组合代数拓扑

出版时间：2011-6  出版社：科学出版社  作者：科兹洛夫  页数：389  
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内容概要

　　该书是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，本书的读者可以是几何，拓扑和代数方向的数学工作者和研究生。

书籍目录

1　OverturePart I Concepts of Algebraic Topology2　Cell Complexes2.1 Abstract Simplicial Complexes2.1.1 Definition of Abstract Simplicial Complexes and Maps Between Them2.1.2 Deletion，Link，Star，and Wedge2.1.3 Simplicial Join2.1.4 Face Posets2.1.5 Barycentric and Stellar Subdivisions2.1.6 Pulling and Pushing Simplicial Structures2.2 Polyhedral Complexes2.2.1 Geometry of Abstract Simplicial Complexes2.2.2 Geometric Meaning of the Combinatorial Constructions2.2.3 Geometric Simplicial Complexes2.2.4 Complexes Whose Cells Belong to a Specified Set of Polyhedra2.3 Trisps2.3.1 Construction Using the Gluing Data2.3.2 Constructions Involving Trisps2.4 CW Complexes2.4.1 Gluing Along a Map2.4.2 Constructive and Intrinsic Definitions2.4.3 Properties and Examples3 Homology Groups3.1　Betti Numbers of Finite Abstract Simplicial Complexes3.2 Simplicial Homology GroupsX Contents3.2.1 Homology Groups of Trisps with Coefficients in Z23.2.2 Orientations3.2.3 Homology Groups of Trisps with Integer Coefficients3.3 Invariants Connected to Homology Groups3.3.1 Betti Numbers and Torsion Coefficients3.3.2　Euler Characteristic and the Euler-Poincar6 Fc'rmula3.4 Variations3.4.1　Augmentation and Reduced Homology Groups3.4.2 Homology Groups with Other Coefficients3.4.3 Simplicial Cohomology Groups3.4.4 Singular Homology3.5 Chain Complexes3.5.1 Definition and Homology of Chain Complexes3.5.2　Maps Between Chain Complexes and Induced Mapson Homology3.5.3 Chain Homotopy3.5.4 Simplicial Homology and Cohomology in the Contextof Chain Complexes3.5.5 Homomorphisms on Homology Induced by Trisp Maps3.6 Cellular Homology3.6.1 An Application of Homology with Integer Coefficients：Winding Number3.6.2 The Definition of Cellular Homology3.6.3 Cellular Maps and Properties of Cellular Homology4　Concepts of Category Theory4.1 The Notion of a Category4.1.1 Definition of a Category.Isomorphisms4.1.2 Examples of Categories4.2 Some Structure Theory of Categories4.2.1　Initial and Terminal Objects4.2.2 Products and Coproducts4.3 Functors4.3.1　The Category Cat4.3.2 Homology and Cohomology Viewed as Functors4.3.3　Group Actions as Functors4.4 Limit C0nstructions4.4.1 Definition of Colimit of a Functor4.4.2　Colimits and Infinite Unions4.4.3 Quotients of Group Actions as Colimits4.4.4 Limits4.5 Comma Categories4.5.1 Objects Below and Above Other Objects4.5.2 The General Construction and Further Examples……PartⅡ Methods of Combinatorial Algebraic TopologyPartⅢ Complexes of Graph HomomorphismsReferencesIndex

编辑推荐

　　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.