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.