij(U; ), for a small regular neighbourhoodU XofXembedded intosome RM. The product, sodened, isinvariantundercontinuousmapsf X Y :f(h1 h2) = f(h1) f(h2) for allh1, h2 H(Y ).It easytoseethat the -pairingequalsthecompositionof theK unnethhomomorphismH(X) H(X) H(X X)withtherestrictiontothediagonalH(X X) H(Xdiag).You can hardly expect to arrive at anything like Serres niteness theoremwithout a linearized (co)homology theory; yet, geometric constructions are of agreat help on the way.Topological and Q-manifolds. The combinatorial proof of the Poincare du-alityis themost transparent for opensubsets X Rnwherethestandarddecomposition S of Rninto cubes is the combinatorial dual of its own translateby a generic vector.Poincare duality remains valid for all oriented topological manifoldsXandalsoforall rational homologyorQ-manifolds, thatarecompacttriangulatedn-spaceswherethelink Lni1Xofeveryi-simplexiinXhasthesamerational homology as the sphereSni1, where it follows from the (special caseof) Alexander duality.Therational homologyof thecomplement toatopologicallyembedded k-sphere as well as of a rational homology sphere, intoSn(or into a Q-manifoldwith the rational homology ofSn) equals that of the (n k 1)-sphere.(The linkLni1(i) is the union of the simplices ni1 Xwhich do notintersect iand for which there exists an simplex inXwhich contains iandni1.)Alternatively, ann-dimensional spaceXcanbeembeddedintosomeRMwhere the duality forXreduces to that for suitably regular neighbourhoodsU RMofXwhich admit Thom isomorphismsHi(X) Hi+Mn(U).If X is a topological manifold, then locally generic cycles of complementarydimension intersect at a discrete set which allows one to dene their geometricintersectionindex. Alsoonecandenetheintersectionof several cycles Cj,j = 1, ...k,with j dim(Ci) =dim(X)astheintersectionindexof jCj XkwithXdiag Xk, but anything more then that can not be done so easily.Possibly, thereisacomprehensiveformulationwithanobviousinvariantproofofthefunctorial Poincaredualitywhichwouldmaketransparent, forexample, themultiplicativityofthesignature(seebelow)andthetopologicalnature of rational Pontryagin classes (see section 10) and which would apply to23cycles of dimensionsNwhereN = and 0 1 in spaces like these weshall meet in section 11.Signature. Theintersectionof(compact)k-cyclesinanoriented, possiblynon-compact and/or disconnected, 2k-manifold X denes a bilinear form on thehomologyHk(X). Ifkis odd, this form is antisymmetric and ifkis even it issymmetric.The signature of the latter, i.e. the number of positive minus the number ofnegative squares in the diagonalized form, is called sig(X). This is well denedifHk(X) has nite rank, e.g. ifXis compact, possibly with a boundary.Geometrically, a diagonalization of the intersection form is achieved with amaximal set of mutuallydisjointk-cycles inXwhere each of them has a non-zero (positive or negative) self-intersection index. (If the cycles are representedbysmoothclosedorientedk-submanifolds, thentheseindicesequal theEulernumbers of the normal bundles of these submanifolds. In fact, such a maximalsystem of submanifolds always exists as it was derived by Thom from the Serreniteness theorem.)Examples. (a)S2k S2khaszerosignature,sincethe2k-homologyisgen-eratedbytheclassesofthetwocoordinatespheres |s1 S2k|and |S2k s2|,which both have zero self-intersections.(b) The complex projective space CP2mhas signature one, since its middlehomology is generated by the class of the complex projective subspace CPmCP2mwith the self-intersection = 1.(c) The tangent bundleT(S2k) has signature 1, sinceHk(T(S2k)) is gener-ated by |S2k| with the self-intersection equal the Euler characteristic (S2k) = 2.Itisobviousthat sig(mX) =m sig(X), wheremXdenotesthedisjointunionof mcopies of X, andthat sig(X) = sig(X), wheresigniesreversion of orientation. FurthermoreThe oriented boundaryXof every compact oriented (4k+1)-manifoldYhaszero signature.(Rokhlin 1952).(Orientedboundariesof non-orientablemanifoldsmayhavenon-zerosig-natures. For examplethedoublecoveringX Xwithsig( X) =2sig(X)non-orientably bounds the corresponding 1-ball bundleYoverX.)Proof. If k-cycles Ci, i = 1, 2, bound relative (k +1)-cycles Di in Y , then the(zero-dimensional) intersection C1 with C2 bounds a relative 1-cycle in Ywhichmakes the index of the intersection zero. Hence,the intersection form vanishes on the kernel kerk Hk(X) of the inclusionhomomorphismHk(X) Hk(Y ).On the other hand, the obvious identity|C D|Y = |C D|XandthePoincaredualityinY showthattheorthogonal complement kerk Hk(X) with respect to the intersection form inXis contained inkerk. QEDObserve that this argument depends entirely on the Poincare duality and itequally applies to the topological and Q-manifolds with boundaries.AlsonoticethattheK unnethformulaandthePoincareduality(trivially)imply the Cartesian multiplicativity of the signature for closed manifolds,sig(X1 X2) = sig(X1) sig(X2).24For example, the products of the complex projective spaces iCP2kihave signa-tures one. (The K unneth formula is obvious here with the cell decompositionsof iCP2kiinto i(2ki + 1) cells.)Amazingly, themultiplicativityof thesignatureof closedmanifoldsundercovering maps can not be seen with comparable clarity.MultiplicativityFormula ifX Xis anl-sheeted covering map, thensign( X) = l sign(X).This can be sometimes proved elementary, e.g. if the fundamental group ofXisfree. Inthiscase, thereobviouslyexistclosedhypersurfacesY XandY X such thatXYis dieomorphic to the disjoint union of l copies of XY .This implies multiplicativity, since signature is additive:removing a closed hypersurface from a manifold does not change the signa-ture.Therefore,sig( X) = sig( X Y ) = l sig(X Y ) = l sig(X).(ThisadditivityofthesignatureeasilyfollowsfromthePoincaredualityasobserved by S. Novikov.)In general, given a nite coveringX X, there exists an immersed hyper-surface Y X (with possible self-intersections) such that the covering trivializesoverX Y ; hence,Xcanbeassembledfromthepiecesof X Y whereeachpieceistakenl times. Onestill hasanadditionformulaforsomestratiedsignature but it is rather involved in the general case.On the other hand, the multiplicativity of the signature can be derived in acouple of lines from the Serre niteness theorem (see below).5 TheSignatureandBordisms.Letusprovethemultiplicativityof thesignaturebyconstructingacompactorientedmanifoldY withaboundary, suchthattheorientedboundary(Y )equalsm X mlXfor some integerm 0.Embed X into Rn+N, N >> n = 2k = dim(X) letX Rn+Nbe an embeddingobtained by a small generic perturbation of the covering mapX X Rn+NandX Rn+Nbe the union ofl generically perturbed copies ofX.LetAandAbe the Atiyah-Thom maps fromSn+N= Rn+Nto the ThomspacesUandUof the normal bundlesU XandUX.LetP X XandP X Xbe the normal projections. These projec-tions, obviously, inducethenormal bundlesUandUofXandXfromthenormal bundleUX. LetP UUandP UUbe the corresponding maps between the Thom spaces and let us look at the twomapsfandffrom the sphereSn+N= RN+nto the Thom spaceU ,f =P A Sn+NU, andf =P A Sn+NU.25Clearly|| f1(X) =Xand (f)1(X) =X.Ontheotherhand, thehomologyhomomorphismsof themaps f andfare related to those ofPandPvia the Thom suspension homomorphismS Hn(X) Hn+N(U ) as followsf|Sn+N| = S P| X| andf|Sn+N| = S P| X|.Sincedeg( P) = deg( P) = l,P| X| =P| X| = l |X| andf|Sn+N| = f|Sn+N| = l S|X| Hn+N(U );hence,somenon-zerom-multiplesofthemapsf, f Sn+N Ucanbejoinedbya(smoothgeneric)homotopyF Sn+N |0, 1| Uby Serres theorem, sincei(U ) = 0,i = 1, ...N 1.Then, becauseof ||, thepullbackF1(X) Sn+N |0, 1|establishesabordism betweenm X Sn+N0 andm X = mlX Sn+N1. This implies thatm sig( X) = ml sig(X) and sincem 0 we getsig( X) = l sig(X). QED.Bordisms and the Rokhlin-Thom-Hirzebruch Formula. Let us mod-ifyourdenitionofhomologyofamanifoldXbyallowingonlynon-singulari-cycles inX, i.e. smooth closed orientedi-submanifolds inXand denote theresulting Abelian group by Boi (X).If 2i n =dim(X)onehasa(minor)problemwithtakingsumsof non-singularcycles, sincegenerici-submanifoldsmayintersectandtheirunionisunavoidably singular. We assume below thati < n]2;otherwise, we replaceXbyX RNforN >> n, where, observe, Boi (X RN) does not depend onNforN >> i.Unlikehomology, thebordismgroups Boi (X)maybenon-trivial evenforacontractiblespace X, e.g. for X =Rn+N. (EverycycleinRnequalstheboundary of any cone over it but this does not work with manifolds due to thesingularityattheapexoftheconewhichisnotallowedbythedenition ofabordism.)In fact,ifN >> n, then the bordism group Bon = Bon(Rn+N) is canonically isomorphicto the homotopy group n+N(V), where V is the Thom space of the tautologicaloriented RN-bundle Vover the Grassmann manifold V = GrorN (Rn+N+1) (Thom,1954).Proof. Let X0 =GrorN (Rn+N)betheGrassmannmanifoldof orientedN-planes andV X0the tautological oriented RNbundle over thisX0.(The space GrorN (Rn+N) equals the double cover of the space GrN(Rn+N) ofnon-orientedN-planes. Forexample, Gror1 (Rn+1)equalsthesphereSn,whileGr1(Rn+1) is the projective space, that isSndivided by the -involution.)Let U

X be the oriented normal bundle of X with the orientation inducedby those ofXand of RN Xand letG X X0be the oriented Gauss mapwhichassignstoeachx XtheorientedN-planeG(x) X0parallel totheoriented normal space toXatx.26SinceGinducesUfromV , itdenestheThommapSn+N= Rn+N VandeverybordismY Sn+N |0, 1|deliversahomotopySn+N |0, 1| Vbetween the Thom maps at the two endsY Sn+N 0 andY Sn+N 1.This dene a homomorphismb Bonn+N(V)since the additive structure in Bon(Ri+N) agrees with that in i+N(Vo ). (Insteadof checking this, which is trivial, one may appeal to the general principle: twonatural Abelian group structures on the same set must coincide.)Also note that one needs the extra 1 in Rn+N+1, since bordismsYbetweenmanifolds in Rn+Nlie in Rn+N+1, or, equivalently, inSn+N+1 |0, 1|.On the other hand, the generic pullback constructionf f1(X0) Rn+N Rn+N= Sn+Ndenesahomomorphismb |f| |f1(X0)|fromn+N(V)to Bon, where,clearlyb bandb bare the identity homomorphisms. QED.Now Serres Q-sphericity theorem implies the followingThomTheorem. The (Abelian) group Boiis nitely generated;BonQ is isomorphic to the rational homology group Hi(X0; Q) = Hi(X0)QforX0 = GrorN (Ri+N+1).Indeed,i(V) = 0 forN >> n, hence, by Serre,n+N(V) Q = Hn+N(V; Q),whileHn+N(V; Q) = Hn(X0; Q)by the Thom isomorphism.In order to apply this, one has to compute the homology Hn(GrorN (RN+n+j)); Q),which, as it is clear from the above, is independent ofN 2n + 2 and ofj > 1;thus, we pass toGror=defj,NGrorN (RN+j).Let us state the answer in the language of cohomology, with the advantage ofthe multiplicative structure (see section 4) where, recall, the cohomology productHi(X) Hj(X) Hi+j(X) for closed orientedn-manifold can be dened viathe Poincare duality H(X) Hn(X) by the intersection product Hni(X)Hnj(X) Hn(i+j)(X).Thecohomologyring H(Gror; Q)isthepolynomial ringinsomedistin-guished integer classes, called Pontryagin classes pk H4k(Gror; Z), k = 1, 2, 3, ...[50], [21].(It would be awkward to express this in the homology language whenN =dim(X) , although the cohomology ringH(X) is canonically isomorphictoHN(X) by Poincare duality.)If X is a smooth oriented n-manifold, its Pontryagin classes pk(X) H4k(X; Z)aredenedas theclasses inducedfrompkbythenormal Gauss mapG GrorN (RN+n) Grorfor an embeddingX Rn+N,N >> n.27Examples (see [50]). (a) The the complex projective spaces havepk(CPn) = _n + 1k _h2kforthegeneratorh H2(CPn)whichisthePoincaredual tothehyperplaneCPn1 CPn1.(b) The rational Pontryagin classes of the Cartesian products X1X2 satisfypk(X1 X2) = i+j=kpi(X1) pj(X2).IfQ is a unitary (i.e. a product of powers) monomial inpi of graded degreen = 4k, then the valueQ(pi)|X| is called the (Pontryagin)Q-number. Equiva-lently, this is the value of Q(pi) H4i(Gror; Z) on the image of (the fundamentalclass) ofXinGrorunder the Gauss map.The Thom theorem now can be reformulated as follows.Two closed oriented n-manifolds are Q-bordant if and only if they have equalQ-numbersforall monomialsQ. Thus, Bon Q = 0, unlessnisdivisibleby4and the rank of BonQ forn = 4kequals the number ofQ-monomials of gradeddegreen, that are ipkiiwith iki = k.(We shall prove this later in this section, also see [50].)For example, if n = 4, then there is a single such monomial, p1; if n=8, theretwoofthem: p2andp21; if n = 12therethreemonomials: p3, p1p2andp31; ifn = 16 there are ve of them.In general, the number of such monomials, say (k) = rank(H4k(Gror; Q)) =rankQ(Bo4k) (obviously) equals the number of the conjugacy classes in the per-mutationgroup(k)(whichcanbeseenasacertainsubgroupintheWeylgroup inSO(4k)), where, by the Euler formula, the generating functionE(t) =1 + k=1,2,...(k)tksatises1]E(t) = k=1,2,...(1 tk) = 022lB2lzl(2l)!,where(2l) = 1 +122l +132l +142l + ... and letB2l = (1)l2l(1 2l) = (1)l+1(2l)!(2l)]22l12lbe the Bernoulli numbers [47],B2 = 1]6, B4 = 1]30, ..., B12 = 691]2730, B14 = 7]6, ..., B30 = 8615841276005]14322, ....WriteR(z1) ... R(zk) = 1 + P1(zj) + ... + Pk(zj) + ...wherePjare homogeneous symmetric polynomials of degreejinz1, ..., zkandrewritePk(zj) = Lk(pi)wherepi = pi(z1, ..., zk) are the elementary symmetric functions inzjof degreei. The Hirzebruch theorem says thatthe above Lk is exactly the polynomial which makes the equality Lk(pi)|X| =sig(X).Asignicant aspect of this formulais that thePontryaginnumbers andthe signature are integers while the Hirzebruch polynomials Lk have non-trivial29denominators. Thisyieldscertainuniversal divisibilitypropertiesofthePon-tryagin numbers (and sometimes of the signatures) for smooth closed orientable4k-manifolds.But despite a heavy integer load carried by the signature formula, its deriva-tion depends only on the rational bordism groups BonQ. This point of elemen-tary linear algebra was overlooked by Thom (isnt it incredible?) who derivedthe signature formula for 8-manifolds from his special and more dicult compu-tation of the true bordism group Bo8. However, the shape given by Hirzebruchto this formula is something more than just linear algebra.Question. Is there an implementation of the analysis/arithmetic encoded inthe Hirzebruch formula by some innite dimensional manifolds?ComputationoftheCohomologyoftheStableGrassmannManifold. First,we show that the cohomology H(Gror; Q) is multiplicatively generated by someclassesei H(Gror; Q) and then we prove that theLi-classes are multiplica-tively independent. (See [50] for computation of the integer cohomology of theGrassmann manifolds.)ThinkoftheunittangentbundleUT(Sn)asthespaceoforthonormal 2-frames in Rn+1, and recall that UT(S2k) is a rational homology (4k1)-sphere.Let Wk = Gror2k+1(R) be the Grassmann manifold of oriented (2k+1)-planesinRN, N , andlet Wkconsistof thepairs (w, u)wherew Wkisan(2k +1)-plane R2k+1w R, and u is an orthonormal frame (pair of orthonormalvectors) in R2k+1w.The mapp Wk Wk1 = Gror2k1(R) which assigns, to every (w, u), the(2k1)-plane uw R2k+1w R normal to u is a bration with contractible bersthat are spaces of orthonormal 2-frames in R uw = R(2k1); hence,p is ahomotopy equivalence.A more interesting bration isq Wk Wkfor (w, u) wwith the bersUT(S2k). SinceUT(S2k) is a rational (4k 1)-sphere, the kernel of the coho-mology homomorphism q H(Wk ; Q) H(Wk; Q) is generated, as a -ideal,by the rational Euler classek H4k(Wk ; Q).It follows by induction onkthat the rational cohomology algebra ofWk =Gror2k+1(R)isgeneratedbycertainei H4i(Wk; Q), i = 0, 1, ..., k, andsinceGror= limkGror2k+1, theseeialso generate the cohomology ofGror.Direct Computation of the L-Classes for the Complex Projective Spaces. LetV X be an oriented vector bundle and, following Rokhlin-Schwartz and Thom,deneL-classesof V , withoutanyreferencetoPontryaginclasses, asfollows.Assume thatXis a manifold with a trivial tangent bundle;otherwise,embedXinto some RMwith largeMand take its small regular neighbourhood. BySerrestheorem,thereexists,foreveryhomologyclassh H4k(X) =H4k(V ),anm = m(h) 0 such that them-multiple ofh is representable by a closed 4k-submanifold Z = Zh Vthat equals the pullback of a generic point in the sphereSM4kunder a generic mapV SM4k= RM4kwith compact support, i.e.where all but a compact subset inVgoes to SM4k. Observe that such aZhas trivial normal bundle inV .DeneL(V ) = 1 + L1(V ) + L2(V ) + ... H4(V ; Q) = kH4k(V ; Q)bytheequalityL(V )(h) = sig(Zh)]m(h) for allh H4k(V ) = H4k(X).If thebundleV isinducedfromW Y byanf X Y thenL(V ) =f(L(W)), since, fordim(W) > 2k (which we may assume), the generic imageof ourZinWhas trivial normal bundle.30It is also clear that the bundleV1 V2 X1 X2hasL(V1 V2) = L(V1) L(V2) by the Cartesian multiplicativity of the signature.Consequently the L-class of the Whitney sum V1V2X of V1 and V2 overX, whichisdenedastherestrictionof V1 V2 X XtoXdiag X X,satisesL(V1 V2) = L(V1) L(V2).Recall that the complex projective space CPk the space of C-lines in Ck+1comes with the canonical C-line bundle represented by these lines and denotedU CPk, while the same bundle with the reversed orientation is denotedU.(We always refer to the canonical orientations of C-objects.)Observe thatU = HomC(U ) for the trivial C-bundle = CPk C CPk= HomC(U U)and that the Euler classe(U) = e(U) equals the generator inH2(CPk) thatisthePoincaredualofthehyperplane CPk1 CPk,andsoelisthedualofCPkl CPk.The Whitney (k + 1)-multiple bundle ofU, denoted (k + 1)U, equals thetangent bundleTk = T(CPk) plus. Indeed, letU CPkbe the Ckbundleof thenormalstothelinesrepresentingthepointsinCPk. ItisclearthatUU = (k+1), i.e. UUis the trivial Ck+1-bundle, and that, tautologically,Tk = HomC(U U).It follows thatTk = HomC(U UU) = HomC(U (k + 1)) = (k + 1)U.Recall thatsig(CP2k) = 1; hence,Lk((k + 1)U) = Lk(Tk) = e2k.Now we compute L(U) = 1+kLk = 1+kl2ke2k, by equating e2kand the2k-degree term in the (k + 1)th power of this sum.(1 + kl2ke2k)k+1= 1 + ... + e2k+ ...Thus,(1 + l1e2)3= 1 + 3l1 + ... = 1 + e2+ ...,which makesl1 = 1]3 andL1(U) = e2]3.Then(1 + l1e2+ l2e4)5= 1 + ... + (10l1 + 5l2)e4+ ... = 1 + ... + e4+ ...which implies thatl2 = 1]5 2l1 = 1]5 2]3 andL2(U) = (7]15)e4, etc.Finally, we compute allL-classesLj(T2k) = (L(U))k+1forT2k = T(CP2k)and thus, allL(jCP2kj).For example,(L1(CP8))2|CP8| = 10]3 while (L1(CP4 CP4))2|CP4 CP4| = 2]331which implies that CP4 CP4and CP8, which have equal signatures, are notrationally bordant, and similarly one sees that the products jCP2kjare mul-tiplicatively independent in the bordism ring Bo Q as we stated earlier.Combinatorial PontryaginClasses. Rokhlin-Schwartz andindepen-dently Thom applied their denition of Lk, and hence of the rational Pontryaginclasses, to triangulated (not necessarily smooth) topological manifoldsXby ob-servingthatthepullbacksofgenericpointss Sn4kunderpiece-wiselinearmapare Q-manifoldsandbypointingoutthatthesignaturesof4k-manifoldsareinvariantunderbordismsbysuch (4k + 1)-dimensional Q-manifoldswithboundaries(bythePoincaredualityissuingfromtheAlexanderduality, seesection 4). Thus, they have shown, in particular, thatrational Pontryaginclassesofsmoothmanifoldsareinvariantunderpiece-wise smooth homeomorphisms between smooth manifolds.The combinatorial pull-back argument breaks down in the topological cate-gory since there is no good notion of a generic continuous map. Yet, S. Novikov(1966) proved that the L-classes and, hence, the rational Pontryagin classes areinvariant under arbitrary homeomorphisms (see section 10).The Thom-Rokhlin-Schwartz argument delivers a denition of rational Pon-tryagin classes for all Q-manifolds which are by far more general objects thansmooth (or combinatorial) manifolds due to possibly enormous (and beautiful)fundamental groups1(Lni1) of their links.Yet, the naturally dened bordism ring QBon of oriented Q-manifolds is onlymarginally dierent from Boin the degreesn 4 where the natural homomor-phisms Bon Q QBon Qareisomorphisms. Thiscanbeeasilyderivedbysurgery(seesection9)fromSerrestheorems. Forexample, if aQ-manifoldXhasasinglesingularityaconeover Q-spherethenaconnectedsumofseveral copies of bounds a smooth Q-ball which implies that a multiple ofXis Q-bordant to a smooth manifold.Onthecontrary, thegroup QBo4 Q, ismuchbiggerthan Bo4 Q = QasrankQ(QBo4) = (see [44], [22], [23] and references therein).(It wouldbeinterestingtohaveanotionof renedbordismsbetweenQ-manifold that would partially keep track of1(Lni1) forn > 4 as well.)The simplest examples of Q-manifolds are one point compacticationsV4kof the tangent bundles of even dimensional spheres,V4k= T(S2k) S2k, sincethe boundaries of the corresponding 2k-ball bundles are Q-homological (2k1)-spheres the unit tangent bundlesUT(S2k) S2k.Observethatthetangentbundlesof spheresarestablytrivial theybe-cometrivial afteraddingtrivial bundlestothem, namelythetangentbundleofS2k R2k+1stabilizes to the trivial bundle upon adding the (trivial) normalbundle ofS2k R2k+1to it. Consequently, the manifoldsV4k= T(S2k) have allcharacteristic classes zero, and V4khave all Q-classes zero except for dimension4k.On the other hand,Lk(V4k ) = sig(V4k ) = 1, since the tangent bundle V4k=T(S2k) S2khas non-zero Euler number. Hence,the Q-manifoldsV4kmultiplicatively generate all of QBo Q except for QBo4.Local Formulae for Combinatorial Pontryagin Numbers. LetXbe a closedoriented triangulated (smooth or combinatorial) 4k-manifold and let S4k1xxX032be the disjoint union of the oriented linksS4k1xof the verticesx inX. Thenthereexists,foreachmonomial Qofthetotal degree4kinthePontryaginclasses, an assignment of rational numbersQ|Sx| to allS4k1x, whereQ|S4k1x|depend only on the combinatorial types of the triangulations ofSx induced fromX, such that the PontryaginQ-number ofXsatises (Levitt-Rourke 1978),Q(pi)|X| = S4k1x[[X]]Q|S4k1x|.Moreover, there is a canonical assignment of real numbers toS4k1xwith thisproperty which also applies to all Q-manifolds (Cheeger 1983).There is no comparable eective combinatorial formulae with a priori ratio-nal numbers Q|Sx| despite many eorts in this direction, see [24] and referencestherein. (Levitt-RourketheoremispurelyexistentialandCheegersdenitiondepends on theL2-analysis of dierential forms on the non-singular locus ofXaway from the codimension 2 skeleton ofX.)Questions. Let |S4k1|beanitecollectionof combinatorial isomor-phismclassesof orientedtriangulated (4k 1)-spheresletQ{[S4k1]}betheQ-vector space of functionsq |S4k1| Q and letXbe a closed orientedtriangulated4k-manifoldhomeomorphictothe4k-torus(oranyparallelizablemanifold for this matter) with all its links in |S4k1|.Denote by q(X) Q{[S4k1]} the function, such that q(X)(|S4k1|) equalsthe number of copies of |S4k1| in S4k1xxX0and let L|S| Q{[S4k1]}be the linear span ofq(X) for all suchX.The above shows that the vectors q(X) of q-numbers satisfy, besides 2k+1Euler-Poincare and Dehn-Somerville equations, aboutexp(

2k3)4k3linear Pon-tryagin relations.Observe that the Euler-Poincare and Dehn-Somerville equations do not de-pend on the -orientations of the links but the Pontryagin relations are anti-symmetric since Q|S4k1| = Q|S4k1|. Both kind of relations are valid forall Q-manifolds.What are the codimensionscodim(L|S4k1| Q{[S4k1]}, i.e. the num-bers of independent relations between the q-numbers, for specic collections|S4k1|?It is pointed out in [23] that the spaces QSiof antisymmetric Q-linear combinations of all combinatorialspheresmakeachaincomplexforthedierential qi QSi QSi1denedbythe linear extension of the operation of taking the oriented links of all verticeson the triangulatedi-spheresSi Si.The operationqiwithvalues inQSi1, whichis obviouslydenedonallclosedorientedcombinatorial i-manifoldsXaswellasoncombinatorial i-spheres, satisesqi1 (qi(X)) = 0, i.e. i-manifolds representi-cycles in this complex.Furthermore, it is shown in [23] (as was pointed out to me by Je Cheeger)that all such anti-symmetric relations are generated/exhausted by the relationsissuing fromqn1 qn = 0, where this identity can be regarded as an oriented(Pontryagininplaceof Euler-Poincare)counterparttotheDehn-Somervilleequations.33Theexhaustiveness of qn1 qn =0andits (easy, [25]) Dehn-Somervillecounterpart, probably, implythat inmost (all?) cases the Euler-Poincare,Dehn-Somerville, Pontryaginandqn1 qn = 0makethefullsetofane(i.e.homogeneous and non-homogeneous linear) relations between the vectors q(X),butitseemshardtoeectively(evenapproximately)evaluatethenumberofindependentrelationsissuingfromfor qn1 qn =0forparticularcollections|Sn1| of allowable links ofXn.Examples. LetD0 = D0() be a Dirichlet-Voronoi (fundamental polyhedral)domainofagenericlattice RMandlet |Sn1|consistofthe(isomor-phismclassesofnaturallytriangulated)boundariesoftheintersectionsof D0with generic anen-planes in RM.What iscodim(L|Sn1| Q|Sn1|) in this case?What are the (ane) relations between the geometricq-numbers i.e. thenumbers of combinatorial types of intersections of -scaled submanifolds X fRM, , (as in the triangulation construction in the previous section) withthe -translates ofD0?Notice, thatsomeof thesearenot convex-like, butthesearenegligiblefor . Ontheotherhand, if issucientlylargeall canbemadeconvex-like by a small perturbationfoffby an argument which is similar tobut slightly more technical than the one used for the triangulation of manifoldsin the previous section.Is there anything special about the geometric q-numbers for distinguishedX, e.g. for roundn-spheres in RN?Observethattheratiosof thegeometric q-numbersareasymptoticallydened for many non-compact complete submanifoldsX RM.For example, if Xis ananesubspace A =AnRM, theseratios are(obviously) expressible in terms of the volumes of the connected regions in D RMobtained by cuttingD along hypersurfaces made of the anen-subspacesA RMwhich are parallel toA and which meet the (M n1)-skeleton ofD.What is the number of our kind of relations between these volumes?There are similar relations/questions for intersection patterns of particularXwithotherfundamental domainsoflatticesinEuclideanandsomenon-Euclideanspaces(wherethenerasymptoticdistributionsof thesepatternshave a slight arithmetic avour).If f Xn RMisagenericmapwithsingularities(whichmayhappenifM 2n) andD RMis a small convex polyhedron in RMwith its faces being-transversal tof (e.g. D =1D0, asinthetriangulationsof theprevious section), then the pullback f1(D) X is not necessarily a topologicalcell. However, some local/additive formulae for certain characteristic numbersmay still be available in the corresponding non-cell decompositions ofX.For instance, one (obviously) has such a formula for the Euler characteristicforallkindofdecompositionsof X. Also, onehassuchalocalformulaforsig(X) and f X R (i.e for M = 1) by Novikovs signature additivity propertymentioned at the end of the previous section.It seems not hardtoshowthat all Pontryaginnumbers canbethus lo-cally/additivelyexpressedforM n, butitisunclearwhatarepreciselytheQ-numberswhicharecombinatorially/locally/additivelyexpressibleforgivenn = 4k andM < n.(For example, ifM = 1, then the Euler characteristic and the signature are,probably, the only locally/additively expressible invariants ofX.)34Bordisms of Immersions. If the allowed singularities of orientedn-cycles inRn+kare those of collections ofn-planes in general position, then the resultinghomologies are the bordism groups of oriented immersed manifoldsXn Rn+k(R.Wells, 1966). For example if k = 1, this group is isomorphic to the stable ho-motopy group stn = n+N(SN), N > n+1, by the Pontryagin pullback construc-tion, since a small generic perturbation of an oriented Xnin Rn+N Rn+1 Xnis embedded into Rn+Nwith a trivial normal bundle, and where every embeddingXn Rn+Nwiththetrivial normal bundlecanbeisotopedtosuchapertur-bationof animmersionXn Rn+1 Rn+NbytheSmale-Hirschimmersiontheorem. (This is obvious forn = 0 andn = 1).Since immersed oriented Xn Rn+1have trivial stable normal bundles, theyhave, forn = 4k, zerosignaturesbytheSerrenitenesstheorem. Conversely,the niteness of the stable groupsstn = n+N(SN) can be (essentially) reducedby a (framed) surgery of Xn(see section 9) to the vanishing of these signatures.Thecomplexityof n+N(SN) shiftsinthispictureonedimensiondowntobordism invariants of the decorated self-intersections of immersed Xn Rn+1,whicharepartiallyreectedinthestructureofthel-sheetedcoveringsoftheloci ofl-multiple points ofXn.TheGalois groupof suchacoveringmaybeequal thefull permutationgroup (l) and the decorated invariants live in certain decorated bordismgroups of the classifying spaces of (l), where the dimension shift suggests aninductive computation of these groups that would imply, in particular, Serresniteness theorem of the stable homotopy groups of spheres. In fact,this canbe implemented in terms of conguration spaces associated to the iterated loopspaces as was pointed out to me by Andras Sz` ucs, also see [1], [75], [76].The simplest bordism invariant of codimension one immersions is the parityofthenumberof (n + 1)-multiplepointsofgenericallyimmersedXn Rn+1.For example, the gure R2with a single double point represents a non-zeroelement in stn=1 = 1+N(SN). The number of (n+1)-multiple points also can beodd for n = 3 (and, trivially, for n = 0) but it is always even for codimension oneimmersionsoforientablen-manifoldswithn 0, 1, 3,whilethenon-orientablecase is more involved [16], [17].Oneknows, (seenextsection)thateveryelementof thestablehomotopygroup stn = n+N(SN), N >> n, can be represented for n 2, 6, 14, 30, 62, 126 byan immersion XnRn+1, where Xnis a homotopy sphere; if n = 2, 6, 14, 30, 62, 126,one can make this with anXnwhererank(H(Xn)) = 4.Whatisthesmallestpossiblesizeof thetopology, e.g. homology, of theimagef(Xn) Rn+1and/or of the homologies of the (natural coverings of the)subsets of thel-multiple points off(Xn)?GeometricQuestionsaboutBordisms. LetXbe a closed oriented Rieman-nian n-manifold with locally bounded geometry, which means that every R-ballinXadmits a-bi-Lipschitz homeomorphism onto the EuclideanR-ball.Suppose Xis bordant to zero and consider all compact Riemannian (n+1)-manifolds Yextending X = (Y ) with its Riemannian tensor and such that thelocal geometries ofYare bounded by some constantsR > withthe obvious precaution near the boundary.One can show that the inmum of the volumes of theseYis bounded byinfYV ol(Y ) F(V ol(X)),35with the power exponent bound on the function F = F(V ). (Falso depends onR, , R, , but this seems non-essential forR > .)What is the true asymptotic behaviour ofF(V ) forV ?Itmaybelinearforall weknowandtheabovedimensionshiftpictureand/or the construction from [23] may be useful here.Is there a better setting of this question with some curvature integrals and/orspectral invariants rather than volumes?The real cohomology of the Grassmann manifolds can be analytically repre-sented by invariant dierential forms. Is there a compatible analytic/geometricrepresentation of BonR?(One may think of a class of measurablen-foliations,see section 10, or, maybe, something more sophisticated than that.)6 ExoticSpheres.In 1956, to everybodys amazement, Milnor found smooth manifolds 7whichwerenot dieomorphictoSN; yet, eachof themwasdecomposableintotheunionof two7-balls B71, B72 7intersectingovertheirboundaries (B71) =(B72) = S6 7like in the ordinary sphereS7.In fact, this decomposition does imply that 7is ordinary in the topologicalcategory: such a 7is (obviously) homeomorphic toS7.The subtlety resides in the equality(B71) = (B72); this identication oftheboundariesisfarfrombeingtheidentitymapfromthepointof viewofeither of the two balls it does not come from any dieomorphismsB71 B72.The equality(B71) = (B72) can be regarded as a self-dieomorphismfoftheroundsphereS6theboundaryofstandardball B7,butthisfdoesnotextend to a dieomorphism ofB7in Milnors example; otherwise, 7would bedieomorphic toS7. (Yet,fradially extends to a piecewise smooth homeomor-phism ofB7which yields a piecewise smooth homeomorphism between 7andS7.)It follows, that such an f can not be included into a family of dieomorphismsbringing it to an isometric transformations ofS6. Thus, any geometric energyminimizingowonthedieomorphismgroupdiff(S6)eithergetsstuckordevelops singularities. (It seems little,if anything at all,is known about suchows and their singularities.)Milnorsspheres7areratherinnocuousspacestheboundariesof (thetotalspacesof)4-ballbundles8 S4insomeinsome R4-bundlesV S4,i.e. 8 Vand, thus, our 7are certainS3-bundles overS4.All 4-ball bundles, or equivalentlyR4-bundles, over S4are easyto de-scribe: eachisdeterminedbytwonumbers: theEulernumbere, thatistheself-intersectionindexof S48, whichassumesall integervalues, andthePontryaginnumberp1(i.e. thevalueofthePontryaginclassp1 H4(S4)on|S4| H4(S4)) which may be an arbitrary even integer.(Milnorexplicitlyconstructhisbrationswithmapsof the3-sphereintothegroupSO(4)of orientationpreservinglinearisometriesofR4asfollows.Decompose S4intotworound4-balls, sayS4=B4+ B4withthecommonboundaryS3 = B4+ B4and letf s O SO(4) be a smooth map. Thengluetheboundariesof B4+ R4andB4 R4bythedieomorphism (s, s) (s, O(s))andobtainV8=B4+ R4fB4 R4whichmakesan R4-brationoverS4.36To construct a specic f, identify R4with the quaternion line H and S3withthemultiplicativegroupofquaternionsofnorm1. Letf(s) =fij(s) SO(4)actbyx sixsjfor x Handtheleftandrightquaternionmultiplication.Then Milnor computes: e = i + jandp1 = 2(i j).)Obviously, all 7are 2-connected, but H3(7) may be non-zero (e.g. for thetrivial bundle). It is not hard to show that 7has the same homology asS7,hence,homotopy equivalent toS7,if and only ife = 1 which means that theselntersection index of the zero section sphereS4 8equals 1; we stick toe = 1 for our candidates for 7.The basic example of 7withe = 1 (the sign depends on the choice of theorientation in 8) is the ordinary 7-sphere which comes with the Hopf brationS7S4, wherethis S7is positionedas theunit sphereinthequaternionplane H2= R8,whereitisfreelyacteduponbythegroupG =S3oftheunitquaternions and whereS7]G equals the sphereS4representing the quaternionprojective line.If 7is dieomorphic toS7one can attach the 8-ball to 8along thisS7-boundary and obtain a smooth closed 8-manifold, say 8+.Milnor observes that the signature of 8+equals 1,since the homology of8+ is represented by a single cycle the sphere S4 8 8+ the selntersectionnumber of which equals the Euler number.Then Milnor invokes the Thom signature theorem45sig(X) + p21|X| = 7p2|X|andconcludesthatthenumber45 + p21mustbedivisibleby7; therefore, theboundaries7of those8whichfail thiscondition, sayforp1 =4, mustbeexotic. (You do not have to know the denition of the Pontryagin classes, justremember they are integer cohomology classes.)Finally, usingquaternions, Milnor explicitlyconstructs aMorsefunction7 Rwithonlytwocritical pointsmaximumandminimumoneach7withe =1; thisyieldsthetwoball decomposition. (Weshall explainthisinsection 8.)(Milnors topological arguments, which he presents with a meticulous care,became a common knowledge and can be now found in any textbook; his lemmaslook apparent to a to-day topology student. The hardest for the modern readeris the nal Milnors lemma claiming that his function 7R is Morse with twocritical points. Milnor is laconic at this point: It is easy to verify is all whathe says.)The 8-manifolds 8+associated with Milnors exotic 7can be triangulatedwithasinglenon-smoothpointinsuchatriangulation. Yet, theyadmitnosmoothstructurescompatiblewiththesetriangulationssincetheir combina-torial Pontryaginnumbers(denedbyRochlin-SchwartzandThom)fail thedivisibility condition issuing from the Thom formula sig(X8) = L2|X8|; in fact,they are not combinatorially bordant to smooth manifolds.Moreover,these 8+are not even topologically bordant,and therefore,theyare non-homeomorphic to smooth manifolds by (slightly rened) Novikovs topo-logical Pontryagin classes theorem.The number of homotopyspheres, i.e. of mutually non-dieomorphic man-ifolds nwhich are homotopy equivalent toSnis not that large. In fact, it isnite forall n 4bytheworkofKervaireandMilnor[39], who, eventually,37derive this from the Serre niteness theorem. (One knows now-a-days that everysmooth homotopy sphere nis homeomorphic toSnaccording to the solutionofthePoincareconjecturebySmaleforn 5, byFreedmanforn = 4andbyPerelmanfor n =3, wherehomeomorphicdieomorphicfor n =3byMoises theorem.)KervaireandMilnorstartbyshowingthatforeveryhomotopyspheren,thereexistsasmoothmapf Sn+NSN, N >>n, suchthatthepullbackf1(s) Sn+Nof a generic points SNis dieomorphic to n. (The existenceofsuchanfwithf1(s) = nisequivalenttotheexistenceofanimmersionnRn+1by the Hirsch theorem.)Then, by applying surgery (see section 9) to thef0-pullback of a point for agiven generic map f0 Sn+NSN, they prove that almost all homotopy classesof mapsSn+NSNcome from homotopyn-spheres. Namely: Ifn 4k +2, then every homotopy class of mapsSn+NSN,N >> n, canbe represented by a n-mapf, i.e. where the pullback of a generic point is ahomotopy sphere.If n = 4k +2, then the homotopy classes of n-maps constitute a subgroupin the corresponding stable homotopy group, say K

n stn = n+N(SN), N >> n,thathasindex1or2andwhichisexpressibleintermsoftheKervaire-(Arf )invariant classifying(similarlytothesignaturefor n =4k)properlydenedself-intersections of (k + 1)-cycles mod 2 in (4k + 2)-manifolds.One knows today by the work of Pontryagin, Kervaire-Milnor and Barratt-Jones-Mahowald see [9] thatIf n =2, 6, 14, 30, 62, thentheKervaireinvariant canbenon-zero, i.e.stn ]K

n = Z2.Furthermore, The Kervaire invariant vanishes, i.e. K

n = stn , forn 2, 6, 14, 30, 62, 126(where it remains unknown ifst126]K

126equals 0 or Z2).In other words,everycontinuousmapSn+NSN, N >>n 2, 6, ..., 126, ishomotopictoasmoothmapf Sn+N SN,suchthatthef-pullbackofagenericpointisahomotopyn-sphere.The case n 2l2 goes back to Browder (1969) and the case n = 2l2, l 8is a recent achievement by Hill,Hopkins and Ravenel [37]. (Their proof relieson a generalized homology theoryHgennwhereHgenn+256 = Hgenn.)If thepullbackof agenericpoint of asmoothmapf Sn+NSN, isdieomorphictoSn, themapf maybenon-contractible. Infact, thesetofthe homotopy classes of such fmakes a cyclic subgroup in the stable homotopygroup of spheres, denotedJn stn = n+N(SN),N >> n (and called the imageof the J-homomorphismn(SO()) stn ). Theorderof Jnis1or2forn 4k1; if n = 4k1, then the order of Jn equals the denominator of |B2k]4k|,whereB2kis the Bernoulli number. The rst non-trivialJareJ1 = Z2,J3 = Z24,J7 = Z240,J8 = Z2,J9 = Z2andJ11 = Z504.Ingeneral, thehomotopyclassesofmapsfsuchthatthef-pullbackofageneric point is dieomorphic to a given homotopy sphere n, make aJn-cosetin the stable homotopy groupstn . Thus the correspondence n fdenes amapfromtheset nofthedieomorphismclassesofhomotopyspheresto38the factor groupstn ]Jn, say n stn ]Jn.The map (which, by the above, is surjective forn 2, 6, 14, 30, 62, 126) isnite-to-oneforn 4,wheretheproofofthisnitenessforn 5dependsonSmalesh-cobordism theorem, (see section 8). In fact, the homotopyn-spheresmake an Abelian group (n 4) under the connected sum operation 1#2(seenext section) and, by applying surgery to manifolds n+1with boundaries n,where these n+1(unlike the above Milnors 8) come as pullbacks of genericpoints under smooth maps from (n + N + 1)-ballsBn+N+1toSN, Kervaire andMilnor show that() n stn ]Jnisahomomorphismwithanite(n 4)kerneldenoted Bn+1 n which is a cyclic group.(The homotopy spheres n Bn+1bound (n+1)-manifolds with trivial tan-gent bundles.)Moreover,() The kernel Bn+1of is zero forn = 2m 4.Ifn + 1 = 4k + 2, then Bn+1is either zero or Z2, depending on the Kervaireinvariant:()Ifnequals1, 5, 13, 29, 61and, possibly, 125, then Bn+1iszero, andBn+1= Z2for the rest ofn = 4k + 1.() ifn = 4k 1, then the cardinality (order) of Bn+1equals 22k2(22k11)times the numerator of |4B2k]k|, whereB2kis the Bernoulli number.The above and the known results on the stable homotopy groupsstnimply,for example, that there are no exotic spheres forn = 5, 6, there are 28 mutuallynon-dieomorphic homotopy 7-spheres,there are 16 homotopy 18-spheres and523264 mutually non-dieomorphic homotopy 19-spheres.By Perelman, there is a single smooth structure on the homotopy 3 sphereand the casen = 4 remains open. (Yet, every homotopy 4-sphere is homeomor-phic toS4by Freedmans solution of the 4D-Poincare conjecture.)7 IsotopiesandIntersections.Besidesconstructing, listingandclassifyingmanifoldsXonewantstounder-stand the topology of spaces of mapsX Y .The space |XY |smth of allC maps carries little geometric load by itselfsince this space is homotopy equivalent to |XY |cont(inuous).Ananalystmaybeconcernedwithcompletionsof |XY |smth, e.g. withSobolev topologies while a geometer is keen to study geometric structures, e.g.Riemannian metrics on this space.But from a dierential topologists point of view the most interesting is thespace of smooth embeddingsF X Ywhich dieomorphically sendXonto asmooth submanifoldX = f(X) Y .Ifdim(Y ) > 2dim(X) then genericfare embeddings,but,in general,youcannotproducethematwill soeasily. However, givensuchanembeddingf0 X Y , thereareplentyofsmoothhomotopies, called(smooth)isotopiesft, t |0, 1|, of it which remain embeddings for every t and which can be obtainedwith the following39IsotopyTheorem. (Thom, 1954.) Let Z Xbeacompact smoothsubmanifold (boundary is allowed) andf0 X Yis an embedding, where theessential case is whereX Yandf0is the identity map.Then every isotopy of Zf0Ycan be extended to an isotopy of all of X. Moregenerally, therestrictionmapRZ |XY |emb |ZY |embis abration;inparticular, theisotopyextensionpropertyholdsforanarbitraryfamilyofembeddingsX Yparametrized by a compact space.Thisissimilartothehomotopyextensionproperty(mentionedinsection1) for spaces of continuous mapsX Y the geometric cornerstone of thealgebraic topology.)Theproof easilyreduceswiththeimplicit functiontheoremtothecase,whereX = Yanddim(Z) = dim(W).Sincedieomorphismsareopeninthespaceof all smoothmaps, onecanextend small isotopies, those which only slightly move Z, and since dieomor-phisms ofYmake a group, the required isotopy is obtained as a composition ofsmall dieomorphisms ofY . (The details are easy.)Bothopenandgrouparecrucial: forexample, homotopiesbylocallydieomorphic maps, say of a disk B2 S2to S2do not extend to S2whenever amap B2S2starts overlapping itself. Also it is much harder (yet possible, [12],[40])toextendtopologicalisotopies,sincehomeomorphismsare,bynomeans,open in the space of all continuous maps.For example ifdim(Y ) 2dim(Z) + 2. then a generic smooth homotopy ofZisanisotopy: Zdoesnot, generically, crossitselfasitmovesinY (unlike,forexample,acirclemovinginthe3-spacewhereself-crossingsarestableun-dersmallperturbationsofhomotopies). Hence, everygenerichomotopyof Zextends to a smooth isotopy ofY .Mazur Swindle and Hauptvermutung.Let U1, U2 be compact n-manifoldswith boundaries and f12 U1U2 and f21 U2U1 be embeddings which landin the interiors of their respective target manifolds.LetW1andW2betheunions(inductivelimits)of theinniteincreasingsequences of spacesW1 = U1 f12U2 f21U1 f12U2 f12...andW2 = U2 f21U1 f12U2 f12U1 f12...Observe thatW1andW2are open manifolds without boundaries and thattheyaredieomorphicsincedroppingtherstterminasequenceU1 U2 U3 ... does not change the union.Similarly, both manifolds are dieomorphic to the unions of the sequencesW11 = U1 f11U1 f11... andW22 = U2 f22U2 f22...forf11 = f12 f21 U1U1andf22 = f21 f12 U2U2.If the self-embeddingf11 is isotopic to the identity map, thenW11 is dieo-morphictotheinteriorof U1bytheisotopytheoremandthesameappliestof22(or any self-embedding for this matter).40Thus we conclude with the above, that, for example,opennormal neighbourhoods Uop1andUop2of twohomotopyequivalent n-manifolds(andtriangulatedspacesingeneral)Z1andZ2in Rn+N, N n + 2,are dieomorphic (Mazur 1961).Anybodymighthaveguessedthattheopenconditionisapuretechni-calityandeverybodybelievedsountil Milnors 1961counterexampletotheHauptvermutung the main conjecture of the combinatorial topology.Milnor has shown that there are two free isometric actions A1 and A2 of thecyclic group Zpon the sphereS3, for every primep 7, such thatthequotient(lens)spacesZ1 = S3]A1andZ2 = S3]A2arehomotopyequiv-alent,buttheirclosednormal neighbourhoodsU1andU2inany R3+Narenotdieomorphic. (ThiscouldnothavehappenedtosimplyconnectedmanifoldsZiby theh-cobordism theorem.)Moreover,thepolyhedraP1andP2obtainedbyattachingtheconestotheboundariesof these manifolds admit no isomorphic simplicial subdivisions.Yet, the interiorsUopiof theseUi, i = 1, 2, are dieomorphic forN 5. Inthis case,P1 and P2 are homeomorphic as the one point compactications of two home-omorphic spacesUop1andUop2.It was previously known that these Z1 and Z2 are homotopy equivalent (J. H.C. Whitehead, 1941); yet, they are combinatorially non-equivalent (Reidemeis-ter, 1936) and, hence, by Moises 1951 positive solution of the Hauptvermutungfor 3-manifolds, non-homeomorphic.There are few direct truly geometric constructions of dieomorphisms, butthoseavailable, areextensivelyused, e.g. berwiselineardieomorphismsofvector bundles. Even the sheer existence of the humble homothety of Rn, x tx,combined with the isotopy theorem, eortlessly yields, for example, the following|BY |-Lemma. Thespaceofembeddingsfofthen-ball(or Rn)intoanarbitraryY = Yn+kis homotopy equivalentto the space of tangentn-frames inY ; in fact the dierentialf Df|0 establishes a homotopy equivalence betweenthe respective spaces.For example,theassignment f J(f)|0of theJacobi matrixat 0 Bnisahomotopyequivalence of the space of embeddingsf B Rnto the linear groupGL(n).Corollary: Ball GluingLemma. Let X1andX2be (n + 1)-dimensionalmanifolds with boundariesY1 andY2, letB1 Y1 be a smooth submanifold dif-feomorphic to then-ball and letf B1B2 Y2 = (A2) be a dieomorphism.If theboundaries Yiof Xiareconnected, thedieomorphismclassof the(n + 1)-manifold X3 =X1 +fX2obtainedbyattaching X1to X2by f and(obviously canonically) smoothed at the corner (or rather the crease) alongthe boundary ofB1, does not depend onB1andf.This X3 is denoted X1#X2. For example, this sum of balls, Bn+1#Bn+1,is again a smooth (n + 1)-ball.ConnectedSum. TheboundaryY3 =(X3)canbedenedwithoutanyreference to Xi Yi, as follows. Glue the manifolds Y1 an Y2 by f B1B2 Y2and then remove the interiors of the ballsB1and of itsf-imageB2.41If themanifolds Yi(not necessarilyanybodys boundaries or evenbeingclosed)areconnected, thentheresultingconnectedsummanifoldisdenotedY1#Y2.Isntitawasteof glue? Youmaybewonderingwhybotherglueingtheinteriors of the balls if you are going to remove them anyway. Wouldnt it beeasierrsttoremovetheseinteriorsfrombothmanifoldsandthengluewhatremains along the spheresSn1i= (Bi)?This is easier but also it is also a wrong thing to do: the result may dependon the dieomorphismSn11Sn12, as it happens forY1 = Y2 = S7in Milnorsexample; buttheconnectedsumdenedwithballsisuniquebythe |BY |-lemma.The ball gluing operation may be used many times in succession;thus, forexample, one builds big (n + 1)-balls from smaller ones, where this lemma inlower dimension may be used for ensuring the ball property of the gluing sites.Gluing and Bordisms. Take two closed orientedn-manifoldX1andX2andletX1 U1fU2 X2beanorientationreversingdieomorphismsbetweencompact n-dimensionalsubmanifolds Ui Xi, i =1, 2withboundaries. If weglue X1andX2byfandremovethe(gluedtogether)interiorsof Uitheresultingmanifold, sayX3 =X1 +UX2isnaturallyorientedand, clearly, itisorientablybordanttothedisjointunionof X1andX2. (Thisissimilartothegeometric/algebraiccancellation of cycles mentioned in section 4.)Conversely, onecangiveanalternativedenitionof theorientedbordismgroup Bonas of the Abelian group generated by orientedn-manifolds with therelationsX3 = X1 + X2for allX3 = X1 +UX2. This gives the same Boneven ifthe onlyUallowed are those dieomorphic toSi Bnias it follows from thehandle decompositions induced by Morse functions.The isotopy theorem is not dimension specic, but the following constructiondue to Haeiger (1961) generalizing the Whitney Lemma of 1944 demonstratessomething special about isotopies in high dimensions.LetYbe a smoothn-manifold andX, X Ybe smooth closed submani-folds in general position. Denote 0 = XX Yand letXbe the (abstract)disjointunion ofXandX. (IfXandXare connected equividimensionalmanifolds, one could say thatXis a smooth manifold with its two connectedcomponentsXandXbeing embedded intoY .)Clearly,dim(0) = nkkforn = dim(Y ), nk = dim(X) andnk = dim(X).Letft X Y , t |0, 1|, be a smooth generic homotopy which disengagesXfromX, i.e. f1(X) does not intersectf1(X), and let = (x, x, t)ft(x)=ft(x) X X |0, 1|,i.e. consists of the triples (x, x, t) for whichft(x) = ft(x).Let X Xbe the unionS S, whereS Xequals the projectionof to theX-factor ofX X |0, 1| andS Xis the projection of toX.42Thus, there is a correspondencexxbetween the points in = S S,where the two points correspond one to another if x S meets x S at somemomenttin the course of the homotopy, i.e.ft(x) = ft(x) for somet |0, 1|.Finally, let W Y betheunionof theft-paths, denoted |x t x| Y ,travelledbythepointsx S andx S until theymeetatsomemomentt. Inotherwords, |x t x| Y consistsoftheunionofthepointsft(x) andft(x) overt |0, t = t(x) = t(x)| andW = xS|x t x| = xS|x t x|.Clearly,dim() = dim(0)+1 = nkk+1 anddim(W) = dim()+1 = nkk+2.TograspthepicturelookatXconsistingof around2-sphereX(wherek = 1) and a round circleX (wherek = 2) in the Euclidean 3-spaceY , whereXandXintersect at two pointsx1, x2 our 0 = x1, x2 in this case.WhenXanXmoveawayonefromtheother byparallel translationsin the opposite directions,their intersection points sweepWwhich equals theintersectionof the3-ball boundedbyXandtheat2-discspannedbyX.The boundary of thisWconsists of two arcsS XandS X, whereSjoinsx1withx2inXandSjoinx1withx2inX.Back to the general case, we want W to be, generically, a smooth submanifoldwithoutdoublepoints as well as without any other singularities, except for theunavoidable corner in its boundary , where S meet S along 0. We need forthis2dim(W) = 2(n k k + 2) < n = dim(Y ) i.e. 2k + 2k > n + 4.Also, we want to avoid an intersection ofWwithXand withXaway from = (W). If we agree thatk k, this, generically, needsdim(W) + dim(X) = (n k k + 2) + (n k) < n i.e. 2k + k > n + 2.Theseinequalitiesimplythatk k 3, andthelowestdimensionwherethey are meaningful is the rst Whitney case: dim(Y ) = n = 6 andk = k = 3.Accordingly, W is called Whitneys disk, although it may be non-homeomorphictoB2with the present denition ofW(due to Haeiger).HaeigerLemma (Whitney fork + k = n). Let the dimensionsn k =dim(X) and nk = dim(X), where k k, of two submanifolds X and Xin the ambientn-manifoldYsatisfy 2k + k > n + 2.Theneveryhomotopyftof(thedisjointunionof )XandXinY whichdisengagesXfromX, can be replaced by a disengaging homotopyfnewtwhichisanisotopy,onbothmanifolds,i.e. fnewt(X)andfnew(X)remansmoothwithout self intersectionpointsinY forall t |0, 1|andfnew1 (X)doesnotintersectfnew1 (X).Proof. Assume ft is smooth generic and take a small neighbourhood U3 Yof W. By genericity, this ft is an isotopy of X as well as of X within U3 Y :43the intersections offt(X) andft(X) withU3, call themX3(t) andX3(t)are smooth submanifolds in U3 for all t, which, moreover, do not intersect awayfromW U3.Hence,by the Thom isotopy theorem,there exists an isotopyFtofY Uwhich equalsftonU2 Uand which is constant int onY U3.Sinceft andFt withinU3 are equal on the overlapU2 U of their deni-tion domains, they make together a homotopy ofXandXwhich, obviously,satises our requirements.There are several immediate generalizations/applications of this theorem.(1) One may allow self-intersections 0within connected components ofX,where the necessary homotopy condition for removing 0 (which was expressedwiththedisengagingftinthepresentcase)isformulatedintermsof mapsf XX Y Ycommuting with the involutions (x1, x2) (x2, x1) in XXand (y1, y2) (y2, y1)inY Y andhavingthepullbacksf1(Ydiag)of thediagonalYdiag Y YequalXdiag X X, [33].(2) One can apply all of the above top parametric families of mapsX Y ,by paying the price of the extrap in the excess ofdim(Y ) overdim(X), [33].If p = 1, this yield an isotopy classication of embeddings X Yfor 3k > n+3by homotopies of the above symmetric mapsX X Y Y, which shows, forexample, that there are no knots for these dimensions (Haeiger, 1961).if3k > n + 3,theneverysmoothembeddingSnk Rnissmoothlyisotopicto the standardSnk Rn.But if 3k = n + 3 andk = 2l + 1 is odd then there areinnitely many isotopy of classes of embeddings S4l1R6l, (Haeiger1962).Non-trivialityof suchaknot S4l1R6lis detectedbyshowingthat amapf0 B4l R6l R+extendingS4l1=(B4l)cannotbeturnedintoanembedding, keeping it transversal to R6l= R6l 0 and with its boundary equalour knotS4l1 R6l.The Whitney-HaeigerWforf0has dimension 6l + 1 2(2l + 1) + 2 = 2l + 1and, generically, it transversally intersectsB4lat several points.The resulting (properly dened) intersection index ofWwithB is non-zero(otherwise one could eliminate these points by Whitney) and it does not dependonf0. In fact, it equals the linking invariant of Haeiger. (This is reminiscentof thehigherlinkingproductsdescribedbySullivansminimal models, seesection 9.)(3) In view of the above, one must be careful if one wants to relax the dimen-sionconstrainbyaninductiveapplicationoftheWhitney-Haeigerdisengag-ing procedure,since obstructions/invariants for removal higher intersectionswhichcomeonthewaymaybenotsoapparent. (Thestructureof higherself-intersections of this kind for Euclidean hypersurfaces carries a signicantinformation on the stable homotopy groups of spheres.)But this is possible, at least on the Q-level, where one has a comprehensivealgebraic control of self-intersections of all multiplicities for maps of codimensionk 3. Also, even without tensoring with Q, the higher intersection obstructionstend to vanish in the combinatorial category.For example,44there is no combinatorial knots of codimensionk 3 (Zeeman, 1963).Theessential mechanismofknottingX =XnY =Yn+2dependsonthefundamental groupofthecomplementU =Y X. Thegroupmaylookanuisancewhenyouwanttountangleaknot, especiallyasurface X2ina4-manifold, butthese = (X)forvariousX Y formbeautifullyintricatepatterns which are poorly understood.For example, the groups = 1(U) capture the etale cohomology of algebraicmanifolds and the Novikov-Pontryagin classes of topological manifolds (see sec-tion 10). Possibly, the groups (X2) for surfacesX2 Y4have much to tell usabout the smooth topology of 4-manifolds.Therearefewsystematicwaysof constructingsimpleX Y , e.g. im-mersed submanifolds, with interesting (e.g. far from being free) fundamentalgroups of their complements.Ohand suggestions are pullbacks of (special singular) divisorsX0in com-plex algebraic manifoldsY0under generic mapsY Y0and immersed subvari-etiesXnin cubically subdividedYn+2, whereXnare made ofn-sub-cubes ninside the cubes n+2 Yn+2and where these interior n n+2are parallel tothen-faces of n+2.It remains equally unclear what is the possible topology of self-intersectionsof immersionsXnYn+2, say forS3S5, where the self-intersection makes alink inS3, and forS4S6where this is an immersed surface inS4.(4)Onecancontrol thepositionof theimageof fnew(X) Y , e.g. bymakingittolandinagivenopensubset W0 W, if thereisnohomotopyobstruction to this.The above generalizes andsimplies inthe combinatorial or piecewisesmooth category, e.g. for unknotting spheres, where the basic constructionis as followsEngulng. LetXbe a piecewise smooth polyhedron in a smooth manifoldY .If n k = dim(X) dim(Y ) 3andif i(Y ) = 0fori = 1, ...dim(Y ),thenthere exists a smooth isotopyFtofYwhich eventually (fort = 1) movesXto agiven (small) neighbourhoodBof a point inY .Sketch of the Proof. Start with a generic ft. This ft does the job away froma certainWwhich hasdim(W) n2k +2. This is < dim(X) under the aboveassumption and the proof proceeds by induction ondim(X).This is called engulng since B, when moved by the time reversed isotopy,engulfsX; engulng was invented by Stallings in his approach to the PoincareConjecture in the combinatorial category, which goes, roughly, as follows.LetY beasmoothn-manifold. Then, withasimpleuseoftwomutuallydual smoothtriangulationsof Y , onecandecomposeY , foreachi, intotheunion of regular neighbourhoodsU1 andU2 of smooth subpolyhedraX1 andX2inY of dimensionsi andn i 1 (similarly to the handlebodydecompositionofa3-manifoldintotheunionoftwothickenedgraphsinit), where, recall, aneighbourhoodUofanX Y isregular ifthere exists anisotopyft U Uwhich brings all ofUarbitrarily close toX.Nowlet Y beahomotopysphereof dimensionn 7, sayn =7, andleti = 3 Then X1 and X2, and hence U1 and U2, can be engulfed by (dieomorphicimages of) balls, say by B1 U1 and B2 U2 with their centers denoted 01 B145and 02 B2.By moving the 6-sphere(B1) B2by the radial isotopy inB2toward 02,one represents Y 02 by the union of an increasing sequence of isotopic copies ofthe ball B1. This implies (with the isotopy theorem) that Y 02 is dieomorphicto R7, hence,Yis homeomorphic toS7.(A rened generalization of this argument delivers the Poincare conjecturein the combinatorial and topological categories for n 5. See [67] for an accountoftechniquesforprovingvariousPoincareconjecturesandforreferencestothe source papers.)8 Handlesandh-Cobordisms.Theoriginal approachof SmaletothePoincareconjecturedependsonhan-dledecompositionsof manifoldscounterpartstocell decompositionsinthehomotopy theory.Such decompositions are more exible, and by far more abundant than tri-angulations and they are better suited for a match with algebraic objects suchashomology. Forexample,onecansometimesrealizeabasisinhomologybysuitably chosen cells or handles which is not even possible to formulate properlyfor triangulations.Recall that ani-handle of dimensionn is the ballBndecomposed into theproductBn= BiBni() where one think of such a handle as an-thickeningof the uniti-ball and whereA() = Si Bn1() Sn1= Bnis seen as an-neighbourhood of its axial (i 1)-sphereSi10 an equatoriali-sphere inSn1.If Xisann-manifoldwithboundaryY andf A() Y asmoothem-bedding, onecanattachBntoXbyfandtheresultingmanifold(withthecorner alongA() made smooth) is denotedX +fBnorX +Si1 Bn, wherethe latter subscript refers to thef-image of the axial sphere inY .The eect of this on the boundary, i.e. modication(X) = Y fY = (X +Si1 Bn)does not depend onXbut only onY andf. It is called ani-surgeryof Y atthe spheref(Si1 0) Y .The manifoldX = Y |0, 1| +Si1 Bn, whereBnis attached toY 1, makesa bordism between Y = Y 0 and Y which equals the surgically modied Y 1-component of the boundary ofX. If the manifoldYis oriented, so isX, unlessi = 1 and the two ends of the 1-handleB1 Bn1() are attached to the sameconnected component ofYwith opposite orientations.When we attach an i-handle to an Xalong a zero-homologous sphere Si1Y , we create a new i-cycle in X+Si1 Bn; when we attach an (i+1)-handle alongani-sphere inXwhich is non-homologous to zero, we kill ani-cycle.These creations/annihilations of homology may cancel each other and a han-dle decomposition of an X may have by far more handles (balls) than the numberof independent homology classes inH(X).46Smales argument proceeds in two steps.(1) The overall algebraic cancellation is decomposed into elementary stepsby reshuing handles (in the spirit of J.H.C. Whiteheads theory of the simplehomotopy type);(2)eachelementarystepisimplementedgeometricallyasintheexamplebelow (which does not elucidate the casen = 6).Cancelling a 3-handle by a 4-handle. LetX = S3B4(0) and let us attachthe 4-handleB7= B4 B3(), 0, meaningthat there is a mapf X1X2of degreem.Every such fbetween closed connected oriented manifolds induces a surjec-tivehomomorphismsfi Hi(X1; Q) Hi(X1; Q)forall i = 0, 1, ..., n, (asweknow from section 4), or equivalently, an injective cohomology homomorphismfi Hi(X2; Q) Hi(X2; Q).Indeed, by the Poincare Q-duality, the cup-product (this the common namefor the product on cohomology) pairing Hi(X2; Q)Hni(X2; Q) Q = Hn(X2; Q)49is faithful; therefore, iffivanishes, then so doesfn. But the latter amountsto multiplication bym = deg(f),Hn(X2; Q) = Q d Q = Hn(X1; Q).(The main advantage of the cohomology product over the intersection producton homology is that the former is preserved by all continuous maps,fi+j(c1 c2) =fi(c1) fj(c2)for all f X Y andall c1 Hi(Y ),c2 Hj(Y ).)If m = 1, then (by the full cohomological Poincare duality) the above remainstrue for all coecient elds F; moreover, the induced homomorphism i(X1) i(X2)issurjectiveasitisseenbylookin