Ⅰ Homology and cohomology theories
1 Foundations
1.1 Preliminaries
1.1.1 Singular homology and cohomology
1.1.2 Borel-Moore homology and cohomology with cornpact support
1.1.3 CW-complexes
1.1.4 Simplicial complexes
1.1.5 Categories of topological spaces
1.1.6 Basic operations on topological spaces
1.2 Eilenberg-Steenrod axioms
1.2.1 Reduced homology and cohomology
1.2.2 First properties
1.2.3 Borel-Moore homology and cohomology with com pact support
1.2 4 Multiplicative cohomology theories
1.3 Thom isomorphism and Gysin map
1.3 Fiber bundles and module structure
1.3.2 Orientability and Thom isomorphism
1.3.3 Gysin map
1.4 Finite CW-complexes
1.4.1 Whitehead axioms
1.4.2 S-Duality
1.4.3 Extension
2 Spectral sequences
2.1 General setting
2.2 Finite filtrations
2.2.1 Preliminaries
2.2.2 First viewpoint
2.2.3 Second viewpoint
2.3 Grading and double complexes
2.3.1 Grading and regular filtrations
2.3.2 Double complexes
2.4 Generalization
2.4.1 Cohomology ofthe quotients
2.4.2 Axiomatization
2.4.3 Generic cohomology theory
3 Atiyah-Hirzebruch spectral sequence
3.1 Description of the spectral sequence
3.1.1 The first step
3.1.2 The second step
3.1.3 The last step
3.1.4 nom the first to the last step
3.2 Gysin map and AHSS
3.2.1 Unit class
3.2.2 Generic cohomology class
4 K-theory
4.1 Basic notions of K-theory
4.1.1 General definitions
4.1.2 Products in K-theory
4.1.3 Bott periodicity
4.1.4 K-theory as a cohomology theory
……
Ⅱ Line bundles and gerbes
Ⅲ Type Ⅱ superstring backgrounds
Ⅳ Pinors and spinors
A Appendices of Part Ⅰ
A.1 Direct sum and direct product
A.2 Compactifications
B Appendices of Part Ⅱ
B.1 12ech Hypercohomology
B.2 Gerbes
C Appendices of Part Ⅲ
C.1 P-Gerbes
C.2 Hodge- with minkowskian signature
编辑手记
