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.