组合代数拓扑

1 Overture

Part I Concepts of Algebraic Topology

2 Cell Complexes

2.1 Abstract Simplicial Complexes

2.1.1 Definition of Abstract Simplicial Complexes and Maps Between Them

2.1.2 Deletion，Link，Star，and Wedge

2.1.3 Simplicial Join

2.1.4 Face Posets

2.1.5 Barycentric and Stellar Subdivisions

2.1.6 Pulling and Pushing Simplicial Structures

2.2 Polyhedral Complexes

2.2.1 Geometry of Abstract Simplicial Complexes

2.2.2 Geometric Meaning of the Combinatorial Constructions

2.2.3 Geometric Simplicial Complexes

2.2.4 Complexes Whose Cells Belong to a Specified Set of Polyhedra

2.3 Trisps

2.3.1 Construction Using the Gluing Data

2.3.2 Constructions Involving Trisps

2.4 CW Complexes

2.4.1 Gluing Along a Map

2.4.2 Constructive and Intrinsic Definitions

2.4.3 Properties and Examples

3 Homology Groups

3.1 Betti Numbers of Finite Abstract Simplicial Complexes

3.2 Simplicial Homology Groups

X Contents

3.2.1 Homology Groups of Trisps with Coefficients in Z2

3.2.2 Orientations

3.2.3 Homology Groups of Trisps with Integer Coefficients

3.3 Invariants Connected to Homology Groups

3.3.1 Betti Numbers and Torsion Coefficients

3.3.2 Euler Characteristic and the Euler-Poincar6 Fc'rmula

3.4 Variations

3.4.1 Augmentation and Reduced Homology Groups

3.4.2 Homology Groups with Other Coefficients

3.4.3 Simplicial Cohomology Groups

3.4.4 Singular Homology

3.5 Chain Complexes

3.5.1 Definition and Homology of Chain Complexes

3.5.2 Maps Between Chain Complexes and Induced Mapson Homology

3.5.3 Chain Homotopy

3.5.4 Simplicial Homology and Cohomology in the Contextof Chain Complexes

3.5.5 Homomorphisms on Homology Induced by Trisp Maps

3.6 Cellular Homology

3.6.1 An Application of Homology with Integer Coefficients：Winding Number

3.6.2 The Definition of Cellular Homology

3.6.3 Cellular Maps and Properties of Cellular Homology

4 Concepts of Category Theory

4.1 The Notion of a Category

4.1.1 Definition of a Category.Isomorphisms

4.1.2 Examples of Categories

4.2 Some Structure Theory of Categories

4.2.1 Initial and Terminal Objects

4.2.2 Products and Coproducts

4.3 Functors

4.3.1 The Category Cat

4.3.2 Homology and Cohomology Viewed as Functors

4.3.3 Group Actions as Functors

4.4 Limit C0nstructions

4.4.1 Definition of Colimit of a Functor

4.4.2 Colimits and Infinite Unions

4.4.3 Quotients of Group Actions as Colimits

4.4.4 Limits

4.5 Comma Categories

4.5.1 Objects Below and Above Other Objects

4.5.2 The General Construction and Further Examples

……

PartⅡ Methods of Combinatorial Algebraic Topology

PartⅢ Complexes of Graph Homomorphisms

References

Index

《组合代数拓扑》是Springer的AlgorithmsandComputationinMathematics丛书系列第21卷，作者多年来从事离散数学，代数拓扑，理论计算机科学。组合代数拓扑是代数拓扑和离散数学的交叉。属于"反映学术前沿进展的优秀学术著作"这一类。比较专门，《组合代数拓扑》的读者可以是几何，拓扑和代数方向的数学工作者和研究生。

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 swft 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 modem research in this active field.