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 compact 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 compact support
1．2．4 Multiplicative cohomology theories
1．3 Thom isomorphism and Gysin map
1．3．1 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-Duajity
1．4．3 Extension
2 Spectral sequences
2．1 Generalsetting
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 of the quotients
2．4．2 Axiomatization
2．4．3 Generic cohomology theory
3 Atiyah-Hirzebruch spectral sequence
4 K-theory
……

II Line bundles and gerbes
III Type II superstring backgrounds
IV Pinors and spinors

A Appendices of Part I

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