Combinatorial Algebraic Topology

By: Dimitry Kozlov

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Paperback | 8 January 2008

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Combinatorial algebraic topology is a fascinating and dynamic field at the crossroads of algebraic topology and discrete mathematics. This volume is the first comprehensive treatment of the subject in book form. The first part of the book constitutes a swift walk through the main tools of algebraic topology, including Stiefel-Whitney characteristic classes, which are needed for the later parts. Readers - graduate students and working mathematicians alike - will probably find particularly useful the second part, which contains an in-depth discussion of the major research techniques of combinatorial algebraic topology. Our presentation of standard topics is quite different from that of existing texts. In addition, several new themes, such as spectral sequences, are included. Although applications are sprinkled throughout the second part, they are principal focus of the third part, which is entirely devoted to developing the topological structure theory for graph homomorphisms. The main benefit for the reader will be the prospect of fairly quickly getting to the forefront of modern research in this active field.

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From the reviews: "This is an introduction to the beautiful world of combinatorial algebraic topology, describing the modern research tools and latest applications in this field. ... This could be used as material for a reading seminar on Chromatic numbers and the Kneser Conjecture, structural theory of morphism complexes, characteristic classes and chromatic numbers, applications of spectral sequence to Hom Complexes. ... an interesting book with large perspective in studying problems on the borderline between discrete mathematics and algebraic topology." (Corina Mohorianu, Zentralblatt MATH, Vol. 1130 (8), 2008) "This monograph offers an introduction to combinatorial algebraic topology, an active field connecting algebraic topology with discrete mathematics and computer science. It is intended to be 'A book to teach from,, providing a self-contained introduction that swiftly guides the reader to the forefront of modern research." (St. Haller, Monatshefte fur Mathematik, Vol. 162 (3), March, 2011)

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Non-Fiction Mathematics Groups & Group Theory Algebra Mathematical Foundation Set Theory Combinatorics & Graph Theory Topology Algebraic Topology Discrete Mathematics

Overture	p. 1
Concepts of Algebraic Topology
Cell Complexes	p. 7
Abstract Simplicial Complexes	p. 7
Definition of Abstract Simplicial Complexes and Maps Between Them	p. 7
Deletion, Link, Star, and Wedge	p. 10
Simplicial Join	p. 12
Face Posets	p. 12
Barycentric and Stellar Subdivisions	p. 13
Pulling and Pushing Simplicial Structures	p. 15
Polyhedral Complexes	p. 16
Geometry of Abstract Simplicial Complexes	p. 16
Geometric Meaning of the Combinatorial Constructions	p. 19
Geometric Simplicial Complexes	p. 23
Complexes Whose Cells Belong to a Specified Set of Polyhedra	p. 25
Trisps	p. 28
Construction Using the Gluing Data	p. 28
Constructions Involving Trisps	p. 30
CW Complexes	p. 33
Gluing Along a Map	p. 33
Constructive and Intrinsic Definitions	p. 34
Properties and Examples	p. 35
Homology Groups	p. 37
Betti Numbers of Finite Abstract Simplicial Complexes	p. 37
Simplicial Homology Groups	p. 39
Homology Groups of Trisps with Coefficients in Z[subscript 2]	p. 39
Orientations	p. 41
Homology Groups of Trisps with Integer Coefficients	p. 41
Invariants Connected to Homology Groups	p. 44
Betti Numbers and Torsion Coefficients	p. 44
Euler Characteristic and the Euler-Poincare Formula	p. 45
Variations	p. 46
Augmentation and Reduced Homology Groups	p. 46
Homology Groups with Other Coefficients	p. 47
Simplicial Cohomology Groups	p. 47
Singular Homology	p. 49
Chain Complexes	p. 51
Definition and Homology of Chain Complexes	p. 51
Maps Between Chain Complexes and Induced Maps on Homology	p. 52
Chain Homotopy	p. 53
Simplicial Homology and Cohomology in the Context of Chain Complexes	p. 54
Homomorphisms on Homology Induced by Trisp Maps	p. 54
Cellular Homology	p. 56
An Application of Homology with Integer Coefficients: Winding Number	p. 56
The Definition of Cellular Homology	p. 57
Cellular Maps and Properties of Cellular Homology	p. 58
Concepts of Category Theory	p. 59
The Notion of a Category	p. 59
Definition of a Category, Isomorphisms	p. 59
Examples of Categories	p. 60
Some Structure Theory of Categories	p. 63
Initial and Terminal Objects	p. 63
Products and Coproducts	p. 64
Functors	p. 68
The Category Cat	p. 68
Homology and Cohomology Viewed as Functors	p. 70
Group Actions as Functors	p. 70
Limit Constructions	p. 71
Definition of Colimit of a Functor	p. 71
Colimits and Infinite Unions	p. 72
Quotients of Group Actions as Colimits	p. 73
Limits	p. 74
Comma Categories	p. 74
Objects Below and Above Other Objects	p. 74
The General Construction and Further Examples	p. 75
Exact Sequences	p. 77
Some Structure Theory of Long and Short Exact Sequences	p. 77
Construction of the Connecting Homomorphism	p. 77
Exact Sequences	p. 79
Deriving Long Exact Sequences from Short Ones	p. 81
The Long Exact Sequence of a Pair and Some Applications	p. 82
Relative Homology and the Associated Long Exact Sequence	p. 82
Applications	p. 84
Mayer-Vietoris Long Exact Sequence	p. 85
Homotopy	p. 89
Homotopy of Maps	p. 89
Homotopy Type of Topological Spaces	p. 90
Mapping Cone and Mapping Cylinder	p. 91
Deformation Retracts and Collapses	p. 93
Simple Homotopy Type	p. 95
Homotopy Groups	p. 96
Connectivity and Hurewicz Theorems	p. 97
Cofibrations	p. 101
Cofibrations and the Homotopy Extension Property	p. 101
NDR-Pairs	p. 103
Important Facts Involving Cofibrations	p. 105
The Relative Homotopy Equivalence	p. 107
Principal [Gamma]-Bundles and Stiefel-Whitney Characteristic Classes	p. 111
Locally Trivial Bundles	p. 111
Bundle Terminology	p. 111
Types of Bundles	p. 112
Bundle Maps	p. 113
Elements of the Principal Bundle Theory	p. 114
Principal Bundles and Spaces with a Free Group Action	p. 114
The Classifying Space of a Group	p. 116
Special Cohomology Elements	p. 119
Z[subscript 2]-Spaces and the Definition of Stiefel-Whitney Classes	p. 120
Properties of Stiefel-Whitney Classes	p. 122
Borsuk-Ulam Theorem, Index, and Coindex	p. 122
Stiefel-Whitney Height	p. 123
Higher Connectivity and Stiefel-Whitney Classes	p. 123
Combinatorial Construction of Stiefel-Whitney Classes	p. 124
Suggested Reading	p. 125
Methods of Combinatorial Algebraic Topology
Combinatorial Complexes Melange	p. 129
Abstract Simplicial Complexes	p. 129
Simplicial Flag Complexes	p. 129
Order Complexes	p. 130
Complexes of Combinatorial Properties	p. 133
The Neighborhood and Lovasz Complexes	p. 133
Complexes Arising from Matroids	p. 134
Geometric Complexes in Metric Spaces	p. 134
Combinatorial Presentation by Minimal Nonsimplices	p. 136
Prodsimplicial Complexes	p. 138
Prodsimplicial Flag Complexes	p. 138
Complex of Complete Bipartite Subgraphs	p. 138
Hom Complexes	p. 140
General Complexes of Morphisms	p. 141
Discrete Configuration Spaces of Generalized Simplicial Complexes	p. 144
The Complex of Phylogenetic Trees	p. 144
Regular Trisps	p. 145
Chain Complexes	p. 147
Bibliographic Notes	p. 148
Acyclic Categories	p. 151
Basics	p. 151
The Notion of Acyclic Category	p. 151
Linear Extensions of Acyclic Categories	p. 152
Induced Subcategories of Cat	p. 153
The Regular Trisp of Composable Morphism Chains in an Acyclic Category	p. 153
Definition and First Examples	p. 153
Functoriality	p. 155
Constructions	p. 156
Disjoint Union as a Coproduct	p. 156
Stacks of Acyclic Categories and Joins of Regular Trisps	p. 156
Links, Stars, and Deletions	p. 158
Lattices and Acyclic Categories	p. 159
Barycentric Subdivision and [Delta]-Functor	p. 160
Intervals in Acyclic Categories	p. 161
Definition and First Properties	p. 161
Acyclic Category of Intervals and Its Structural Functor	p. 164
Topology of the Category of Intervals	p. 167
Homeomorphisms Associated with the Direct Product Construction	p. 168
Simplicial Subdivision of the Direct Product	p. 168
Further Subdivisions	p. 171
The Mobius Function	p. 173
Mobius Function for Posets	p. 173
Mobius Function for Acyclic Categories	p. 174
Bibliographic Notes	p. 178
Discrete Morse Theory	p. 179
Discrete Morse Theory for Posets	p. 179
Acyclic Matchings in Hasse Diagrams of Posets	p. 179
Poset Maps with Small Fibers	p. 182
Universal Object Associated to an Acyclic Matching	p. 183
Poset Fibrations and the Patchwork Theorem	p. 185
Discrete Morse Theory for CW Complexes	p. 187
Attaching Cells to Homotopy Equivalent Spaces	p. 187
The Main Theorem of Discrete Morse Theory for CW Complexes	p. 189
Examples	p. 192
Algebraic Morse Theory	p. 201
Acyclic Matchings on Free Chain Complexes and the Morse Complex	p. 201
The Main Theorem of Algebraic Morse Theory	p. 203
An Example	p. 205
Bibliographic Notes	p. 208
Lexicographic Shellability	p. 211
Shellability	p. 211
The Basics	p. 211
Shelling Induced Subcomplexes	p. 214
Shelling Nerves of Acyclic Categories	p. 215
Lexicographic Shellability	p. 216
Labeling Edges as a Way to Order Chains	p. 216
EL-Labeling	p. 217
General Lexicographic Shellability	p. 219
Lexicographic Shellability and Nerves of Acyclic Categories	p. 223
Bibliographic Notes	p. 224
Evasiveness and Closure Operators	p. 225
Evasiveness	p. 225
Evasiveness of Graph Properties	p. 225
Evasiveness of Abstract Simplicial Complexes	p. 229
Closure Operators	p. 232
Collapsing Sequences Induced by Closure Operators	p. 232
Applications	p. 234
Monotone Poset Maps	p. 236
The Reduction Theorem and Implications	p. 237
Further Facts About Nonevasiveness	p. 238
NE-Reduction and Collapses	p. 238
Nonevasiveness of Noncomplemented Lattices	p. 240
Other Recursively Defined Classes of Complexes	p. 242
Bibliographic Notes	p. 243
Colimits and Quotients	p. 245
Quotients of Nerves of Acyclic Categories	p. 245
Desirable Properties of the Quotient Construction	p. 245
Quotients of Simplicial Actions	p. 245
Formalization of Group Actions and the Main Question	p. 248
Definition of the Quotient and Formulation of the Main Problem	p. 248
An Explicit Description of the Category C/G	p. 249
Conditions on Group Actions	p. 250
Outline of the Results and Surjectivity of the Canonical Map	p. 250
Condition for Injectivity of the Canonical Projection	p. 251
Conditions for the Canonical Projection to be an Isomorphism	p. 252
Conditions for the Categories to be Closed Under Taking Quotients	p. 255
Bibliographic Notes	p. 257
Homotopy Colimits	p. 259
Diagrams over Trisps	p. 259
Diagrams and Colimits	p. 259
Arrow Pictures and Their Nerves	p. 260
Homotopy Colimits	p. 262
Definition and Some Examples	p. 262
Structural Maps Associated to Homotopy Colimits	p. 263
Deforming Homotopy Colimits	p. 265
Nerves of Coverings	p. 266
Nerve Diagram	p. 266
Projection Lemma	p. 267
Nerve Lemmas	p. 269
Gluing Spaces	p. 271
Gluing Lemma	p. 271
Quillen Lemma	p. 272
Bibliographic Notes	p. 273
Spectral Sequences	p. 275
Filtrations	p. 275
Contriving Spectral Sequences	p. 276
The Objects to be Constructed	p. 276
The Actual Construction	p. 278
Questions of Convergence and Interpretation of the Answer	p. 280
An Example	p. 280
Maps Between Spectral Sequences	p. 281
Spectral Sequences and Nerves of Acyclic Categories	p. 283
A Class of Filtrations	p. 283
Mobius Function and Inequalities for Betti Numbers	p. 285
Bibliographic Notes	p. 288
Complexes of Graph Homomorphisms
Chromatic Numbers and the Kneser Conjecture	p. 293
The Chromatic Number of a Graph	p. 293
The Definition and Applications	p. 293
The Complexity of Computing the Chromatic Number	p. 294
The Hadwiger Conjecture	p. 295
State Graphs and the Variations of the Chromatic Number	p. 298
Complete Graphs as State Graphs	p. 298
Kneser Graphs as State Graphs and Fractional Chromatic Number	p. 298
The Circular Chromatic Number	p. 300
Kneser Conjecture and Lovasz Test	p. 301
Formulation of the Kneser Conjecture	p. 301
The Properties of the Neighborhood Complex	p. 302
Lovasz Test for Graph Colorings	p. 303
Simplicial and Cubical Complexes Associated to Kneser Graphs	p. 304
The Vertex-Critical Subgraphs of Kneser Graphs	p. 306
Chromatic Numbers of Kneser Hypergraphs	p. 307
Bibliographic Notes	p. 307
Structural Theory of Morphism Complexes	p. 309
The Scope of Morphism Complexes	p. 309
The Morphism Complexes and the Prodsimplicial Flag Construction	p. 309
Universality	p. 311
Special Families of Hom Complexes	p. 312
Coloring Complexes of a Graph	p. 312
Complexes of Bipartite Subgraphs and Neighborhood Complexes	p. 313
Functoriality of Hom (-, -)	p. 315
Functoriality on the Right	p. 315
Aut (G) Action on Hom (T, G)	p. 316
Functoriality on the Left	p. 316
Aut (T) Action on Hom (T, G)	p. 318
Commuting Relations	p. 318
Products, Compositions, and Hom Complexes	p. 320
Coproducts	p. 320
Products	p. 320
Composition of Hom Complexes	p. 322
Folds	p. 323
Definition and First Properties	p. 323
Proof of the Folding Theorem	p. 324
Bibliographic Notes	p. 326
Characteristic Classes and Chromatic Numbers	p. 327
Stiefel-Whitney Characteristic Classes and Test Graphs	p. 327
Powers of Stiefel-Whitney Classes and Chromatic Numbers of Graphs	p. 327
Stiefel-Whitney Test Graphs	p. 328
Examples of Stiefel-Whitney Test Graphs	p. 329
Complexes of Complete Multipartite Subgraphs	p. 329
Odd Cycles as Stiefel-Whitney Test Graphs	p. 334
Homology Tests for Graph Colorings	p. 337
The Symmetrizer Operator and Related Structures	p. 338
The Topological Rationale for the Tests	p. 338
Homology Tests	p. 340
Examples of Homology Tests with Different Test Graphs	p. 341
Bibliographic Notes	p. 346
Applications of Spectral Sequences to Hom Complexes	p. 349
Hom[subscript +] Construction	p. 349
Various Definitions	p. 349
Connection to Independence Complexes	p. 351
The Support Map	p. 352
An Example: Hom[subscript +](C[subscript m], K[subscript n])	p. 353
Setting up the Spectral Sequence	p. 354
Filtration Induced by the Support Map	p. 354
The 0th and the 1st Tableaux	p. 355
The First Differential	p. 355
Encoding Cohomology Generators by Arc Pictures	p. 356
The Language of Arcs	p. 356
The Corresponding Cohomology Generators	p. 356
The First Reduction	p. 357
Topology of the Torus Front Complexes	p. 358
Reinterpretation of H(A[subscript t], d[subscript 1]) Using a Family of Cubical Complexes {[Phi subscript m,n,g]}	p. 358
The Torus Front Interpretation	p. 360
Grinding	p. 362
Thin Fronts	p. 364
The Implications for the Cohomology Groups of Hom (C[subscript m], K[subscript n])	p. 366
Euler Characteristic Formula	p. 367
Cohomology with Integer Coefficients	p. 368
Fixing Orientations on Hom and Hom[subscript +] Complexes	p. 368
Signed Versions of Formulas for Generators [sigma subscript V superscript S]	p. 370
Completing the Calculation of the Second Tableau	p. 371
Summary: the Full Description of the Groups H* (Hom (C[subscript m], K[subscript n]); Z)	p. 374
Bibliographic Notes and Conclusion	p. 376
References	p. 377
Index	p. 385
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Combinatorial Algebraic Topology

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