# Topology II: Introduction to Homology and Cohomology Theory

The emphasis of the course is on the interconnections of modern
topology with other branches of mathematics and on concrete
topological spaces (manifolds, vector bundles) rather than the
most general abstract categories.

Prerequisites: Topology: topological spaces, continuity, connectedness, compactness,homeomorphisms;

Linear Algebra;
Algebra: abelian groups, subgroups, and quotient (factor) groups.

### Curriculum:

1. Chain complexes, cycles, boundaries, and homology groups.
Polyhedra, triangulations, and simplicial homology groups of
topological spaces. Betti numbers and Euler characteristic.
Homology groups of classical surfaces: sphere, torus, Klein
bottle, projective plane.

2. Cell spaces or CW-complexes. Cell chains, cycles, boundaries,
and incidence coefficients. Cell homology groups and their
coincidence with simplicial homology groups (without proof).
Examples: multidimensional spheres, tori and projective spaces.

3. The long exact sequence associated with a short sequence of chain
complexes. Relative homology groups. The exact sequence of a pair
of topological spaces. Mayer–Vietoris exact sequence. Computing
the homology groups of some topological spaces.

4. Main topological constructions: product, quotient space, cone,
wedge, suspension, join, loop space and their homology groups.

5. Homotopy. Classification of mappings from a circle to itself and
their degrees or rotation numbers. The index of an isolated
singular point of a plane vector field. Poincaré index theorem:
the sum of the indices of a vector field on a surface is equal to
its Euler characteristic. Brushing a sphere. Brouwer fixed point
theorem.

6. Singular homology groups of topological spaces. Homology groups of
a point. Homotopy invariance of singular homology groups. Exact
sequences of pairs and triples. Homology groups of spheres. The
coincidence of singular, cell, and simplicial homology groups.

7. Intersection of submanifolds and cycles. Homological
interpretation of the index of a vector field. Lefschetz fixed
point theorem and its applications.

8. Cohomology groups and Poincaré duality. De Rham cohomology
groups. Multiplication of cocycles and its applications.

### Textbooks

• S.Matveev, Lectures on Algebraic Topology, AMS, 1999. 