Characteristic Classes and Smooth Structures on Manifolds Edited by S
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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. -
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. -
K-Theoryand Characteristic Classes
MATH 6530: K-THEORY AND CHARACTERISTIC CLASSES Taught by Inna Zakharevich Notes by David Mehrle [email protected] Cornell University Fall 2017 Last updated November 8, 2018. The latest version is online here. Contents 1 Vector bundles................................ 3 1.1 Grassmannians ............................ 8 1.2 Classification of Vector bundles.................... 11 2 Cohomology and Characteristic Classes................. 15 2.1 Cohomology of Grassmannians................... 18 2.2 Characteristic Classes......................... 22 2.3 Axioms for Stiefel-Whitney classes................. 26 2.4 Some computations.......................... 29 3 Cobordism.................................. 35 3.1 Stiefel-Whitney Numbers ...................... 35 3.2 Cobordism Groups.......................... 37 3.3 Geometry of Thom Spaces...................... 38 3.4 L-equivalence and Transversality.................. 42 3.5 Characteristic Numbers and Boundaries.............. 47 4 K-Theory................................... 49 4.1 Bott Periodicity ............................ 49 4.2 The K-theory spectrum........................ 56 4.3 Some properties of K-theory..................... 58 4.4 An example: K-theory of S2 ..................... 59 4.5 Power Operations............................ 61 4.6 When is the Hopf Invariant one?.................. 64 4.7 The Splitting Principle........................ 66 5 Where do we go from here?........................ 68 5.1 The J-homomorphism ........................ 70 5.2 The Chern Character and e invariant............... -
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. -
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. -
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. -
Characteristic Classes and Bounded Cohomology
DISS. ETH Nr. 15636 Characteristic Classes and Bounded Cohomology A dissertation submitted to the SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH for the degree of Doctor of Mathematics presented by Michelle Karlsson Dipl. Math. Univ. Gen`eve born on the 25th of July 1976 citizen of Kerns (OW) accepted on the recommendation of Prof. Dr. Marc Burger, examiner Prof. Dr. Wolfgang L¨uck, co-examiner 2004 Acknowledgments I wish to express my sincere gratitude to Prof. Marc Burger for having taken me under his directorship and suggested to me a fascinating subject which brought me into several areas of mathematics. I am also grateful to Prof. Wolfgang L¨uck for kindly accepting to be the external examiner of this dissertation. For fruitful discussions related to my work I owe special thanks to Prof. Johan L. Dupont, PD. Oliver Baues and Prof. Alessandra Iozzi. Many thanks to the whole of Group 1 for providing a productive working environment. Espe- cially to Prof. Guido Mislin for organizing enlightening seminars and reading a short version of this work. Finally, I heartily thank my family and friends for their patience and constant support. Contents Introduction vii 1 Bundles 1 1.1 Principal bundles and classifying spaces . 1 1.1.1 Principal bundles . 1 1.1.2 Classifying spaces . 8 1.2 Elements of Differential geometry . 12 1.2.1 Connections . 12 1.2.2 Curvature . 17 1.3 Flat bundles . 20 1.3.1 Definition . 20 1.3.2 Transition functions . 21 1.3.3 The space of representations . 23 2 Simplicial complexes 25 2.1 Definitions . -
Manifolds: Where Do We Come From? What Are We? Where Are We Going
Manifolds: Where Do We Come From? What Are We? Where Are We Going Misha Gromov September 13, 2010 Contents 1 Ideas and Definitions. 2 2 Homotopies and Obstructions. 4 3 Generic Pullbacks. 9 4 Duality and the Signature. 12 5 The Signature and Bordisms. 25 6 Exotic Spheres. 36 7 Isotopies and Intersections. 39 8 Handles and h-Cobordisms. 46 9 Manifolds under Surgery. 49 1 10 Elliptic Wings and Parabolic Flows. 53 11 Crystals, Liposomes and Drosophila. 58 12 Acknowledgments. 63 13 Bibliography. 63 Abstract Descendants of algebraic kingdoms of high dimensions, enchanted by the magic of Thurston and Donaldson, lost in the whirlpools of the Ricci flow, topologists dream of an ideal land of manifolds { perfect crystals of mathematical structure which would capture our vague mental images of geometric spaces. We browse through the ideas inherited from the past hoping to penetrate through the fog which conceals the future. 1 Ideas and Definitions. We are fascinated by knots and links. Where does this feeling of beauty and mystery come from? To get a glimpse at the answer let us move by 25 million years in time. 25 106 is, roughly, what separates us from orangutans: 12 million years to our common ancestor on the phylogenetic tree and then 12 million years back by another× branch of the tree to the present day orangutans. But are there topologists among orangutans? Yes, there definitely are: many orangutans are good at "proving" the triv- iality of elaborate knots, e.g. they fast master the art of untying boats from their mooring when they fancy taking rides downstream in a river, much to the annoyance of people making these knots with a different purpose in mind. -
Characteristic Classes on Grassmannians
Turkish Journal of Mathematics Turk J Math (2014) 38: 492 { 523 http://journals.tubitak.gov.tr/math/ ⃝c TUB¨ ITAK_ Research Article doi:10.3906/mat-1302-54 Characteristic classes on Grassmannians Jin SHI1, Jianwei ZHOU2;∗ 1Suzhou Senior Technical Institute, Suzhou, P.R. China 2Department of Mathematics, Suzhou University, Suzhou, P.R. China Received: 27.02.2013 • Accepted: 03.04.2013 • Published Online: 14.03.2014 • Printed: 11.04.2014 Abstract: In this paper, we study the geometry and topology on the oriented Grassmann manifolds. In particular, we use characteristic classes and the Poincar´eduality to study the homology groups of Grassmann manifolds. We show that for k = 2 or n ≤ 8, the cohomology groups H∗(G(k; n); R) are generated by the first Pontrjagin class, the Euler classes q of the canonical vector bundles. In these cases, the Poincar´eduality: H (G(k; n); R) ! Hk(n−k)−q(G(k; n); R) can be expressed explicitly. Key words: Grassmann manifold, fibre bundle, characteristic class, homology group, Poincar´eduality 1. Introduction Let G(k; n) be the Grassmann manifold formed by all oriented k -dimensional subspaces of Euclidean space n R . For any π 2 G(k; n), there are orthonormal vectors e1; ··· ; ek such that π can be represented by V k n e1 ^ · · · ^ ek . Thus G(k; n) becomes a submanifold of the space (R ); then we can use moving frame to study the Grassmann manifolds. There are 2 canonical vector bundles E = E(k; n) and F = F (k; n) over G(k; n) with fibres generated by vectors of the subspaces and the vectors orthogonal to the subspaces, respectively. -
Topological Hochschild Homology of Thom Spectraand the Free Loop Space
Geometry & Topology 14 (2010) 1165–1242 1165 Topological Hochschild homology of Thom spectra and the free loop space ANDREW JBLUMBERG RALPH LCOHEN CHRISTIAN SCHLICHTKRULL We describe the topological Hochschild homology of ring spectra that arise as Thom spectra for loop maps f X BF , where BF denotes the classifying space for W ! stable spherical fibrations. To do this, we consider symmetric monoidal models of the category of spaces over BF and corresponding strong symmetric monoidal Thom spectrum functors. Our main result identifies the topological Hochschild homology as the Thom spectrum of a certain stable bundle over the free loop space L.BX /. This leads to explicit calculations of the topological Hochschild homology for a large class of ring spectra, including all of the classical cobordism spectra MO, MSO, MU, etc, and the Eilenberg–Mac Lane spectra HZ=p and H Z. 19D55, 55N20; 18G55, 55P43, 55P47, 55R25 1 Introduction Many interesting ring spectra arise naturally as Thom spectra. It is well-known that one may associate a Thom spectrum T .f / to any map f X BF , where BF W ! denotes the classifying space for stable spherical fibrations. This construction is homotopy invariant in the sense that the stable homotopy type of T .f / only depends on the homotopy class of f ; see Lewis, May, Steinberger and McClure[23] and Mahowald[27]. Furthermore, if f is a loop map, then it follows from a result of Lewis[23] that T .f / inherits the structure of an A ring spectrum. In this case the topological Hochschild homology spectrum THH.1T .f // is defined. -
The First and Second Stiefel-Whitney Classes; Orientation and Spin Structure
The first and second Stiefel-Whitney classes; orientation and spin structure Robert Hines December 6, 2017 Classifying spaces and universal bundles Theorem 1. Given a topological group G, there is a space BG and a principal G-bundle EG such that for any space B (homotopy equivalent to a CW-complex), the principal G-bundles on B are in bijection with homotopy classes of maps f : B ! BG via pullback. I.e., given a principal G-bundle E ! B, there exists a map f : B ! BG such that f ∗EG =∼ E and f ∗EG =∼ g∗EG if and only if f and g are homotopy equivalent. 1 k For G = O(n), we can take BG = Grn(R ), a limit of the Grassmann manifolds Grn(R ) k 1 k of n-planes in R . Above this we have EG = Vn(R ), a limit of the Stiefel manifolds Vn(R ) of k orthogonal n-frames in R . The map EG ! BG is given by forgetting the framing. The fiber is clearly O(n). Stiefel-Whitney classes Vaguely, characteristic classes are cohomology classes associated to vector bundles (functorially) ∗ over a space B. We will be concerned with the Stiefel-Whitney classes in H (B; F2) associated to real vector bundles over B. These are mod 2 reductions of obstructions to finding (n − i + 1) linearly independent sections of an n-dimensional vector bundle over the i skeleton of B. i Theorem 2 ([M], Chapter 23). There are characteristic classes wi(ξ) 2 H (B; F2) associated to an n-dimensional real vector bundle ξ : E ! B that satisfy and are uniquely determined by • w0(ξ) = 1 and wi(ξ) = 0 for i > dim ξ, • wi(ξ ⊕ ) = wi(ξ) where is the trivial line bundle R × B, 1 • w1(γ1) 6= 0 where γ1 is the universal line bundle over RP , Pi • wi(ξ ⊕ ζ) = j=0 wj(ξ) [ wi−j(ζ). -
An Introduction to Characteristic Classes
An Introduction to Characteristic Classes Derek Sorensen December 18, 2017 When studying principal G-bundles, a useful tool for studying \how nontrivial" the bundles are (whatever that means) is to study its classifying map. If you look at the notes on principal G-bundles, we know that principal G-bundles over a CW-complex X are in bijective correspondence with the set [X; BG] of homotopy classes of maps. However, in most cases it is not at all straightforward to find and study these maps. Instead, we use the ever-useful tool of cohomology to study the maps on cohomology which are induced by the maps in [X; BG]. This gives us the definition of a characteristic class of a principal G-bundle. Definition 0.1. Let π : E ! B be a principal G-bundle with classifying map ['] 2 [B; BG]. If c 2 H∗(BG), then the characteristic class c(E) 2 H∗(B) is the image of c under the map '∗ : H∗(BG) ! H∗(B). Immediately from this definition, we have that if E1 and E2 are isomorphic principal G-bundles over X, then their characteristic classes are isomorphic. Likewise, if E = B × G is the trivial principal G-bundle, then its classifying map is nullhomotopic, and so its characteristic classes must be trivial. Theorem 0.2. Given any fiber bundle π : E ! B with fiber F and structure group Aut(F ), there exists a principal Aut(F )-bundle P such that E = P ×G F . Characteristic classes can also be defined for arbitrary fiber bundles with fiber F and structure group Aut(F ).