DOCSLIB.ORG

Topics in Topology. Fall 2008. the Signature Theorem and Some of Its Applications1 Liviu I. Nicolaescu

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Topics in Topology. Fall 2008. the Signature Theorem and Some of Its Applications1 Liviu I. Nicolaescu Topics in topology. Fall 2008. The signature theorem and some of its applications1 Liviu I. Nicolaescu DEPT. OF MATH,UNIVERSITY OF NOTRE DAME,NOTRE DAME, IN 46556-4618. E-mail address: [email protected] URL: http://www.nd.edu/˜lnicolae/ 1Last modified on December 8, 2013. Please feel free to email me corrections. Contents Introduction v Chapter 1. Singular homology and cohomology1 x1.1. The basic properties of singular homology and cohomology1 x1.2. Products6 x1.3. Local homology and cohomology8 Chapter 2. Poincare´ duality 11 x2.1. Manifolds and orientability 11 x2.2. Various versions of Poincare´ duality 14 x2.3. Intersection theory 18 Chapter 3. Vector bundles and classifying spaces 25 x3.1. Definition and examples of vector bundles 25 x3.2. Functorial operations with vector bundles 30 x3.3. The classification of vector bundles 35 Chapter 4. The Thom isomorphism, the Euler class and the Gysin sequence 39 x4.1. The Thom isomorphism 39 x4.2. Fiber bundles 44 x4.3. The Gysin sequence and the Euler class 47 x4.4. The Leray-Hirsch isomorphism 51 Chapter 5. The construction of Chern classes a` la Grothendieck 53 x5.1. The first Chern class 53 x5.2. The Chern classes: uniqueness and a priori investigations 55 x5.3. Chern classes: existence 59 x5.4. The localization formula and some of its applications 61 Chapter 6. Pontryagin classes and Pontryagin numbers 69 iii iv Contents x6.1. The Pontryagin classes of a real vector bundle 69 x6.2. Pontryagin numbers 72 Chapter 7. The Hirzebruch signature formula 77 x7.1. Multiplicative sequences 77 x7.2. Genera 81 x7.3. The signature formula 85 x7.4. The Lagrange inversion formula 86 Chapter 8. Milnor’s exotic spheres 89 x8.1. An invariant of smooth rational homology 7-spheres 89 x8.2. Disk bundles over the 4-sphere 91 Chapter 9. Thom’s work on cobordisms 99 x9.1. The Pontryagin-Thom construction 99 x9.2. The cohomology of the Grassmannians of oriented subspaces 105 Chapter 10. The Chern-Weil construction 109 x10.1. Principal bundles 109 x10.2. Connections on vector bundles 112 x10.3. Connections on principal bundles 120 x10.4. The Chern-Weil construction 121 x10.5. The Chern classes 125 Appendix A. Homework assignments 131 xA.1. Homework 1 131 xA.2. Homework 2 131 xA.3. Homework 3. 132 xA.4. Homework 4 133 Appendix B. Solutions to selected problems 137 xB.1. Solution to Exercise A.3.2. 137 xB.2. Solution to Exercise A.4.1 140 xB.3. Solution to Exercise A.4.2. 141 xB.4. Solution to Exercise A.4.3 142 xB.5. Solution to Exercise A.4.4. 145 Bibliography 149 Index 151 Introduction These are the notes for the Fall 2008 Topics in topology class at the University of Notre Dame. I view them as a “walk down memory lane” since I discuss several exciting topological developments that took place during the fifties decade which radically changed the way we think about many issues. The pretext for this journey is Milnor’s surprising paper [Mi56] in which he constructs exotic smooth structures on the 7-sphere. The main invariant used by Milnor to distinguish between two smooth structures on the same manifold was based on a formula that F. Hirzebruch had recently discovered that related the signature of a smooth manifold to the integral of a certain polynomial in the Pontryagin classes of the tangent bundle. The goal of this class is easily stated: understand all the details of the very densely packed, beautiful paper [Mi56]. From an academic point of view this has many very useful benefits because it forces the aspiring geometer to learn a large chunk of the basic notions and tools that are part of the everyday arsenal of the modern geometer/topologist: Poincare´ duality, Thom isomorphism, Euler, Chern and Pontryagin classes, cobordisms groups, signature formula. Moreover, such a journey has to include some beautiful side-trips into the inner making of several concrete fundamental examples. These can only enhance the appreciation and understanding of the subject. The goal is a bit ambitious for a one-semester course, and much like the classic [MS] from which I have drawn a lot of inspiration, I had to make some choices. Here is what I decided to leave out: • The proofs of the various properties of [, \ and × products. • The proofs of the Poincare´ duality theorem, the Thom isomorphism theorem, and of the Leray-Hirsch theorem. • The beautiful work of J.P. Serre on the cohomology of loops spaces, Serre classes of Abelian groups and their applications to homotopy theory. The first topic ought to be part of a first year graduate course in algebraic topology, and is pre- sented in many easily accessible sources. The right technology for dealing with the last two topics is that of spectral sequences, but given that the spectral sequences have a rather different flavor than the rest of the arguments which are mostly geometrical, I decided it would be too much for anyone to absorb in one semester. v Chapter 1 Singular homology and cohomology For later use we survey a few basic facts from algebraic topology. For more details we refer to [Bre, Do, Hatch1, Spa]. 1.1. The basic properties of singular homology and cohomology A ring R is called convenient if it is isomorphic to one of the rings Z; Q; R or Fp := Z=p, p prime. In the sequel we will work exclusively with convenient rings . We denote by ∆n the standard affine n-simplex n n+1 X ∆n := (t0; : : : ; tn) 2 R≥0 ; ti = 1 : i=0 For every topological space X we denote by Sk(X) the set of singular k-simplices in X, i.e., the set of continuous maps σ : ∆k ! X: We denote by C•(X; R) the singular chain complex of X with coefficients in the ring R, M M M C•(X; R) = Ck(X; R);Ck(X; R) = R = Rjσi: k≥0 σ2Sk(X) σ2Sk(X) Above, for every singular simplex σ 2 Sk(X) we denoted by jσi the generator of Ck(X; R) associ- ated to this singular simplex. For every k ≥ 1, and every i = 0; : : : ; k we have a natural map @i : Sk(X) ! Sk−1(X); Sk(X) 3 σ 7! @iσ 2 Sk−1(X); where for any (t0; : : : ; tk−1) 2 ∆k−1 we have @iσ(t0; : : : ; tk−1) = σ(t0; : : : ; ti−1; 0; ti; : : : ; tk−1): 1 2 Singular homology and cohomology The boundary operator @ : Ck(X; R) ! Ck−1(X; R) is then uniquely determined by k X i @jσi = (−1) j@iσi: i=0 We denote by H•(X; R) the homology of this complex. It is called the singular homology of X with coefficients in R. Every subspace A ⊂ X defines a subcomplex C•(A; R) ⊂ C•(X; R) and we denote by C•(X; A; R) the quotient complex C•(X; A; R) := C•(X; R)=C•(A; R): We denote by H•(X; A; R) the homology of the complex C•(X; A; R). It is called the singular homology of the pair (X; A) with coefficients in R. When R = Z we will drop the ring R from the notation. The singular homology enjoys several remarkable properties. • If ∗ denotes the topological space consisting of a single point, then ( R k = 0 Hk(∗;R) = 0 k > 0: • A continuous map of pairs f :(X; A) ! (Y; B) induces a morphism f∗ : H•(X; A; R) ! H•(Y; B; R): Two homotopic maps f0; f1 :(X; A) ! (Y; B) induce identical morphisms, (f0)∗ = (f1)∗. More- over 1 = 1 : (X;A) ∗ H•(X;A;R) f g • If (X; A) ! (Y; B) ! (Z; C) are continuous maps then the induced maps in homology satisfy (g ◦ f)∗ = g∗ ◦ f∗: • For any topological pair (X; A) we have a long exact sequence @ i∗ j∗ @ ··· ! Hk(A; R) ! Hk(X; R) ! Hk(X; A; R) ! Hk−1(A; R) !··· ; i j where i∗ and j∗ are induced by the natural maps (A; ;) ,! (X; ;) ,! (X; A). This long exact sequence is natural in the sense that given any continuous maps f :(X; A) ! (Y; B) the diagram below is commutative for any k ≥ 0. @ Hk+1(X; A; R) w Hk(A; R) f∗ f∗ (1.1.1) u u Hk+1(Y; B; R) w Hk(B; R) @ • If X is a topological space and B ⊂ A are subspaces of X, then we have a natural long exact sequence @ i∗ j∗ @ ··· ! Hk(A; B; R) ! Hk(X; B; R) ! Hk(X; A; R) ! Hk−1(A; B; R) !··· Liviu I. Nicolaescu 3 • (The universal coefficients theorem) For any topological pair (X; A) we have a natural isomorphism H•(X; A) ⊗Z R ! H•(X; A; R): Moreover, for every prime number p > 1 we have a non-canonical isomorphism of vector spaces over Fp Hk(X; A; Fp) = Hk(X; A) ⊗ Fp ⊕ Torp Hk−1(X; A) ; where for any Abelian group G, and any positive integer n we set Torn(G) := g 2 G; ng = 0 : • (Kunneth¨ theorem) If X; Y are topological spaces, A ⊂ X, B ⊂ Y , then for every k ≥ 0 there exists a natural injection M × : Hi(X; A; R) ⊗ Hj(Y; B; R) ! Hk (X; A) × (Y; B); R i+j=k where (X; A) × (Y; B) := (X × Y; A × Y [ X × B): ∼ If R is a field, then the cross product × defined above is an isomorphism. If R = Z, then we have a short exact sequence M × 0 ! Hi(X; A; R) ⊗ Hj(Y; B; R) −! Hk (X; A) × (Y; B); R ! i+j=k M ! Hi(X) ∗ Hj(Y ) ! 0; (1.1.2) i+j=k−1 where the torsion product G∗H of two Abelian groups G; H is uniquely determined by the following properties.
Load more

Recommended publications
  • Poincar\'E Duality for Spaces with Isolated Singularities
    POINCARÉ DUALITY FOR SPACES WITH ISOLATED SINGULARITIES MATHIEU KLIMCZAK Abstract. In this paper we assign, under reasonable hypothesis, to each pseudomanifold with isolated singularities a rational Poincaré duality space. These spaces are constructed with the formalism of intersection spaces defined by Markus Banagl and are indeed related to them in the even dimensional case. 1. Introduction We are concerned with rational Poincaré duality for singular spaces. There is at least two ways to restore it in this context : • As a self-dual sheaf. This is for instance the case with rational intersection homology. • As a spatialization. That is, given a singular space X, trying to associate to it a new topological space XDP that satisfies Poincaré duality. This strategy is at the origin of the concept of intersection spaces. Let us briefly recall this two approaches. While seeking for a theory of characteristic numbers for complex analytic vari- eties and other singular spaces, Mark Goresky and Robert MacPherson discovered (and then defined in [9] for PL pseudomanifolds and in [8] for topological pseudo- p manifolds) a family of groups IH∗ (X) called intersection homology groups of X. These groups depend on a multi-index p called a perversity. Intersection homology is able to restore Poincaré duality on topological stratified pseudomanifolds. If X is a compact oriented pseudomanifold of dimension n and p, q are two complementary perversities, over Q we have an isomorphism p ∼ n−r IHr (X) = IHq (X), n−r q With IHq (X) := hom(IHn−r(X), Q). Intersection spaces were defined by Markus Banagl in [2] as an attempt to spa- tialize Poincaré duality for singular spaces.
    [Show full text]
  • Homotopy Properties of Thom Complexes (English Translation with the Author’S Comments) S.P.Novikov1
    Homotopy Properties of Thom Complexes (English translation with the author’s comments) S.P.Novikov1 Contents Introduction 2 1. Thom Spaces 3 1.1. G-framed submanifolds. Classes of L-equivalent submanifolds 3 1.2. Thom spaces. The classifying properties of Thom spaces 4 1.3. The cohomologies of Thom spaces modulo p for p > 2 6 1.4. Cohomologies of Thom spaces modulo 2 8 1.5. Diagonal Homomorphisms 11 2. Inner Homology Rings 13 2.1. Modules with One Generator 13 2.2. Modules over the Steenrod Algebra. The Case of a Prime p > 2 16 2.3. Modules over the Steenrod Algebra. The Case of p = 2 17 1The author’s comments: As it is well-known, calculation of the multiplicative structure of the orientable cobordism ring modulo 2-torsion was announced in the works of J.Milnor (see [18]) and of the present author (see [19]) in 1960. In the same works the ideas of cobordisms were extended. In particular, very important unitary (”complex’) cobordism ring was invented and calculated; many results were obtained also by the present author studying special unitary and symplectic cobordisms. Some western topologists (in particular, F.Adams) claimed on the basis of private communication that J.Milnor in fact knew the above mentioned results on the orientable and unitary cobordism rings earlier but nothing was written. F.Hirzebruch announced some Milnors results in the volume of Edinburgh Congress lectures published in 1960. Anyway, no written information about that was available till 1960; nothing was known in the Soviet Union, so the results published in 1960 were obtained completely independently.
    [Show full text]
  • The Signature Operator at 2 Jonathan Rosenberga,∗,1, Shmuel Weinbergerb,2
    Topology 45 (2006) 47–63 www.elsevier.com/locate/top The signature operator at 2 Jonathan Rosenberga,∗,1, Shmuel Weinbergerb,2 aDepartment of Mathematics, University of Maryland, College Park, MD 20742, USA bDepartment of Mathematics, University of Chicago, Chicago, IL 60637, USA Received 5 March 2004; received in revised form 24 February 2005; accepted 8 June 2005 Abstract It is well known that the signature operator on a manifold defines a K-homology class which is an orientation after inverting 2. Here we address the following puzzle: What is this class localized at 2, and what special properties does it have? Our answers include the following: • the K-homology class M of the signature operator is a bordism invariant; • the reduction mod 8 of the K-homology class of the signature operator is an oriented homotopy invariant; • the reduction mod 16 of the K-homology class of the signature operator is not an oriented homotopy invariant. ᭧ 2005 Elsevier Ltd. All rights reserved. Keywords: K-homology; Signature operator; Surgery theory; Homotopy eqvivalence; Lens space 0. Introduction The motivation for this paper comes from a basic question, of how to relate index theory (studied analytically) with geometric topology. More specifically, if M is a manifold (say smooth and closed), then the machinery of Kasparov theory [5,12,13] associates a K-homology class with any elliptic differential operator on M.IfM is oriented, then in particular one can do this construction with the signature operator ∗ Corresponding author. Tel.: +1 3014055166; fax: +1 3013140827. E-mail addresses: [email protected] (J. Rosenberg), [email protected] (S.
    [Show full text]
  • Visible L-Theory
    Forum Math. 4 (1992), 465-498 Forum Mathematicum de Gruyter 1992 Visible L-theory Michael Weiss (Communicated by Andrew Ranicki) Abstract. The visible symmetric L-groups enjoy roughly the same formal properties as the symmetric L-groups of Mishchenko and Ranicki. For a fixed fundamental group n, there is a long exact sequence involving the quadratic L-groups of a, the visible symmetric L-groups of n, and some homology of a. This makes visible symmetric L-theory computable for fundamental groups whose ordinary (quadratic) L-theory is computable. 1991 Mathematics Subject Classification: 57R67, 19J25. 0. Introduction The visible symmetric L-groups are a refinement of the symmetric L-groups of Mishchenko [I21 and Ranicki [15]. They are much easier to compute, although they hardly differ from the symmetric L-groups in their formal properties. Cappell and Shaneson [3] use the visible symmetric L-groups in studying 4-dimensional s-cobordisms; see also Kwasik and Schultz [S]. The visible symmetric L-groups are related to the Ronnie Lee L-groups of Milgram [Ill. The reader is assumed to be familiar with the main results and definitions of Ranicki [I 53. Let R be a commutative ring (with unit), let n be a discrete group, and equip the group ring R [n] with the w-twisted involution for some homomorphism w: n -* E,. An n-dimensional symmetric algebraic Poincare complex (SAPC) over R[n] is a finite dimensional chain complex C of finitely generated free left R [n]-modules, together with a nondegenerate n-cycle 4 E H~mz[z,~(W7 C' OR,,,C) where W is the standard resolution of E over Z [H,] .
    [Show full text]
  • Notes on the Atiyah-Singer Index Theorem Liviu I. Nicolaescu
    Notes on the Atiyah-Singer Index Theorem Liviu I. Nicolaescu Notes for a topics in topology course, University of Notre Dame, Spring 2004, Spring 2013. Last revision: November 15, 2013 i The Atiyah-Singer Index Theorem This is arguably one of the deepest and most beautiful results in modern geometry, and in my view is a must know for any geometer/topologist. It has to do with elliptic partial differential opera- tors on a compact manifold, namely those operators P with the property that dim ker P; dim coker P < 1. In general these integers are very difficult to compute without some very precise information about P . Remarkably, their difference, called the index of P , is a “soft” quantity in the sense that its determination can be carried out relying only on topological tools. You should compare this with the following elementary situation. m n Suppose we are given a linear operator A : C ! C . From this information alone we cannot compute the dimension of its kernel or of its cokernel. We can however compute their difference which, according to the rank-nullity theorem for n×m matrices must be dim ker A−dim coker A = m − n. Michael Atiyah and Isadore Singer have shown in the 1960s that the index of an elliptic operator is determined by certain cohomology classes on the background manifold. These cohomology classes are in turn topological invariants of the vector bundles on which the differential operator acts and the homotopy class of the principal symbol of the operator. Moreover, they proved that in order to understand the index problem for an arbitrary elliptic operator it suffices to understand the index problem for a very special class of first order elliptic operators, namely the Dirac type elliptic operators.
    [Show full text]
  • The Total Surgery Obstruction Revisited
    THE TOTAL SURGERY OBSTRUCTION REVISITED PHILIPP KUHL,¨ TIBOR MACKO AND ADAM MOLE November 15, 2018 Abstract. The total surgery obstruction of a finite n-dimensional Poincar´e complex X is an element s(X) of a certain abelian group Sn(X) with the property that for n ≥ 5 we have s(X) = 0 if and only if X is homotopy equivalent to a closed n-dimensional topological manifold. The definitions of Sn(X) and s(X) and the property are due to Ranicki in a combination of results of two books and several papers. In this paper we present these definitions and a detailed proof of the main result so that they are in one place and we also add some of the details not explicitly written down in the original sources. Contents 1. Introduction 1 2. Algebraic complexes 10 3. Normal complexes 18 4. Algebraic bordism categories and exact sequences 27 5. Categories over complexes 29 6. Assembly 34 7. L-Spectra 35 8. Generalized homology theories 37 9. Connective versions 41 10. Surgery sequences and the structure groups Sn(X) 42 11. Normal signatures over X 44 12. Definition of s(X) 47 13. Proof of the Main Technical Theorem (I) 48 14. Proof of the Main Technical Theorem (II) 55 15. Concluding remarks 70 References 70 arXiv:1104.5092v2 [math.AT] 21 Sep 2011 1. Introduction An important problem in the topology of manifolds is deciding whether there is an n-dimensional closed topological manifold in the homotopy type of a given n-dimensional finite Poincar´ecomplex X.
    [Show full text]
  • Characteristic Classes and K-Theory Oscar Randal-Williams
    Characteristic classes and K-theory Oscar Randal-Williams https://www.dpmms.cam.ac.uk/∼or257/teaching/notes/Kthy.pdf 1 Vector bundles 1 1.1 Vector bundles . 1 1.2 Inner products . 5 1.3 Embedding into trivial bundles . 6 1.4 Classification and concordance . 7 1.5 Clutching . 8 2 Characteristic classes 10 2.1 Recollections on Thom and Euler classes . 10 2.2 The projective bundle formula . 12 2.3 Chern classes . 14 2.4 Stiefel–Whitney classes . 16 2.5 Pontrjagin classes . 17 2.6 The splitting principle . 17 2.7 The Euler class revisited . 18 2.8 Examples . 18 2.9 Some tangent bundles . 20 2.10 Nonimmersions . 21 3 K-theory 23 3.1 The functor K ................................. 23 3.2 The fundamental product theorem . 26 3.3 Bott periodicity and the cohomological structure of K-theory . 28 3.4 The Mayer–Vietoris sequence . 36 3.5 The Fundamental Product Theorem for K−1 . 36 3.6 K-theory and degree . 38 4 Further structure of K-theory 39 4.1 The yoga of symmetric polynomials . 39 4.2 The Chern character . 41 n 4.3 K-theory of CP and the projective bundle formula . 44 4.4 K-theory Chern classes and exterior powers . 46 4.5 The K-theory Thom isomorphism, Euler class, and Gysin sequence . 47 n 4.6 K-theory of RP ................................ 49 4.7 Adams operations . 51 4.8 The Hopf invariant . 53 4.9 Correction classes . 55 4.10 Gysin maps and topological Grothendieck–Riemann–Roch . 58 Last updated May 22, 2018.
    [Show full text]
  • Characteristic Classes and Smooth Structures on Manifolds Edited by S
    7102 tp.fh11(path) 9/14/09 4:35 PM Page 1 SERIES ON KNOTS AND EVERYTHING Editor-in-charge: Louis H. Kauffman (Univ. of Illinois, Chicago) The Series on Knots and Everything: is a book series polarized around the theory of knots. Volume 1 in the series is Louis H Kauffman’s Knots and Physics. One purpose of this series is to continue the exploration of many of the themes indicated in Volume 1. These themes reach out beyond knot theory into physics, mathematics, logic, linguistics, philosophy, biology and practical experience. All of these outreaches have relations with knot theory when knot theory is regarded as a pivot or meeting place for apparently separate ideas. Knots act as such a pivotal place. We do not fully understand why this is so. The series represents stages in the exploration of this nexus. Details of the titles in this series to date give a picture of the enterprise. Published*: Vol. 1: Knots and Physics (3rd Edition) by L. H. Kauffman Vol. 2: How Surfaces Intersect in Space — An Introduction to Topology (2nd Edition) by J. S. Carter Vol. 3: Quantum Topology edited by L. H. Kauffman & R. A. Baadhio Vol. 4: Gauge Fields, Knots and Gravity by J. Baez & J. P. Muniain Vol. 5: Gems, Computers and Attractors for 3-Manifolds by S. Lins Vol. 6: Knots and Applications edited by L. H. Kauffman Vol. 7: Random Knotting and Linking edited by K. C. Millett & D. W. Sumners Vol. 8: Symmetric Bends: How to Join Two Lengths of Cord by R.
    [Show full text]
  • FOLIATIONS Introduction. the Study of Foliations on Manifolds Has a Long
    BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 80, Number 3, May 1974 FOLIATIONS BY H. BLAINE LAWSON, JR.1 TABLE OF CONTENTS 1. Definitions and general examples. 2. Foliations of dimension-one. 3. Higher dimensional foliations; integrability criteria. 4. Foliations of codimension-one; existence theorems. 5. Notions of equivalence; foliated cobordism groups. 6. The general theory; classifying spaces and characteristic classes for foliations. 7. Results on open manifolds; the classification theory of Gromov-Haefliger-Phillips. 8. Results on closed manifolds; questions of compact leaves and stability. Introduction. The study of foliations on manifolds has a long history in mathematics, even though it did not emerge as a distinct field until the appearance in the 1940's of the work of Ehresmann and Reeb. Since that time, the subject has enjoyed a rapid development, and, at the moment, it is the focus of a great deal of research activity. The purpose of this article is to provide an introduction to the subject and present a picture of the field as it is currently evolving. The treatment will by no means be exhaustive. My original objective was merely to summarize some recent developments in the specialized study of codimension-one foliations on compact manifolds. However, somewhere in the writing I succumbed to the temptation to continue on to interesting, related topics. The end product is essentially a general survey of new results in the field with, of course, the customary bias for areas of personal interest to the author. Since such articles are not written for the specialist, I have spent some time in introducing and motivating the subject.
    [Show full text]
  • THOM COBORDISM THEOREM 1. Introduction in This Short Notes We
    THOM COBORDISM THEOREM MIGUEL MOREIRA 1. Introduction In this short notes we will explain the remarkable theorem of Thom that classifies cobordisms. This theorem was originally proven by Ren´eThom in [] but we will follow a more modern exposition. We want to give a precise idea of the everything involved but we won't give complete details at some points. Let's define the concept of cobordism. Definition 1. Let M; N be two smooth closed n-manifolds. We say that M; N are cobordant if there exists a compact (n + 1)-manifold W such that @W = M t N. This is an equivalence relation. Given a space X and f : M ! X we call the pair (M; f) a singular manifold. We say that (M; f) and (N; g) are cobordant if there is a cobordism W between M and N and f t g extends to F : W ! X. We denote by Nn(X) the set of cobordism classes of singular n-manifolds (M; f). In particular Nn ≡ Nn(∗) is the set of cobordism sets. We'll already state the two amazing theorems by Thom. The first gives a homotopy-theoretical interpretation of Nn and the second actually computes it. Theorem 1. N∗ is a generalized homology theory associated to the Thom spectrum MO, i.e. Nn(X) = colim πn+k(MO(k) ^ X+): k!1 Theorem 2. We have an isomorphism of graded Z=2-algebras ∼ ` π∗MO = Z=2[xi : 0 < i 6= 2 − 1] = Z=2[x2; x4; x5; x6;:::] where xi has grading i.
    [Show full text]
  • Floer Homotopy Theory, Realizing Chain Complexes by Module Spectra, And
    Floer homotopy theory, realizing chain complexes by module spectra, and manifolds with corners Ralph L. Cohen ∗ Department of Mathematics Stanford University Stanford, CA 94305 February 13, 2013 Abstract In this paper we describe and continue the study begun in [5] of the homotopy theory that underlies Floer theory. In that paper the authors addressed the question of realizing a Floer complex as the celluar chain complex of a CW -spectrum or pro-spectrum, where the attaching maps are determined by the compactified moduli spaces of connecting orbits. The basic obstructions to the existence of this realization are the smoothness of these moduli spaces, and the existence of compatible collections of framings of their stable tangent bundles. In this note we describe a generalization of this, to show that when these moduli spaces are smooth, ∗ and are oriented with respect to a generalized cohomology theory E , then a Floer E∗-homology theory can be defined. In doing this we describe a functorial viewpoint on how chain complexes can be realized by E-module spectra, generalizing the stable homotopy realization criteria given in [5]. Since these moduli spaces, if smooth, will be manifolds with corners, we give a discussion about the appropriate notion of orientations of manifolds with corners. Contents 1 Floer homotopy theory 5 arXiv:0802.2752v1 [math.AT] 20 Feb 2008 1.1 PreliminariesfromMorsetheory . ......... 5 1.2 SmoothFloertheories ............................. ...... 9 2 Realizing chain complexes by E-module spectra 10 ∗ 3 Manifolds with corners, E -orientations of flow categories, and Floer E∗ - homol- ogy 14 ∗The author was partially supported by a grant from the NSF 1 Introduction In [5], the authors began the study of the homotopy theoretic aspects of Floer theory.
    [Show full text]
  • A Combinatorial Computation of the First Pontryagin Class of the Complex Projective Plane
    LAZAR MILIN A COMBINATORIAL COMPUTATION OF THE FIRST PONTRYAGIN CLASS OF THE COMPLEX PROJECTIVE PLANE ABSTRACT. This paper carries out an explicit computation of the combinatorial formula of Gabrielov, Gel'fand, and Losik for the first Pontryagin class of the complex projective plane with the 9-vertex triangulation discovered by Wolfgang Kfihnel. The conditions of the original formula must be modified since the 8-vertex triangulation of the 3-sphere link of each vertex cannot be realized as the complex of faces of a convex polytope in 4-space, but it can be so realized by a star-shaped polytope, and the space of all such realizations is not connected. 0. INTRODUCTION In a series of papers [24]-[26] in the 1940s, L. S. Pontryagin defined and studied 'characteristic cycles' on smooth manifolds. In the introduction to the first paper of the series, he stressed the importance of having 'a definition of characteristic cycles which would be applicable to combinatorial mani- folds' since it would provide an algorithm for their calculation from 'the combinatorial structure of the manifolds'. In the late 1950s, Rdne Thom [33], and independently Rohlin and Sarc [28], proved that the rational Pontryagin classes are combinatorial invariants, and in the 1960s. S. P. Novikov [23] proved that these classes are topological invariants. No combinatorial formula for Pontryagin classes was proved until the mid-1970s, when Gabrielov et al. [6], [7] established a formula for the first Pontryagin class pl(X) of a combinatorial manifold X. The formula ex- presses Pl(X) in terms of the simplicial structure of X and some additional structure imposed on X, so in this sense the formula is not purely combinatorial.
    [Show full text]