DOCSLIB.ORG

Equivariant Homotopy and Cohomology

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

EQUIVARIANT HOMOTOPY AND COHOMOLOGY THEORY May JP The author acknowledges the supp ort of the NSF Mathematics Subject Classication Revision Primary L M N N N N N P P P P Q R R T R Secondary E G G A P P P Q Q R R S S S U U R R S Author addresses University of Chicago Chicago Il Email address maymathuchicagoedu ii Contents Intro duction Chapter I Equivariant Cellular and Homology Theory Some basic denitions and adjunctions Analogs for based Gspaces GCW complexes Ordinary homology and cohomology theories Obstruction theory ersal co ecient sp ectral sequences Univ Chapter I I P ostnikov Systems Lo calization and Completion Gspaces and Postnikov systems Eilenb ergMacLane lo calizations of spaces Summary Lo calizations of Gspaces Summary completions of spaces Completions of Gspaces Chapter I I I Equivariant Rational Homotopy Theory Summary the theory of minimal mo dels Equivariant minimal mo dels Rational equivariant Hopf spaces Chapter IV Smith Theory cohomology Smith theory via Bredon iii iv CONTENTS Borel cohomology lo calization and Smith theory Equivariant Applications Chapter V Categorical Constructions Co ends and geometric realization Homotopy colimits and limits Elmendorf s theorem on diagrams of xed p oint spaces spaces Eilenb ergMac Lane Gspaces and universal F Chapter VI The Homotopy Theory of Diagrams Elementary homotopy theory of diagrams Homotopy Groups Cellular Theory The homology and cohomology theory of diagrams J The closed mo del structure on U Another pro of of Elmendorf s theorem Chapter VI I Equivariant Bundle theory and Classifying Spaces The denition of equivariant bundles The classication of equivariant bundles Some examples of classifying spaces Chapter VI I I The Sullivan Conjecture Statements of versions of the Sullivan conjecture and Fix Algebraic preliminaries Lannes functors T Lannes generalization of the Sullivan conjecture Sketch pro of of Lannes theorem Maps b etw een classifying spaces An intro duction to equivariant stable homotopy Chapter IX Gspheres in homotopy theory GUniverses and stable Gmaps Euler characteristic and transfer Gmaps Mackey functors and coMackey functors Ggraded homology and cohomology RO CONTENTS v The Conner conjecture V complexes and RO Ggraded cohomology Chapter X GCW Motivation for cellular theories based on representations GCWV complexes Homotopy theory of GCWV complexes Ordinary RO Ggraded homology and cohomology Chapter XI The equivariant Hurewicz and Susp ension Theorems Background on the classical theorems Formulation of the problem and counterexamples An oversimplied description of the results The statements of the theorems Sketch pro ofs of the theorems Chapter XI I The Equivariant Stable Homotopy Category An intro ductory overview Presp ectra and sp ectra Smash pro ducts Function sp ectra The equivariant case Spheres and homotopy groups GCW sp ectra Stability of the stable category Getting into the stable category Chapter XI I I RO Ggraded homology and cohomology theories Axioms for ROGgraded cohomology theories Representing RO Ggraded theories by Gsp ectra Browns theorem and RO Ggraded cohomology Equivariant Eilenb ergMacLane sp ectra Ring Gsp ectra and pro ducts theory Chapter XIV An intro duction to equivariant K vi CONTENTS The denition and basic prop erties of K theory G Bundles over a p oint the representation ring Equivariant Bott p erio dicity Equivariant K theory sp ectra The AtiyahSegal completion theorem The generalization to families Chapter XV An intro duction to equivariant cob ordism A review of nonequivariant cob ordism Equivariant cob ordism and Thom sp ectra Computations the use of families Sp ecial cases o dd order groups and Z Chapter XVI Sp ectra and Gsp ectra change of groups duality Fixed p oint sp ectra and orbit sp ectra Split Gsp ectra and free Gsp ectra Geometric xed p oint sp ectra uller isomorphism Change of groups and the Wirthm Quotien t groups and the Adams isomorphism sp ectra from Gsp ectra The construction of GN SpanierWhitehead duality V duality of Gspaces and Atiyah duality Poincar e duality Chapter XVI I The Burnside ring Generalized Euler characteristics and transfer maps G The Burnside ring AG and the zero stem S 0 Prime ideals of the Burnside ring Idemp otent elements of the Burnside ring Lo calizations of the Burnside ring Lo calization of equivariant homology and cohomology Chapter XVI I I Transfer maps in equivariant bundle theory CONTENTS vii The transfer and a dimensionshifting variant Basic prop erties of transfer maps Smash pro ducts and Euler characteristics The double coset formula and its applications Transitivity of the transfer Chapter XIX Stable homotopy and Mackey functors The splitting of equivariant stable homotopy groups Generalizations of the splitting theorems Equivalent denitions of Mackey functors Induction theorems Splittings of rational Gsp ectra for nite groups G Chapter XX The Segal conjecture The statement in terms of completions of Gsp ectra A calculational reformulation groups A generalization and the reduction to p The pro of of the Segal conjecture for nite pgroups Approximations of singular subspaces of Gspaces An inverse limit of Adams sp ectral sequences Further generalizations maps b etween classifying spaces Chapter XXI Generalized Tate cohomology Denitions and basic prop erties Ordinary theories AtiyahHirzebruch sp ectral sequences Cohomotopy p erio dicity and ro ot invariants The generalization to families Equivariant K theory Further calculations and applications Chapter XXI I Brave new algebra The category of S mo dules Categories of Rmo dules viii CONTENTS The algebraic theory of Rmo dules The homotopical theory of Rmo dules Categories of Ralgebras Bouseld lo calizations of Rmo dules and algebras Top ological Ho chschild homology and cohomology Chapter XXI I I Brave new equivariant foundations Twisted halfsmash pro ducts sp ectra The category of L A ring sp ectra and S algebras and E 1 1 Alternative p ersp ectives on equivariance The construction of equivariant algebras and mo dules Comparisons of categories of LGsp ectra Equivariant Algebra Chapter XXIV Brave New Intro duction Cech cohomology in algebra Lo cal and Brave new versions of lo cal and Cech cohomology Lo calization theorems in equivariant homology Completions completion theorems and lo cal homology A pro of and generalization of the lo calization theorem The application to K theory Lo cal Tate cohomology Chapter XXV Lo calization and completion in complex b ordism The lo calization theorem for stable complex b ordism An outline of the pro of The norm map and its prop erties The idea b ehind the construction of norm maps functors with smash pro duct Global I The denition of the norm map The splitting of M U as an algebra G CONTENTS ix Loers completion conjecture Chapter XXVI Some calculations in complex equivariant b ordism Notations and terminology Stably almost complex structures and b ordism Tangential structures Calculational to ols Statements of the main results 1 S Preliminary lemmas and families in G 1 On the families F in G S i 1 Passing from G to G S and G Z k Bibliography Intro duction This volume b egan with Bob Piacenzas suggestion that I b e the principal lecturer Regional Conference in Fairbanks Alaska That event to ok at an NSFCBMS place in August of and the interim has seen very substantial progress in this general area of mathematics The scop e of this volume has grown accordingly ariant algebraic top ology to sta The original fo cus was an intro duction to equiv ble homotopy theory and to equivariant stable homotopy theory that was geared towards graduate students with a reasonably go o d understanding of nonequivari ant algebraic top ology More recent material is changing the direction of the last by allowing the intro duction of p ointset top ological algebra into sta two sub jects ble homotopy theory b oth equivariant and nonequivariant and the last p ortion of the b o ok fo cuses on an intro duction to these new developments There is a progression with the later p ortions of the b o ok on the whole b eing more dicult than the earlier p ortions Equivariant algebraic top ology concerns the study of algebraic invariants of two chapters intro duce the basic structural spaces with group actions The rst foundations of the sub ject cellular theory ordinary homology and cohomology theory Eilenb ergMac Lane Gspaces Postnikov systems lo calizations of Gspaces and completions of Gspaces In most of this work G can b e any top ological group but we restrict attention to compact Lie groups in the rest of the b o ok Chapter I I I on equivariant rational homotopy theory was written by Georgia Triantallou In it she shows how to generalize Sullivans theory of minimal mo dels to obtain an algebraization of the homotopy category of nilp otent G spaces for a nite group G This chapter contains a rst surprise rational Hopf Gspaces need not split as pro ducts of Eilenb ergMac Lane Gspaces This is a hint INTRODUCTION that the calculational b ehavior of equivariant algebraic top ology is more intricate and dicult to determine than that of the classical nonequivariant theory gives two pro ofs of the rst main theorem of equivariant algebraic Chapter IV theory any xed p oint space
Recommended publications
  • Topological Pattern Recognition for Point Cloud Data∗

    Topological Pattern Recognition for Point Cloud Data∗

    Acta Numerica (2014), pp. 289–368 c Cambridge University Press, 2014 doi:10.1017/S0962492914000051 Printed in the United Kingdom Topological pattern recognition for point cloud data∗ Gunnar Carlsson† Department of Mathematics, Stanford University, CA 94305, USA E-mail: [email protected] In this paper we discuss the adaptation of the methods of homology from algebraic topology to the problem of pattern recognition in point cloud data sets. The method is referred to as persistent homology, and has numerous applications to scientific problems. We discuss the definition and computation of homology in the standard setting of simplicial complexes and topological spaces, then show how one can obtain useful signatures, called barcodes, from finite metric spaces, thought of as sampled from a continuous object. We present several different cases where persistent homology is used, to illustrate the different ways in which the method can be applied. CONTENTS 1 Introduction 289 2 Topology 293 3 Shape of data 311 4 Structures on spaces of barcodes 331 5 Organizing data sets 343 References 365 1. Introduction Deriving knowledge from large and complex data sets is a fundamental prob- lem in modern science. All aspects of this problem need to be addressed by the mathematical and computational sciences. There are various dif- ferent aspects to the problem, including devising methods for (a) storing massive amounts of data, (b) efficiently managing it, and (c) developing un- derstanding of the data set. The past decade has seen a great deal of devel- opment of powerful computing infrastructure, as well as methodologies for ∗ Colour online for monochrome figures available at journals.cambridge.org/anu.
  • Algebraic K-Theory and Equivariant Homotopy Theory

    Algebraic K-Theory and Equivariant Homotopy Theory

    Algebraic K-Theory and Equivariant Homotopy Theory Vigleik Angeltveit (Australian National University), Andrew J. Blumberg (University of Texas at Austin), Teena Gerhardt (Michigan State University), Michael Hill (University of Virginia), and Tyler Lawson (University of Minnesota) February 12- February 17, 2012 1 Overview of the Field The study of the algebraic K-theory of rings and schemes has been revolutionized over the past two decades by the development of “trace methods”. Following ideas of Goodwillie, Bokstedt¨ and Bokstedt-Hsiang-¨ Madsen developed topological analogues of Hochschild homology and cyclic homology and a “trace map” out of K-theory that lands in these theories [15, 8, 9]. The fiber of this map can often be understood (by work of McCarthy and Dundas) [27, 13]. Topological Hochschild homology (THH) has a natural circle action, and topological cyclic homology (TC) is relatively computable using the methods of equivariant stable homotopy theory. Starting from Quillen’s computation of the K-theory of finite fields [28], Hesselholt and Madsen used TC to make extensive computations in K-theory [16, 17], in particular verifying certain cases of the Quillen-Lichtenbaum conjecture. As a consequence of these developments, the modern study of algebraic K-theory is deeply intertwined with development of computational tools and foundations in equivariant stable homotopy theory. At the same time, there has been a flurry of renewed interest and activity in equivariant homotopy theory motivated by the nature of the Hill-Hopkins-Ravenel solution to the Kervaire invariant problem [19]. The construction of the norm functor from H-spectra to G-spectra involves exploiting a little-known aspect of the equivariant stable category from a novel perspective, and this has begun to lead to a variety of analyses.
  • Sheaves and Homotopy Theory

    Sheaves and Homotopy Theory

    SHEAVES AND HOMOTOPY THEORY DANIEL DUGGER The purpose of this note is to describe the homotopy-theoretic version of sheaf theory developed in the work of Thomason [14] and Jardine [7, 8, 9]; a few enhancements are provided here and there, but the bulk of the material should be credited to them. Their work is the foundation from which Morel and Voevodsky build their homotopy theory for schemes [12], and it is our hope that this exposition will be useful to those striving to understand that material. Our motivating examples will center on these applications to algebraic geometry. Some history: The machinery in question was invented by Thomason as the main tool in his proof of the Lichtenbaum-Quillen conjecture for Bott-periodic algebraic K-theory. He termed his constructions `hypercohomology spectra', and a detailed examination of their basic properties can be found in the first section of [14]. Jardine later showed how these ideas can be elegantly rephrased in terms of model categories (cf. [8], [9]). In this setting the hypercohomology construction is just a certain fibrant replacement functor. His papers convincingly demonstrate how many questions concerning algebraic K-theory or ´etale homotopy theory can be most naturally understood using the model category language. In this paper we set ourselves the specific task of developing some kind of homotopy theory for schemes. The hope is to demonstrate how Thomason's and Jardine's machinery can be built, step-by-step, so that it is precisely what is needed to solve the problems we encounter. The papers mentioned above all assume a familiarity with Grothendieck topologies and sheaf theory, and proceed to develop the homotopy-theoretic situation as a generalization of the classical case.
  • Topology and Data

    Topology and Data

    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 46, Number 2, April 2009, Pages 255–308 S 0273-0979(09)01249-X Article electronically published on January 29, 2009 TOPOLOGY AND DATA GUNNAR CARLSSON 1. Introduction An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that the nature of the data we are obtaining is significantly different. For example, it is now often the case that we are given data in the form of very long vectors, where all but a few of the coordinates turn out to be irrelevant to the questions of interest, and further that we don’t necessarily know which coordinates are the interesting ones. A related fact is that the data is often very high-dimensional, which severely restricts our ability to visualize it. The data obtained is also often much noisier than in the past and has more missing information (missing data). This is particularly so in the case of biological data, particularly high throughput data from microarray or other sources. Our ability to analyze this data, both in terms of quantity and the nature of the data, is clearly not keeping pace with the data being produced. In this paper, we will discuss how geometry and topology can be applied to make useful contributions to the analysis of various kinds of data.
  • 4 Homotopy Theory Primer

    4 Homotopy Theory Primer

    4 Homotopy theory primer Given that some topological invariant is different for topological spaces X and Y one can definitely say that the spaces are not homeomorphic. The more invariants one has at his/her disposal the more detailed testing of equivalence of X and Y one can perform. The homotopy theory constructs infinitely many topological invariants to characterize a given topological space. The main idea is the following. Instead of directly comparing struc- tures of X and Y one takes a “test manifold” M and considers the spacings of its mappings into X and Y , i.e., spaces C(M, X) and C(M, Y ). Studying homotopy classes of those mappings (see below) one can effectively compare the spaces of mappings and consequently topological spaces X and Y . It is very convenient to take as “test manifold” M spheres Sn. It turns out that in this case one can endow the spaces of mappings (more precisely of homotopy classes of those mappings) with group structure. The obtained groups are called homotopy groups of corresponding topological spaces and present us with very useful topological invariants characterizing those spaces. In physics homotopy groups are mostly used not to classify topological spaces but spaces of mappings themselves (i.e., spaces of field configura- tions). 4.1 Homotopy Definition Let I = [0, 1] is a unit closed interval of R and f : X Y , → g : X Y are two continuous maps of topological space X to topological → space Y . We say that these maps are homotopic and denote f g if there ∼ exists a continuous map F : X I Y such that F (x, 0) = f(x) and × → F (x, 1) = g(x).
  • The Real Projective Spaces in Homotopy Type Theory

    The Real Projective Spaces in Homotopy Type Theory

    The real projective spaces in homotopy type theory Ulrik Buchholtz Egbert Rijke Technische Universität Darmstadt Carnegie Mellon University Email: [email protected] Email: [email protected] Abstract—Homotopy type theory is a version of Martin- topology and homotopy theory developed in homotopy Löf type theory taking advantage of its homotopical models. type theory (homotopy groups, including the fundamen- In particular, we can use and construct objects of homotopy tal group of the circle, the Hopf fibration, the Freuden- theory and reason about them using higher inductive types. In this article, we construct the real projective spaces, key thal suspension theorem and the van Kampen theorem, players in homotopy theory, as certain higher inductive types for example). Here we give an elementary construction in homotopy type theory. The classical definition of RPn, in homotopy type theory of the real projective spaces as the quotient space identifying antipodal points of the RPn and we develop some of their basic properties. n-sphere, does not translate directly to homotopy type theory. R n In classical homotopy theory the real projective space Instead, we define P by induction on n simultaneously n with its tautological bundle of 2-element sets. As the base RP is either defined as the space of lines through the + case, we take RP−1 to be the empty type. In the inductive origin in Rn 1 or as the quotient by the antipodal action step, we take RPn+1 to be the mapping cone of the projection of the 2-element group on the sphere Sn [4].
  • Floer Homology, Gauge Theory, and Low-Dimensional Topology

    Floer Homology, Gauge Theory, and Low-Dimensional Topology

    Floer Homology, Gauge Theory, and Low-Dimensional Topology Clay Mathematics Proceedings Volume 5 Floer Homology, Gauge Theory, and Low-Dimensional Topology Proceedings of the Clay Mathematics Institute 2004 Summer School Alfréd Rényi Institute of Mathematics Budapest, Hungary June 5–26, 2004 David A. Ellwood Peter S. Ozsváth András I. Stipsicz Zoltán Szabó Editors American Mathematical Society Clay Mathematics Institute 2000 Mathematics Subject Classification. Primary 57R17, 57R55, 57R57, 57R58, 53D05, 53D40, 57M27, 14J26. The cover illustrates a Kinoshita-Terasaka knot (a knot with trivial Alexander polyno- mial), and two Kauffman states. These states represent the two generators of the Heegaard Floer homology of the knot in its topmost filtration level. The fact that these elements are homologically non-trivial can be used to show that the Seifert genus of this knot is two, a result first proved by David Gabai. Library of Congress Cataloging-in-Publication Data Clay Mathematics Institute. Summer School (2004 : Budapest, Hungary) Floer homology, gauge theory, and low-dimensional topology : proceedings of the Clay Mathe- matics Institute 2004 Summer School, Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary, June 5–26, 2004 / David A. Ellwood ...[et al.], editors. p. cm. — (Clay mathematics proceedings, ISSN 1534-6455 ; v. 5) ISBN 0-8218-3845-8 (alk. paper) 1. Low-dimensional topology—Congresses. 2. Symplectic geometry—Congresses. 3. Homol- ogy theory—Congresses. 4. Gauge fields (Physics)—Congresses. I. Ellwood, D. (David), 1966– II. Title. III. Series. QA612.14.C55 2004 514.22—dc22 2006042815 Copying and reprinting. Material in this book may be reproduced by any means for educa- tional and scientific purposes without fee or permission with the exception of reproduction by ser- vices that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given.
  • The Seifert-Van Kampen Theorem Via Covering Spaces

    The Seifert-Van Kampen Theorem Via Covering Spaces

    Treball final de grau GRAU DE MATEMÀTIQUES Facultat de Matemàtiques i Informàtica Universitat de Barcelona The Seifert-Van Kampen theorem via covering spaces Autor: Roberto Lara Martín Director: Dr. Javier José Gutiérrez Marín Realitzat a: Departament de Matemàtiques i Informàtica Barcelona, 29 de juny de 2017 Contents Introduction ii 1 Category theory 1 1.1 Basic terminology . .1 1.2 Coproducts . .6 1.3 Pushouts . .7 1.4 Pullbacks . .9 1.5 Strict comma category . 10 1.6 Initial objects . 12 2 Groups actions 13 2.1 Groups acting on sets . 13 2.2 The category of G-sets . 13 3 Homotopy theory 15 3.1 Homotopy of spaces . 15 3.2 The fundamental group . 15 4 Covering spaces 17 4.1 Definition and basic properties . 17 4.2 The category of covering spaces . 20 4.3 Universal covering spaces . 20 4.4 Galois covering spaces . 25 4.5 A relation between covering spaces and the fundamental group . 26 5 The Seifert–van Kampen theorem 29 Bibliography 33 i Introduction The Seifert-Van Kampen theorem describes a way of computing the fundamen- tal group of a space X from the fundamental groups of two open subspaces that cover X, and the fundamental group of their intersection. The classical proof of this result is done by analyzing the loops in the space X and deforming them into loops in the subspaces. For all the details of such proof see [1, Chapter I]. The aim of this work is to provide an alternative proof of this theorem using covering spaces, sets with actions of groups and category theory.
  • Lecture Notes in Mathematics

    Lecture Notes in Mathematics

    Lecture Notes in Mathematics For information about Vols. 1-1145 please contact your bookseller Vol. 1173: H. DeHs, M. Knebusch, Locally Semialgebraic Spaces. XVI, or Springer-Verlag. 329 pages. 1g95, Vol. 1146: 5eminaire d'Aigebre Paul Dubreil et Marie-Paula Malliavin. Vol. 1174: Categories in Continuum Physics, Buffalo 1982. Seminar. Proceedings, 1g63-1984. Edite par M.-P. Malliavin. IV, 420 pages. Edited by F.W. Lawvere and S.H. Schanuel. V, t26 pages. t986. 1985. Vol. 1175: K. Mathiak, Valuations of Skew Fields and Projective Vol. 1147: M. Wschebor, Surfaces Aleatoires. VII, 11t pages. 1985. Hjelmslev Spaces. VII, 116 pages. 1986. Vol. 1t48: Mark A. Kon, Probability Distributions in Quantum Statistical Vol. 1176: R.R. Bruner, J.P. May, J.E. McClure, M. Steinberger, Mechanics. V, 12t pages. 1985. Hoo Ring Spectra and their Applications. VII, 388 pages. 1988. Vol. 1149: Universal Algebra and Lattice Theory. Proceedings, 1984. Vol. 1t77: Representation Theory I. Finite Dimensional Algebras. Edited by S. D. Comer. VI, 282 pages. 1985. Proceedings, t984. Edited by V. Dlab, P. Gabriel and G. Michler. XV, 340 pages. 1g86. Vol. 1150: B. Kawohl, Rearrangements and Convexity of Level Sets in Vol. 1178: Representation Theory II. Groups and Orders. Proceed­ PDE. V, 136 pages. 1985. ings, 1984. Edited by V. Dlab, P. Gabriel and G. Michler. XV, 370 Vol 1151: Ordinary and Partial Differential Equations. Proceedings, pages. 1986. 1984. Edited by B.D. Sleeman and R.J. Jarvis. XIV, 357 pages. 1985. Vol. 1179: Shi J .-Y. The Kazhdan-Lusztig Cells in Certain Affine Weyl Vol. 1152: H. Widom, Asymptotic Expansions for Pseudodifferential Groups.
  • Rational Homotopy Theory: a Brief Introduction

    Rational Homotopy Theory: a Brief Introduction

    Contemporary Mathematics Rational Homotopy Theory: A Brief Introduction Kathryn Hess Abstract. These notes contain a brief introduction to rational homotopy theory: its model category foundations, the Sullivan model and interactions with the theory of local commutative rings. Introduction This overview of rational homotopy theory consists of an extended version of lecture notes from a minicourse based primarily on the encyclopedic text [18] of F´elix, Halperin and Thomas. With only three hours to devote to such a broad and rich subject, it was difficult to choose among the numerous possible topics to present. Based on the subjects covered in the first week of this summer school, I decided that the goal of this course should be to establish carefully the founda- tions of rational homotopy theory, then to treat more superficially one of its most important tools, the Sullivan model. Finally, I provided a brief summary of the ex- tremely fruitful interactions between rational homotopy theory and local algebra, in the spirit of the summer school theme “Interactions between Homotopy Theory and Algebra.” I hoped to motivate the students to delve more deeply into the subject themselves, while providing them with a solid enough background to do so with relative ease. As these lecture notes do not constitute a history of rational homotopy theory, I have chosen to refer the reader to [18], instead of to the original papers, for the proofs of almost all of the results cited, at least in Sections 1 and 2. The reader interested in proper attributions will find them in [18] or [24]. The author would like to thank Luchezar Avramov and Srikanth Iyengar, as well as the anonymous referee, for their helpful comments on an earlier version of this article.
  • [Math.AT] 24 Aug 2005

    [Math.AT] 24 Aug 2005

    QUADRATIC FUNCTIONS IN GEOMETRY, TOPOLOGY, AND M-THEORY M. J. HOPKINS AND I. M. SINGER Contents 1. Introduction 2 2. Determinants, differential cocycles and statement of results 5 2.1. Background 5 2.2. Determinants and the Riemann parity 7 2.3. Differential cocycles 8 2.4. Integration and Hˇ -orientations 11 2.5. Integral Wu-structures 13 2.6. The main theorem 16 2.7. The fivebrane partition function 20 3. Cheeger–Simons cohomology 24 3.1. Introduction 24 3.2. Differential Characters 25 3.3. Characteristic classes 27 3.4. Integration 28 3.5. Slant products 31 4. Generalized differential cohomology 32 4.1. Differential function spaces 32 4.2. Naturality and homotopy 36 4.3. Thom complexes 38 4.4. Interlude: differential K-theory 40 4.5. Differential cohomology theories 42 4.6. Differential function spectra 44 4.7. Naturality and Homotopy for Spectra 46 4.8. The fundamental cocycle 47 arXiv:math/0211216v2 [math.AT] 24 Aug 2005 4.9. Differential bordism 48 4.10. Integration 53 5. The topological theory 57 5.1. Proof of Theorem 2.17 57 5.2. The topological theory of quadratic functions 65 5.3. The topological κ 69 5.4. The quadratic functions 74 Appendix A. Simplicial methods 82 A.1. Simplicial set and simplicial objects 82 A.2. Simplicial homotopy groups 83 The first author would like to acknowledge support from NSF grant #DMS-9803428. The second author would like to acknowledge support from DOE grant #DE-FG02-ER25066. 1 2 M.J.HOPKINSANDI.M.SINGER A.3.
  • Topology - Wikipedia, the Free Encyclopedia Page 1 of 7

    Topology - Wikipedia, the Free Encyclopedia Page 1 of 7

    Topology - Wikipedia, the free encyclopedia Page 1 of 7 Topology From Wikipedia, the free encyclopedia Topology (from the Greek τόπος , “place”, and λόγος , “study”) is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation. Ideas that are now classified as topological were expressed as early as 1736. Toward the end of the 19th century, a distinct A Möbius strip, an object with only one discipline developed, which was referred to in Latin as the surface and one edge. Such shapes are an geometria situs (“geometry of place”) or analysis situs object of study in topology. (Greek-Latin for “picking apart of place”). This later acquired the modern name of topology. By the middle of the 20 th century, topology had become an important area of study within mathematics. The word topology is used both for the mathematical discipline and for a family of sets with certain properties that are used to define a topological space, a basic object of topology. Of particular importance are homeomorphisms , which can be defined as continuous functions with a continuous inverse. For instance, the function y = x3 is a homeomorphism of the real line. Topology includes many subfields. The most basic and traditional division within topology is point-set topology , which establishes the foundational aspects of topology and investigates concepts inherent to topological spaces (basic examples include compactness and connectedness); algebraic topology , which generally tries to measure degrees of connectivity using algebraic constructs such as homotopy groups and homology; and geometric topology , which primarily studies manifolds and their embeddings (placements) in other manifolds.