Homology
S.V. Duzhin
A program of the course at the 2nd Italian Diffiety School,
Forino, February 22 - March 3, 1999
1. Simplicial homology.
Simplicial complex. Chains, cycles, boundaries. Homology. The problem of topological invariance of homology groups. Homology of 1- and 2-dimensional manifolds.

2. Algebraic homology.
General algebraic definition of a chain and cochain complex. Morphisms of complexes. Algebraic homotopy. The long exact sequence of homologies arising from a short exact sequence of chain complexes.

3. Cell homology.
Cell complexes and their homology. Relation with simplicial homology. Examples of computation: Sn, RPn, CPn.

4. Singular homology.
Singular chain complex. Equivalence of singular homology and cell homology. Homotopy invariance of homology groups.

5. Cohomology.
Dual complex. Cohomology. Relation between homology and cohomology. Multiplication in cohomologies.

6. Fibre bundles.
Fibre bundles. Examples.

7. Spectral sequences.
General definition of a spectral sequence.

8. Leray-Serre spectral sequence.
Filtration in the chain complex of a fibre space induced by the fibering. Spectral sequence of a fibre bundle. Examples: S3, S2´S1.

Questions and suggestions should go to Jet Nestruev, jet @ diffiety.ac.ru.