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Algebraic Topology
Andries Brouwer,
aeb@cwi.nl
v1.0, 991111
Some fragments of algebraic topology.
1.
Introduction
2.
Topology
2.1 Topological spaces
2.2 Operations on topological spaces
2.3 Separation
2.4 Covering
2.5 Convergence
2.6 Connectedness
2.7 Local properties
2.8 Function spaces
3.
Homotopy
3.1 Homotopic maps
3.2 Homotopy classes
3.3 Fundamental group
3.4 Morphisms
3.5 Simple connectedness
3.6 The fundamental group of a product
3.7 The fundamental group of a retract
3.8 The fundamental group of a union
3.9 The fundamental group of a topological group is Abelian
3.10 The fundamental group of orthogonal groups
3.11 Hopf spaces
3.12 Higher homotopy groups
3.13 Fiber spaces
3.14 Some results for spheres
4.
Two-dimensional manifolds
4.1 Classification
4.2 The Poincaré Conjecture
4.3 2-Manifolds with boundary
5.
Knots, Links, Braids
5.1 Wild embeddings
5.2 Knot diagrams and Reidemeister moves
5.3 Prime knots and Seifert surfaces
5.4 Catalog
5.5 Invariants - the Kauffman bracket - the Jones polynomial
5.6 Links
5.7 Braids
6.
Homology
6.1 Homology of chain complexes
6.2 Homology of exterior algebra
6.3 Simplicial homology
6.4 Singular homology
6.5 .Cech homology
6.6 Chain homotopy
6.7 Singular Homology and Homotopy
6.8 Exact sequences
6.9 The Maier-Vietoris sequence
6.10 Relative homology
6.11 Betti numbers, Euler characteristic, Lefschetz number,
6.12 Axiomatic homology
7.
Cohomology
7.1 The cup product
7.2 The cap product
8.
Products
8.1 The product of two chain complexes
8.2 The Eilenberg-Zilber theorem
8.3 Künneth formula
8.4 Homology with coefficients
8.5 The cross product
9.
Homological algebra
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