DOCSLIB.ORG

Equivariant Geometric K-Homology with Coefficients

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Equivariant Geometric K-Homology with Coefficients Equivariant geometric K-homology with coefficients Michael Walter Equivariant geometric K-homology with coefficients Diplomarbeit vorgelegt von Michael Walter geboren in Lahr angefertigt am Mathematischen Institut der Georg-August-Universität zu Göttingen 2010 v Equivariant geometric K-homology with coefficients Michael Walter Abstract K-homology is the dual of K-theory. Kasparov’s analytic version, where cycles are given by (ab- stract) elliptic operators over (not necessarily commutative) spaces, has proved to be an extremely powerful tool which, together with its bivariant generalization KK-theory, lies at the heart of many important results at the intersection of algebraic topology, functional analysis and geome- try. Independently, Baum and Douglas have proposed a geometric version of K-homology inspired by singular bordism. Cycles for this theory are given by vector bundles over compact Spinc- manifolds with boundary which map to the target, i.e. E f (M, BM) / (X, Y). There is a natural transformation to analytic K-homology defined by sending such a cycle to the pushforward of the class determined by the twisted Dirac operator. It is well-known to be an isomorphism, although a rigorous proof has appeared only recently. While both theories have obvious generalizations to the equivariant case and coefficients, the question whether these remain isomorphic is far from trivial (and has negative answer in the general case). In their work on equivariant correspondences Emerson and Meyer have isolated a useful sufficient condition for their theory which, while vastly more general, only deals with the absolute case. Our focus is not so much to construct a geometric theory in the most general situation, but to show that in the presence of a group action and coefficients the above picture still gives a generalized homology theory in a very geometrical way, isomorphic to Kasparov’s theory. In this thesis we will show that equivariant geometric and analytic K-homology for compact Lie group actions and unital coefficient algebras are naturally isomorphic generalized homology theories on a broad category of spaces, following an approach by Baum, Oyono-Oyono and Schick. Chapter 1 gives a modern overview of both analytic K-theory and K-homology in terms of Kas- parov’s equivariant KK-theory. On the way we generalize several well-known results and con- structions to the equivariant case and coefficients, in particular Swan’s theorem, Poincaré duality, Thom isomorphisms and functorial Gysin maps. For the latter we work out in some detail the relationship between its geometrical and analytical descriptions. Chapter 2 introduces geometric K-homology groups by defining cycles and equivalence relations. By translating the latter into analytical terms we show that the natural transformation from geo- metric to analytic cycles is well-defined on the level of homology classes. Using the Mostow embedding theorem we then prove that this map is in fact an isomorphism. vi Acknowledgements. It is a pleasure to acknowledge Prof. Thomas Schick for inspiring discus- sions and helpful advice, and for sharing his enthusiasm for doing mathematics. Contents Abstract v Chapter 1. Analytic K-theory and K-homology 1 ¦ 1.1. C -algebras and topological spaces with group actions 1 ¦ 1.2. Hilbert G-C -modules 5 1.3. Equivariant KK-theory 9 1.4. Analytic K-theory 15 1.5. Topological K-theory 18 1.6. Analytic K-homology 25 1.7. Products 25 1.8. Elliptic operators and analytic K-homology 29 1.9. Dirac operators 34 1.10. Spinc-structure and spinor bundles 40 1.11. Fundamental classes and Poincaré duality 51 1.12. The double of a manifold 60 1.13. Equivariant Spinc-structure of the spheres 62 1.14. Bott periodicity and Thom isomorphism 65 1.15. Gysin maps 70 Chapter 2. Geometric K-homology 75 2.1. Cycles and equivalence relations 75 2.2. Geometric K-homology groups 79 2.3. Isomorphism between geometric and analytic K-homology 82 Outlook 87 Bibliography 89 Notation Index 91 Subject Index 95 vii CHAPTER 1 Analytic K-theory and K-homology In his landmark article [Kas81], Kasparov introduced a bivariant theory, KK-theory, which uni- fied both K-theory and K-homology. Its main feature is the existence of the Kasparov product, an associative pairing KK(A, B) ¢ KK(B, C) Ñ KK(A, C) which unifies and generalizes most constructions of the preceding theories. Pushforwards and pullbacks, boundary maps and the pairing of K-theory and K-homology, as well also more in- volved constructions such as Bott periodicity and the Thom isomorphism can all be expressed in terms of Kasparov products with suitable elements. In the subsequent paper [Kas88], KK-theory was extended to the equivariant setting. In this chapter, we start by giving a modern account of analytic K-theory and K-homology which we define in terms of Kasparov’s equivariant KK-bifunctor (Sections 1.1–1.7). We shall also gen- eralize the classical topological picture of K-theory to the equivariant case and coefficients (Section 1.5). Key ingredient to this latter result is a version of Swan’s theorem for equivariant bundles of finitely generated, projective Hilbert modules. The second part of the chapter focuses on connecting analysis with geometry. We recall how symmetric elliptic operators, in particular Dirac operators, define cycles in K-homology and show that it is possible to include into these the action of more general algebras which are similarly well-behaved as C0(M) (Section 1.8). Kasparov’s Dirac element fits naturally into this setting, as do the fundamental classes determined by manifolds equipped with an equivariant Spinc-structure, the K-theory equivalent of an orientation (Sections 1.9–1.11). We will in fact see that both elements are but different sides of the same medal. This will be our way of deducing Poincaré duality. With this machinery in place, we proceed to re-derive Bott periodicity and the Thom isomorphism by a careful analysis of the Spinc-structure on even-dimensional spheres (Sections 1.13 and 1.14). Our proof will give both a topological and an analytic description of these results which will be crucial for the second part of the thesis. In the last section we similarly define “wrong-way” functorial Gysin maps first in an analytical way and then derive concrete geometric descriptions for them (Section 1.15). Although Kasparov’s theory can be developed in a more general context, it is sufficient for our ¦ purposes to only consider separable complex C -algebras, and we can avoid much technical trou- ¦ ble that way. See [Bla98, Chapter 14] for a convenient review of the basic theory of C -algebras and [Bla06] for a more detailed account. We shall also require that all topological spaces be locally compact, second-countable and Hausdorff and that topological groups even be compact. ¦ 1.1. C -algebras and topological spaces with group actions ¦ ¦ In this section we will recall the fundamental notion of a G-C -algebra, that is, a C -algebra equipped with a continuous group action by automorphisms. We shall always identify a group element with the automorphism by which it acts. ¦ ¦ 1.1.1 DEFINITION (G-CONTINUITY, G-C -ALGEBRA). Let A be a C -algebra equipped with a (left) action of a (compact topological) group G by automorphisms. An element a P A is called G- continuous if the orbit map g ÞÑ g(a) is continuous. 1 2 1. ANALYTIC K-THEORY AND K-HOMOLOGY ¦ The category of G-C -algebras (G-algebra in [Kas88], covariant system in [Bla98]) is then defined ¦ as follows. Its objects are (separable) C -algebras with an action of G by automorphisms such that every element is G-continuous. In other words, the associated group homomorphism G Ñ Aut(A) is strongly continuous. Morphisms are G-equivariant ¦-homomorphisms. ¦ The category of (Z/2-)graded G-C -algebras is defined similarly except that we require both the ¦ group actions and morphisms to preserve the grading. Unless noted otherwise, all G-C -algebras are assumed to be ungraded (or, equivalently, trivially graded). We will always denote the degree of homogeneous elements a by Ba. ¦ We will always consider the complex numbers C as a trivial G-C -algebra. ¦ Recall that the tensor product A bp B of graded G-C -algebras A and B can be formed in different ways by completing the skew-commutative algebraic tensor product A bp B. We will always use the minimal/spatial tensor product from [Kas81, Section 2]. There exist canonical braiding iso- B B morphisms A bp B B bp A sending a tensor a b b to (¡1) a bb b a such that the class of graded ¦ G-C -algebras forms a symmetric monoidal category. We write A b B if both algebras are trivially ¦ graded. A detailed treatment of tensor products of C -algebras can be found in [WO93, Appen- dix T]; for the graded case see [Bla98, Section 14]. ¦ Most of the time, one of the factors in our tensor products will be a nuclear C -algebra so that all ¦ C -tensor products are in fact equal. We will make repeated use of the following facts: ¦ 1.1.2 PROPOSITION ([Bla06, Theorem 15.8.2]). The following classes of C -algebras are nuclear: Com- ¦ ¦ ¦ mutative C -algebras, finite-dimensional C -algebras, C -algebras of compact operators on a separable Hilbert space. ¦ Moreover, the class of nuclear C -algebras is closed under direct limits, extensions, ideals and quotients. 1.1.3 PROPOSITION ([WO93, Theorem T.6.26]). Let 1 2 0 / A / A / A / 0 ¦ ¦ ¦ 2 be a short exact sequence of C -algebras and B another C -algebra.
Load more

Recommended publications
  • Algebraic Topology
    ALGEBRAIC TOPOLOGY C.R. F. MAUNDER ALGEBRAIC TOPOLOGY C. R. F. MAUNDER Fellow of Christ's College and University Lecturer in Pure Mathematics, Cambridge CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE LONDON NEW YORK NEW ROCHELLE MELBOURNE SYDNEY Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge C82inP 32East 57th Street, New York, NYzoozz,USA 296 Beaconsfield Parade, Middle Park, Melbourne 3206, Australia CC. R. F. Miunder '970 CCambridge University Press 1980 First published by VanNostrandReinhold (UK) Ltd First published by the Cambridge University Press 1980 Firstprinted in Great Britain by Lewis Reprints Ltd, London and Tonbridge Reprinted in Great Britain at the University Press, Cambridge British Library cataloguing in publication data Maunder, Charles Richard Francis Algebraic topology. r.Algebraic topology I. Title 514'.2QA6!2 79—41610 ISBN 0521 231612 hard covers ISBN 0 521298407paperback INTRODUCTION Most of this book is based on lectures to third-year undergraduate and postgraduate students. It aims to provide a thorough grounding in the more elementary parts of algebraic topology, although these are treated wherever possible in an up-to-date way. The reader interested in pursuing the subject further will find ions for further reading in the notes at the end of each chapter. Chapter 1 is a survey of results in algebra and analytic topology that will be assumed known in the rest of the book. The knowledgeable reader is advised to read it, however, since in it a good deal of standard notation is set up. Chapter 2 deals with the topology of simplicial complexes, and Chapter 3 with the fundamental group.
    [Show full text]
  • Combinatorial Topology and Applications to Quantum Field Theory
    Combinatorial Topology and Applications to Quantum Field Theory by Ryan George Thorngren A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Vivek Shende, Chair Professor Ian Agol Professor Constantin Teleman Professor Joel Moore Fall 2018 Abstract Combinatorial Topology and Applications to Quantum Field Theory by Ryan George Thorngren Doctor of Philosophy in Mathematics University of California, Berkeley Professor Vivek Shende, Chair Topology has become increasingly important in the study of many-body quantum mechanics, in both high energy and condensed matter applications. While the importance of smooth topology has long been appreciated in this context, especially with the rise of index theory, torsion phenomena and dis- crete group symmetries are relatively new directions. In this thesis, I collect some mathematical results and conjectures that I have encountered in the exploration of these new topics. I also give an introduction to some quantum field theory topics I hope will be accessible to topologists. 1 To my loving parents, kind friends, and patient teachers. i Contents I Discrete Topology Toolbox1 1 Basics4 1.1 Discrete Spaces..........................4 1.1.1 Cellular Maps and Cellular Approximation.......6 1.1.2 Triangulations and Barycentric Subdivision......6 1.1.3 PL-Manifolds and Combinatorial Duality........8 1.1.4 Discrete Morse Flows...................9 1.2 Chains, Cycles, Cochains, Cocycles............... 13 1.2.1 Chains, Cycles, and Homology.............. 13 1.2.2 Pushforward of Chains.................. 15 1.2.3 Cochains, Cocycles, and Cohomology.........
    [Show full text]
  • 256B Algebraic Geometry
    256B Algebraic Geometry David Nadler Notes by Qiaochu Yuan Spring 2013 1 Vector bundles on the projective line This semester we will be focusing on coherent sheaves on smooth projective complex varieties. The organizing framework for this class will be a 2-dimensional topological field theory called the B-model. Topics will include 1. Vector bundles and coherent sheaves 2. Cohomology, derived categories, and derived functors (in the differential graded setting) 3. Grothendieck-Serre duality 4. Reconstruction theorems (Bondal-Orlov, Tannaka, Gabriel) 5. Hochschild homology, Chern classes, Grothendieck-Riemann-Roch For now we'll introduce enough background to talk about vector bundles on P1. We'll regard varieties as subsets of PN for some N. Projective will mean that we look at closed subsets (with respect to the Zariski topology). The reason is that if p : X ! pt is the unique map from such a subset X to a point, then we can (derived) push forward a bounded complex of coherent sheaves M on X to a bounded complex of coherent sheaves on a point Rp∗(M). Smooth will mean the following. If x 2 X is a point, then locally x is cut out by 2 a maximal ideal mx of functions vanishing on x. Smooth means that dim mx=mx = dim X. (In general it may be bigger.) Intuitively it means that locally at x the variety X looks like a manifold, and one way to make this precise is that the completion of the local ring at x is isomorphic to a power series ring C[[x1; :::xn]]; this is the ring where Taylor series expansions live.
    [Show full text]
  • Algebraic Topology
    Algebraic Topology John W. Morgan P. J. Lamberson August 21, 2003 Contents 1 Homology 5 1.1 The Simplest Homological Invariants . 5 1.1.1 Zeroth Singular Homology . 5 1.1.2 Zeroth deRham Cohomology . 6 1.1.3 Zeroth Cecˇ h Cohomology . 7 1.1.4 Zeroth Group Cohomology . 9 1.2 First Elements of Homological Algebra . 9 1.2.1 The Homology of a Chain Complex . 10 1.2.2 Variants . 11 1.2.3 The Cohomology of a Chain Complex . 11 1.2.4 The Universal Coefficient Theorem . 11 1.3 Basics of Singular Homology . 13 1.3.1 The Standard n-simplex . 13 1.3.2 First Computations . 16 1.3.3 The Homology of a Point . 17 1.3.4 The Homology of a Contractible Space . 17 1.3.5 Nice Representative One-cycles . 18 1.3.6 The First Homology of S1 . 20 1.4 An Application: The Brouwer Fixed Point Theorem . 23 2 The Axioms for Singular Homology and Some Consequences 24 2.1 The Homotopy Axiom for Singular Homology . 24 2.2 The Mayer-Vietoris Theorem for Singular Homology . 29 2.3 Relative Homology and the Long Exact Sequence of a Pair . 36 2.4 The Excision Axiom for Singular Homology . 37 2.5 The Dimension Axiom . 38 2.6 Reduced Homology . 39 1 3 Applications of Singular Homology 39 3.1 Invariance of Domain . 39 3.2 The Jordan Curve Theorem and its Generalizations . 40 3.3 Cellular (CW) Homology . 43 4 Other Homologies and Cohomologies 44 4.1 Singular Cohomology .
    [Show full text]
  • Ambidexterity in K(N)-Local Stable Homotopy Theory
    Ambidexterity in K(n)-Local Stable Homotopy Theory Michael Hopkins and Jacob Lurie December 19, 2013 Contents 1 Multiplicative Aspects of Dieudonne Theory 4 1.1 Tensor Products of Hopf Algebras . 6 1.2 Witt Vectors . 12 1.3 Dieudonne Modules . 19 1.4 Disconnected Formal Groups . 28 2 The Morava K-Theory of Eilenberg-MacLane Spaces 34 2.1 Lubin-Tate Spectra . 35 2.2 Cohomology of p-Divisible Groups . 41 2.3 The Spectral Sequence of a Filtered Spectrum . 47 2.4 The Main Calculation . 50 3 Alternating Powers of p-Divisible Groups 63 3.1 Review of p-Divisible Groups . 64 3.2 Group Schemes of Alternating Maps . 68 3.3 The Case of a Field . 74 3.4 Lubin-Tate Cohomology of Eilenberg-MacLane Spaces . 82 3.5 Alternating Powers in General . 86 4 Ambidexterity 89 4.1 Beck-Chevalley Fibrations and Norm Maps . 91 4.2 Properties of the Norm . 96 4.3 Local Systems . 103 4.4 Examples . 107 5 Ambidexterity of K(n)-Local Stable Homotopy Theory 112 5.1 Ambidexterity and Duality . 113 5.2 The Main Theorem . 121 5.3 Cartier Duality . 126 5.4 The Global Sections Functor . 135 1 Introduction Let G be a finite group, and let M be a spectrum equipped with an action of G. We let M hG denote the homotopy fixed point spectrum for the action of G on M, and MhG the homotopy orbit spectrum for the hG action of G on X. These spectra are related by a canonical norm map Nm : MhG ! M .
    [Show full text]
  • Algebraic Topology 565 2016
    Algebraic Topology 565 2016 February 21, 2016 The general format of the course is as in Fall quarter. We'll use Hatcher Chapters 3-4, and we'll start using selected part of Milnor's Characteristic classes. One global notation change: From now on we fix a principal ideal domain R, and let H∗X denote H∗(X; R). This change extends in the obvious way to other notations: C∗X means RS·X (singular chains with coefficients in R), tensor products and Tor are over R, cellular homology is with coefficients in R, and so on. Course outline Note: Applications and exercises are still to appear. Some parts of the outline below won't make sense yet, but still serve to give the general idea. 1. Homology of products. There are two steps to compute the homology of a product ∼ X × Y : (i) The Eilenberg-Zilber theorem that gives a chain equivalence C∗X ⊗ C∗Y = C∗(X × Y ), and (ii) the purely algebraic Kunneth theorem that expresses the homology of a tensor product of chain complexes in terms of tensor products and Tor's of the homology of the individual complexes. I plan to only state part (ii) and leave the proof to the text. On the other hand, I'll take a very different approach to (i), not found in Hatcher: the method of acyclic models. This more categorical approach is very slick and yields easy proofs of some technical theorems that are crucial for the cup product in cohomology later. There are explicit formulas for certain choices of the above chain equivalence and its inverse; in particular there is a very simple and handy formula for an inverse, known as the Alexander- Whitney map.
    [Show full text]
  • Positivity in Algebraic Geometry I
    Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 48 Positivity in Algebraic Geometry I Classical Setting: Line Bundles and Linear Series Bearbeitet von R.K. Lazarsfeld 1. Auflage 2004. Buch. xviii, 387 S. Hardcover ISBN 978 3 540 22533 1 Format (B x L): 15,5 x 23,5 cm Gewicht: 1650 g Weitere Fachgebiete > Mathematik > Geometrie > Elementare Geometrie: Allgemeines Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. Introduction to Part One Linear series have long stood at the center of algebraic geometry. Systems of divisors were employed classically to study and define invariants of pro- jective varieties, and it was recognized that varieties share many properties with their hyperplane sections. The classical picture was greatly clarified by the revolutionary new ideas that entered the field starting in the 1950s. To begin with, Serre’s great paper [530], along with the work of Kodaira (e.g. [353]), brought into focus the importance of amplitude for line bundles. By the mid 1960s a very beautiful theory was in place, showing that one could recognize positivity geometrically, cohomologically, or numerically. During the same years, Zariski and others began to investigate the more complicated be- havior of linear series defined by line bundles that may not be ample.
    [Show full text]
  • Cambridge University Press 978-1-107-15074-4 — Singular Intersection Homology Greg Friedman Index More Information
    Cambridge University Press 978-1-107-15074-4 — Singular Intersection Homology Greg Friedman Index More Information Index KEY: Numbers in bold indicate where terms are defined ∂-pseudomanifold, see pseudomanifold, effects of simplicial subdivision on, 96–98 ∂-pseudomanifold in small degrees, 130 ∂-stratified pseudomanifold, see stratified on regular strata, 96, 130 pseudomanifold, ∂-stratified pseudomanifold PL chain, 121 abstract simplicial complex, see simplicial complex, simplex, 92 abstract singular chain, 129 acknowledgments, xiv singular simplex, 129 acyclic model, 199 strata vs. skeleta, 105–107 Acyclic Model Theorem, 368 Aluffi, Paolo, 707 admissible triangulation, see triangulation, admissible assembly map, 697, 702 agreeable triple of perversities, 372, 376 AT map, 751 ( , ) AT space, 744, 751 Qp¯ ,t¯Y Qt¯X ,q¯ ; Q , 434 (0¯, t¯; 0¯), 373 augmentation cocycle (1), xxi (n¯, n¯; 0¯), 374 augmentation map (a), xx see (p¯, Dp¯; 0¯), 373 augmented intersection chain complex, (p¯, t¯;¯p), 374 intersection chain complex, augmented (t¯, 0;¯ 0¯), 373 balls, 189 and diagonal maps, 372 Banagl, Markus, xiv, 695, 697, 699, 701, 704, 707, 708 and projection maps, 398 basic sets, 607 in terms of dual perversities, 373 Bernstein, Joseph, 710 Akin, Ethan, 692, 706 Beshears, Aaron, 59 Albin, Pierre, 706, 707, 709 Be˘ılinson, Alexander, 710 Alexander–Whitney map, 199, 365–367 bilinear pairing, see pairing intersection version, see intersection Bockstein map, 233 Alexander–Whitney map (IAW) bordism, 689–702 algebra background, 713–738 bordism group, 690
    [Show full text]
  • UC Berkeley UC Berkeley Electronic Theses and Dissertations
    UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Combinatorial Topology and Applications to Quantum Field Theory Permalink https://escholarship.org/uc/item/7r44w49f Author Thorngren, Ryan George Publication Date 2018 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Combinatorial Topology and Applications to Quantum Field Theory by Ryan George Thorngren A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Vivek Shende, Chair Professor Ian Agol Professor Constantin Teleman Professor Joel Moore Fall 2018 Abstract Combinatorial Topology and Applications to Quantum Field Theory by Ryan George Thorngren Doctor of Philosophy in Mathematics University of California, Berkeley Professor Vivek Shende, Chair Topology has become increasingly important in the study of many-body quantum mechanics, in both high energy and condensed matter applications. While the importance of smooth topology has long been appreciated in this context, especially with the rise of index theory, torsion phenomena and dis- crete group symmetries are relatively new directions. In this thesis, I collect some mathematical results and conjectures that I have encountered in the exploration of these new topics. I also give an introduction to some quantum field theory topics I hope will be accessible to topologists. 1 To my loving parents, kind friends, and patient teachers. i Contents I Discrete Topology Toolbox1 1 Basics4 1.1 Discrete Spaces..........................4 1.1.1 Cellular Maps and Cellular Approximation.......6 1.1.2 Triangulations and Barycentric Subdivision......6 1.1.3 PL-Manifolds and Combinatorial Duality........8 1.1.4 Discrete Morse Flows...................9 1.2 Chains, Cycles, Cochains, Cocycles..............
    [Show full text]
  • Spanier-Whitehead K-Duality for $ C^* $-Algebras
    SPANIER-WHITEHEAD K-DUALITY FOR C∗-ALGEBRAS JEROME KAMINKER AND CLAUDE L. SCHOCHET Abstract. Classical Spanier-Whitehead duality was introduced for the stable homotopy category of finite CW complexes. Here we provide a comprehensive treatment of a noncommutative version, termed Spanier-Whitehead K-duality, which is defined on the category of C∗-algebras whose K-theory is finitely generated and that satisfy the UCT, with morphisms the KK-groups. We explore what happens when these assumptions are relaxed in various ways. In particular, we consider the relationship between Paschke duality and Spanier- Whitehead K-duality. Contents 1. Introduction 1 2. Spanier-Whitehead K- Duality 4 3. Fitting Classical Spanier-Whitehead duality into the Spanier-Whitehead K-duality framework 8 4. Examples of noncommutative duality 10 4.1. Hyperbolic dynamics 10 4.2. Baum-Connes conjecture 11 4.3. Mukai transform 12 5. Poincar´eduality 12 6. Existence of Spanier-Whitehead K-Duals 14 7. Non-existence of Spanier-Whitehead K-Duals 15 8. Mod-p K-theory 16 9. Paschke Duality 17 ∗ 10. C -substitutes I: K∗(A) countable 19 11. C∗-substitutesII:bootstrapentries 22 References 27 arXiv:1609.00409v4 [math.OA] 13 Jun 2017 1. Introduction Classical Spanier-Whitehead duality is a generalization of Alexander duality, which relates the homology of a space to the cohomology of its complement in a sphere. Ed Spanier and J.H.C. Whitehead [42], [43], noting that the dimension of the sphere did not play an essential role, adapted it to the context of stable homo- topy theory. Its history and its relation to other classical duality ideas are described in depth by Becker and Gottlieb [4].
    [Show full text]
  • Arxiv:1606.04233V2 [Math.AT] 21 Sep 2016 Ls.I 7,M Oek N .Mchro Rv Htthe That Prove Macpherson R
    SINGULAR DECOMPOSITIONS OF A CAP PRODUCT DAVID CHATAUR, MARTINTXO SARALEGI-ARANGUREN, AND DANIEL TANRE´ Abstract. In the case of a compact orientable pseudomanifold, a well-known theorem of M. Goresky and R. MacPherson says that the cap product with a fundamental class factorizes through the intersection homology groups. In this work, we show that this classical cap product is compatible with a cap product in intersection (co)- homology, that we have previously introduced. If the pseudomanifold is also normal, for any commutative ring of coefficients, the existence of a classical Poincar´eduality isomorphism is equivalent to the existence of an isomorphism between the intersection homology groups corresponding to the zero and the top perversities. Let X be a compact oriented pseudomanifold and [X] ∈ Hn(X; Z) be its fundamental class. In [7], M. Goresky and R. MacPherson prove that the Poincar´eduality map k defined by the cap product −∩ [X]: H (X; Z) → Hn−k(X; Z) can be factorized as p p k α p β H (X; Z) −→ Hn−k(X; Z) −→ Hn−k(X; Z), (1) p where the groups Hi (X; Z) are the intersection homology groups for the perversity p. The study of the Poincar´eduality map via a filtration on homology classes is also considered in the thesis of C. McCrory [10], [11], using a Zeeman’s spectral sequence. In [5, Section 8.1.6], G. Friedman asks for a factorization of the Poincar´eduality map through a cap product defined in the context of an intersection cohomology recalled in Section 4. He proves it with a restriction on the torsion part of the intersection coho- mology.
    [Show full text]
  • Introduction to Algebraic Topology and Algebraic
    INTERNATIONAL SCHOOL FOR ADVANCED STUDIES Trieste U. Bruzzo INTRODUCTION TO ALGEBRAIC TOPOLOGY AND ALGEBRAIC GEOMETRY Notes of a course delivered during the academic year 2002/2003 La filosofia `escritta in questo grandissimo libro che continuamente ci sta aperto innanzi a gli occhi (io dico l'universo), ma non si pu`o intendere se prima non si impara a intender la lingua, e conoscer i caratteri, ne' quali `escritto. Egli `escritto in lingua matematica, e i caratteri son triangoli, cerchi, ed altre figure geometriche, senza i quali mezi `eimpossibile a intenderne umanamente parola; senza questi `eun aggirarsi vanamente per un oscuro laberinto. Galileo Galilei (from \Il Saggiatore") i Preface These notes assemble the contents of the introductory courses I have been giving at SISSA since 1995/96. Originally the course was intended as introduction to (complex) algebraic geometry for students with an education in theoretical physics, to help them to master the basic algebraic geometric tools necessary for doing research in algebraically integrable systems and in the geometry of quantum field theory and string theory. This motivation still transpires from the chapters in the second part of these notes. The first part on the contrary is a brief but rather systematic introduction to two topics, singular homology (Chapter 2) and sheaf theory, including their cohomology (Chapter 3). Chapter 1 assembles some basics fact in homological algebra and develops the first rudiments of de Rham cohomology, with the aim of providing an example to the various abstract constructions. Chapter 5 is an introduction to spectral sequences, a rather intricate but very pow- erful computation tool.
    [Show full text]