S.V. Duzhin

A program of the course at the 2nd Italian Diffiety School,

Forino, February 22 - March 3, 1999

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: S^{n}, **R**P^{n}, **C**P^{n}.

**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:
S^{3}, S^{2}´S^{1}.

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