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Christian Caron Springer Heidelberg Physics Editorial Department I Tiergartenstrasse 17 69121 Heidelberg / Germany [email protected] D. Husemöller M. Joachim B. Jurcoˇ M. Schottenloher

Basic Bundle Theory and K-Cohomology Invariants

With contributions by Siegfried Echterhoff, Stefan Fredenhagen and Bernhard Krötz Authors D. Husemöller B. Jurcoˇ MPI für Mathematik MPI für Mathematik Vivatsgasse 7 Vivatsgasse 7 53111 Bonn, Germany 53111 Bonn, Germany [email protected] [email protected]

M. Joachim M. Schottenloher Universität Münster Universität München Mathematisches Institut Mathematisches Institut Einsteinstr. 62 Theresienstr. 39 48149 Münster, Germany 80333 München, Germany [email protected] Martin.Schottenloher@Mathematik. Uni-Muenchen.de Contributors Siegfried Echterhoff (Chap. 17, Appendix) Bernhard Krötz (Chap. 16, Appendix) Mathematisches Institut MPI für Mathematik, Bonn, Germany Westfälische Wilhelms-Universität Münster, Germany Stefan Fredenhagen (Physical Background to the K-Theory Classification of D-Branes), MPI für Gravitationsphysik, Potsdam, Germany

D. Husemöller et al., Basic Bundle Theory and K-Cohomology Invariants, Lect. Notes Phys. 726 (Springer, Berlin Heidelberg 2008), DOI 10.1007/ 978-3-540-74956-1

Library of Congress Control Number: 2007936164 ISSN 0075-8450 ISBN 978-3-540-74955-4 Springer Berlin Heidelberg New York

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This lecture notes volume has its origins in a course by Husem¨oller on fibre bundles and twisted K-theory organized by Brano Jurˇco for physics students at the LMU in M¨unchen, summer term 2003. The fact that K-theory invariants, and in particular twisted K-theory invariants, were being used in the geometric aspects of mathemat- ical physics created the need for an accessible treatment of the subject. The course surveyed the book Fibre Bundles, 3rd. Ed. 1994, Springer-Verlag by Husem¨oller, and covered topics used in mathematical physics related to K-theory invariants. This book is referred to just by its title throughout the text. The idea of lecture notes came up by J. Wess in 2003 in order to serve several purposes. Firstly, they were to be a supplement to the book Fibre Bundles providing companion reading and alternative approaches to certain topics; secondly, they were to survey some of the basic results of background to K-theory, for example operator algebra K-theory, not covered in the Fibre Bundles; and finally the notes would contain information on the relation to physics. This we have done in the survey following this introduction “Physical background to the K-theory classification of D-branes: Introduction and references” tracing the papers how and where K-theory invariants started to play a role in string theory. The basic references to physics are given at the end of this survey, while the mathematics references are at the end of the volume. Other lectures of Husem¨oller had contributed to the text of the notes. During 2001/2002 in M¨unster resp. during Summer 2002 in M¨unchen, Husem¨oller gave Graduate College courses on the topics in the notes, organized by Joachim Cuntz resp. Martin Schottenloher, and in the Summer 2001, he had a regular course on C*-algebras and K-theory in M¨unchen. The general question of algebra bundles was studied with the support of Professor Cuntz in M¨unster during short periods from 2003 to 2005. Finally, Husem¨oller lectured on these topics during a workshop at IPM, Tehran, Iran, September 2005. It is with a great feeling of gratitude that these lecture opportunities are remembered here. The notes are organized into five parts. The first part on basic bundle theory emphasizes the concept of bundle as one treats the concepts of set, space, homotopy, group, or ring in basic mathematics. A bundle is just a map called the projection from the total space to its base space. As with commutative groups, topological

vii viii Preface groups, transformation groups, and Lie groups, the concept of bundle is enhanced or enriched with additional axioms and structures leading toetale ´ bundles, principal bundles, fibre bundles, vector bundles, and algebra bundles. A topic discussed in the first part, which is not taken up in the Fibre Bundles, is the Serre–Swan theorem which relates vector bundles on a compact space X with finitely generated projective modules over the ring C(X) of continuous complex valued functions on the space X. This is one of the points where topological, algebraic, and operator K-theory come together. The second part of the notes takes up the homotopy classification of principal bundles and fibre bundles. Applications to the case of vector bundles are considered and the role of homotopy theory in K-theory is developed. This is related to the fact that K-theory is a representable functor on the homotopy category. The theory of characteristic classes in describing orientation and spin structures on vector bundles is carried out in detail, also leading to the notion of a string structure on a bundle and on a manifold. There are various versions of topological K-theory, and their relation to Bott peri- odicity is considered in the third part of the notes. An advanced version of operator K-theory, called KK-theory which integrates K-cohomology and K-homology, is introduced, and various features are sketched. The fourth part of the notes begins with algebra bundles with fibres that are either matrix algebras or algebras of bounded operators on a separable Hilbert space. The infinite dimensional algebra bundles are classified by only one characteristic class in the integral third cohomology group of the base space along the lines of the classification of complex line bundles with its first Chern class in the integral second cohomology group of the base space. The twisting of twisted K-theory is given by an infinite dimensional algebra bundle, and the twisted K-theory is defined in terms of cross sections of Fredholm bundles related to the algebra bundle describing the twist under consideration. A fundamental theme in bundle theory centers around the gluing of local bundle data related to bundles into a global object. In the fifth part we return to this theme and study gluing on open sets in a topological space of not just simple bundle data but also data in a more general category where the gluing data may satisfy transitiv- ity conditions only up to an isomorphism. The resulting objects are gerbes or stacks.

August 2007 Dale Husem¨oller Contents

Physical Background to the K-Theory Classification of D-Branes: Introduction and References ...... 1

Part I Bundles over a Space and Modules over an Algebra

1 Generalities on Bundles and Categories ...... 9 1 Bundles Over a Space ...... 9 2 Examples of Bundles ...... 11 3 Two Operations on Bundles ...... 13 4 Category Constructions Related to Bundles ...... 14 5 FunctorsBetweenCategories...... 16 6 MorphismsofFunctorsorNaturalTransformations...... 18 7 EtaleMapsandCoverings...... ´ 20 References...... 22

2 Vector Bundles...... 23 1 Bundles of Vector Spaces and Vector Bundles ...... 23 2 Isomorphisms of Vector Bundles and Induced Vector Bundles ..... 25 3 Image and Kernel of Vector Bundle Morphisms ...... 26 4 The Canonical Bundle Over the Grassmannian Varieties ...... 28 5 Finitely Generated Vector Bundles ...... 29 6 Vector Bundles on a Compact Space ...... 31 7 Collapsing and Clutching Vector Bundles on Subspaces ...... 31 8 Metrics on Vector Bundles ...... 33 Reference...... 34

3 Relation Between Vector Bundles, Projective Modules, and Idempotents ...... 35 1 Local Coordinates of a Vector Bundle Given by Global Functions overaNormalSpace...... 36 2 TheFullEmbeddingPropertyoftheCrossSectionFunctor ...... 37 3 Finitely Generated Projective Modules ...... 38 4 TheSerre–SwanTheorem...... 40

ix x Contents

5 Idempotent Classes Associated to Finitely Generated Projective Modules ...... 42

4 K-Theory of Vector Bundles, of Modules, and of Idempotents ...... 45 1 Generalities on Adding Negatives ...... 45 2 K-Groups of Vector Bundles ...... 47 3 K-Groups of Finitely Generated Projective Modules...... 48 4 K-Groups of Idempotents ...... 50 5 K-Theory of Topological Algebras ...... 51 References...... 54

5 Principal Bundles and Sections of Fibre Bundles: Reduction of the Structure and the Gauge Group I ...... 55 1 Bundles Defined by Transformation Groups ...... 55 2 Definition and Examples of Principal Bundles...... 57 3 Fibre Bundles...... 58 4 Local Coordinates for Fibre Bundles ...... 58 5 ExtensionandRestrictionofStructureGroup...... 60 6 Automorphisms of Principal Bundles and Gauge Groups...... 62 Reference...... 62

Part II Homotopy Classification of Bundles and Cohomology: Classifying Spaces

6 Homotopy Classes of Maps and the Homotopy Groups ...... 65 1 The Space Map(X,Y) ...... 65 2 Continuity of Substitution and Map(X×T,Y ) ...... 66 3 FreeandBasedHomotopyClassesofMaps ...... 67 4 HomotopyCategories...... 68 5 Homotopy Groups of a Pointed Space ...... 69 6 Bundles on a Cylinder B×[0,1] ...... 72

7 The Milnor Construction: Homotopy Classification of Principal Bundles ...... 75 1 Basic Data from a Numerable Principal Bundle ...... 75 2 TotalSpaceoftheMilnorConstruction...... 76 3 UniquenessuptoHomotopyoftheClassifyingMap...... 78 4 The Infinite Sphere as the Total Space of the Milnor Construction . . 80 References...... 81

8 Fibrations and Bundles: Gauge Group II ...... 83 1 Factorization,Lifting,andExtensioninSquareDiagrams...... 84 2 FibrationsandCofibrations...... 85 3 FibresandCofibres:LoopSpaceandSuspension...... 88 4 Relation Between Loop Space and Suspension Group Structures on Homotopy Classes of Maps [X,Y]* ...... 90 Contents xi

5 Outline of the Fibre Mapping Sequence and Cofibre Mapping Sequence ...... 91 6 FromBasetoFibreandFromFibretoBase...... 93 7 Homotopy Characterization of the Universal Bundle ...... 95 8 ApplicationtotheClassifyingSpaceoftheGaugeGroup...... 95 9 The Infinite Sphere as the Total Space of a Universal Bundle ...... 96 Reference...... 96

9 Cohomology Classes as Homotopy Classes: CW-Complexes ...... 97 1 Filtered Spaces and Cell Complexes ...... 98 2 Whitehead’sCharacterizationofHomotopyEquivalences...... 99 3 Axiomatic Properties of Cohomology and Homology ...... 100 4 Construction and Calculation of Homology and Cohomology ...... 103 5 HurewiczTheorem...... 105 6 Representability of Cohomology by Homotopy Classes ...... 105 7 Products of Cohomology and Homology ...... 106 8 Introduction to Morse Theory ...... 107 References...... 109

10 Basic Characteristic Classes ...... 111 1 Characteristic Classes of Line Bundles ...... 111 2 Projective Bundle Theorem and Splitting Principle ...... 113 3 Chern Classes and Stiefel–Whitney Classes of Vector Bundles . . . . . 114 4 ElementaryPropertiesofCharacteristicClasses...... 117 5 Chern Character and Related Multiplicative Characteristic Classes . 118 6 EulerClass...... 121 7 Thom Space, Thom Class, and Thom Isomorphism ...... 122 8 Stiefel–WhitneyClassesinTermsofSteenrodOperations...... 122 9 Pontrjaginclasses...... 125 References...... 125

11 Characteristic Classes of Manifolds ...... 127 1 OrientationinEuclideanSpaceandonManifolds...... 127 2 Poincar´e Duality on Manifolds ...... 129 3 Thom Class of the Tangent Bundle and Duality ...... 130 4 EulerClassandEulerCharacteristicofaManifold...... 131 5 Wu’sFormulafortheStiefel–WhitneyClassesofaManifold...... 132 6 Cobordism and Stiefel–Whitney Numbers ...... 133 7 Introduction to Characteristic Classes and Riemann–Roch ...... 134 Reference...... 135 xii Contents

12 Spin Structures ...... 137 1 The Groups Spin(n) and Spinc(n)...... 137 2 OrientationandtheFirstStiefel–WhitneyClass...... 139 3 SpinStructuresandtheSecondStiefel–WhitneyClass...... 140 4 Spinc StructuresandtheThirdIntegralStiefel–WhitneyClass.....141 5 Relation Between Characteristic Classes of Real and Complex Vector Bundles ...... 142 6 Killing Homotopy Groups in a Fibration ...... 142

Part III Versions of K-Theory and Bott Periodicity

13 G-Spaces, G-Bundles, and G-Vector Bundles ...... 149 1 Relations Between Spaces and G-Spaces: G-Homotopy...... 149 2 Generalities on G-Bundles ...... 152 3 Generalities on G-Vector Bundles ...... 153 4 Special Examples of G-Vector Bundles ...... 155 5 Extension and Homotopy Problems for G-Vector Bundles forGaCompactGroup ...... 157 6 Relations Between Complex and Real G-Vector Bundles...... 158 7 KRG-Theory...... 159 References...... 161

14 Equivariant K-Theory Functor KG : Periodicity, Thom Isomorphism, Localization, and Completion ...... 163 1 Associated Projective Space Bundle to a G-Equivariant Bundle . . . . 163 2 Assertion of the Periodicity Theorem for a Line Bundle ...... 164 3 ThomIsomorphism...... 167 4 LocalizationTheoremofAtiyahandSegal ...... 170 5 Equivariant K-Theory Completion Theorem of Atiyah and Segal . . . 172 References...... 173

15 Bott Periodicity Maps and Clifford Algebras ...... 175 1 Vector Bundles and Their Principal Bundles and Metrics ...... 175 2 Homotopy Representation of K-Theory ...... 176 3 TheBottMapsinthePeriodicitySeries...... 179 ∗ ( ) ( ) 4 KRG X and the Representation Ring RR G ...... 180 5 Generalities on Clifford Algebras and Their Modules...... 181 −q 6 KRG (*) and Modules Over Clifford Algebras ...... 184 7 BottPeriodicityandMorseTheory...... 185 8 The Graded Rings KU∗(∗) and KO∗(∗) ...... 187 References...... 188 Contents xiii

16 GramÐSchmidt Process, Iwasawa Decomposition, and Reduction of Structure ...... 189 1 ClassicalGram–SchmidtProcess ...... 189 2 Definition of Basic Linear Groups...... 190 3 Iwasawa Decomposition for GL and SL ...... 191 4 Applications to Structure Group Reduction for Principal Bundles Related to Vector Bundles ...... 192 5 The Special Case of SL2(R) andtheUpperHalfPlane ...... 193 6 Relation Between SL2(R) and SL2(C) with the Lorentz Groups . . . . 194 A Appendix: A Novel Characterization of the Iwasawa Decomposition of a Simple Lie Group (by B. Krotz¨ ) ...... 195 References...... 201

17 Topological Algebras: G-Equivariance and KK-Theory ...... 203 1 The Module of Cross Sections for a G-Equivariant Vector Bundle . . 204 2 G-Equivariant K-Theory and the K-Theory of Cross Products .....205 3 Generalities on Topological Algebras: Stabilization ...... 207 4 Ell(X)andExt(X) Pairing with K-Theory to Z ...... 209 5 Extensions:UniversalExamples...... 212 6 Basic Examples of Extensions for K-Theory...... 215 7 HomotopyInvariant,HalfExact,andStableFunctors...... 219 8 The Bivariant Functor kk∗(A, B) ...... 220 9 BottMapandBottPeriodicity...... 221 A Appendix: The Green–Julg Theorem (by S. Echterhoff )...... 223 References...... 226

Part IV Algebra Bundles: Twisted K-Theory

18 Isomorphism Classification of Operator Algebra Bundles ...... 229 1 Vector Bundles and Algebra Bundles ...... 230 2 Principal Bundle Description and Classifying Spaces ...... 231 3 Homotopy Classification of Principal Bundles ...... 233 4 Classification of Operator Algebra Bundles ...... 235 References...... 239

19 Brauer Group of Matrix Algebra Bundles and K-Groups ...... 241 1 Properties of the Morphism αn ...... 241 2 From Brauer Groups to Grothendieck Groups ...... 243 3 Stability I: Vector Bundles ...... 244 4 Stability II: Characteristic Classes of Algebra Bundles and Projective K-Group...... 245 5 Rational Class Groups ...... 246 6 SheafTheoryInterpretation...... 247 Reference...... 249 xiv Contents

20 Analytic Definition of Twisted K-Theory ...... 251 1 CrossSectionsandFibreHomotopyClassesofCrossSections.....251 2 Two Basic Analytic Results in Bundle Theory and K-Theory...... 252 3TwistedK-Theory in Terms of Fredholm Operators ...... 253

21 The AtiyahÐHirzebruch Spectral Sequence in K-Theory ...... 255 1 Exact Couples: Their Derivation and Spectral Sequences ...... 255 2 Homological Spectral Sequence for a Filtered Object ...... 256 3 K-Theory Exact Couples for a Filtered Space ...... 258 4 Atiyah–Hirzebruch Spectral Sequence for K-Theory...... 260 5 FormulasforDifferentials...... 262 6 Calculations for Products of Real Projective Spaces ...... 263 7TwistedK-TheorySpectralSequence...... 264 Reference...... 264

22 Twisted Equivariant K-Theory and the Verlinde Algebra ...... 265 1 The Verlinde Algebra as the Quotient of the Representation Ring. . . 266 2 The Verlinde Algebra for SU(2) and sl(2) ...... 268 3TheG-Bundles on G with the Adjoint G–Action...... 271 4 A Version of the Freed–Hopkins–Teleman Theorem ...... 273 References...... 274

Part V Gerbes and the Three Dimensional Integral Cohomology Classes

23 Bundle Gerbes ...... 277 1 Notation for Gluing of Bundles ...... 277 2 Definition of Bundle Gerbes ...... 280 3 TheGerbeCharacteristicClass...... 281 4 Stability Properties of Bundle Gerbes ...... 283 5 Extensions of Principal Bundles Over a Central Extension ...... 284 6 Modules Over Bundle Gerbes and Twisted K-Theory...... 284 Reference...... 286

24 Category Objects and Groupoid Gerbes ...... 287 1 SimplicialObjectsinaCategory...... 287 2 CategoriesinaCategory...... 290 3 The Nerve of the Classifying Space Functor and Definition of Algebraic K-Theory...... 293 4 Groupoids in a Category ...... 295 5 The Groupoid Associated to a Covering ...... 297 6 Gerbes on Groupoids ...... 298 7 The Groupoid Gerbe Characteristic Class ...... 300 Contents xv

25 Stacks and Gerbes ...... 303 1 PresheavesandSheaveswithValuesinCategory...... 304 2 Generalities on Adjoint Functors ...... 306 3 Categories Over Spaces (Fibred Categories) ...... 309 4 PrestacksOveraSpace...... 311 5 DescentData...... 315 6 TheStackAssociatedtoaPrestack...... 316 7 Gerbes as Stacks of Groupoids ...... 318 8 Cohomological Classification of Principal G-Sheaves ...... 318 9 Cohomological Classification of Bands Associated with a Gerbe . . . 320

Bibliography ...... 323

Index of Notations ...... 327

Notation for Examples of Categories ...... 333

Index ...... 335