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Realizing Cohomology Classes As Euler Classes 1
Ó DOI: 10.2478/s12175-012-0057-2 Math. Slovaca 62 (2012), No. 5, 949–966 REALIZING COHOMOLOGY CLASSES AS EULER CLASSES Aniruddha C Naolekar (Communicated by J´ulius Korbaˇs ) s ◦ ABSTRACT. For a space X,letEk(X), Ek(X)andEk (X) denote respectively the set of Euler classes of oriented k-plane bundles over X, the set of Euler classes of stably trivial k-plane bundles over X and the spherical classes in Hk(X; Z). s ◦ We prove some general facts about the sets Ek(X), Ek(X)andEk (X). We also compute these sets in the cases where X is a projective space, the Dold manifold P (m, 1) and obtain partial computations in the case that X is a product of spheres. c 2012 Mathematical Institute Slovak Academy of Sciences 1. Introduction Given a topological space X, we address the question of which classes x ∈ Hk(X; Z) can be realized as the Euler class e(ξ)ofanorientedk-plane bundle ξ over X. We also look at the question of which integral cohomology classes are spherical. It is convenient to make the following definition. s º ≥ ÒØÓÒ 1.1 For a space X and an integer k 1, the sets Ek(X), Ek(X) ◦ and Ek (X) are defined to be k Ek(X)= e(ξ) ∈ H (X; Z) | ξ is an oriented k-plane bundle over X s k Ek(X)= e(ξ) ∈ H (X; Z) | ξ is a stably trivial k-plane bundle over X ◦ k k ∗ Ek (X)= x ∈ H (X; Z) | there exists f : S −→ X with f (x) =0 . -
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. -
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. -
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. -
The Atiyah-Singer Index Theorem*
CHAPTER 5 The Atiyah-Singer Index Theorem* Peter B. Gilkey Mathematics Department, University of Oregon, Eugene, OR 97403, USA E-mail: gilkey@ math. uo regon, edu Contents 0. Introduction ................................................... 711 1. Clifford algebras and spin structures ..................................... 711 2. Spectral theory ................................................. 718 3. The classical elliptic complexes ........................................ 725 4. Characteristic classes of vector bundles .................................... 730 5. Characteristic classes of principal bundles .................................. 737 6. The index theorem ............................................... 739 References ..................................................... 745 *Research partially supported by the NSF (USA) and MPIM (Germany). HANDBOOK OF DIFFERENTIAL GEOMETRY, VOL. I Edited by EJ.E. Dillen and L.C.A. Verstraelen 2000 Elsevier Science B.V. All fights reserved 709 The Atiyah-Singer index theorem 711 O. Introduction Here is a brief outline to the paper. In Section 1, we review some basic facts concerning Clifford algebras and spin structures. In Section 2, we discuss the spectral theory of self- adjoint elliptic partial differential operators and give the Hodge decomposition theorem. In Section 3, we define the classical elliptic complexes: de Rham, signature, spin, spin c, Yang-Mills, and Dolbeault; these elliptic complexes are all of Dirac type. In Section 4, we define the various characteristic classes for vector bundles that we shall need: Chern forms, Pontrjagin forms, Chern character, Euler form, Hirzebruch L polynomial, A genus, and Todd polynomial. In Section 5, we discuss the characteristic classes for principal bundles. In Section 6, we give the Atiyah-Singer index theorem; the Chern-Gauss-Bonnet formula, the Hirzebruch signature formula, and the Riemann-Roch formula are special cases of the index theorem. We also discuss the equivariant index theorem and the index theorem for manifolds with boundary. -
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. -
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. -
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 . -
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. -
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.