Professor Alexandru I. Suciu

MTH 3107 Topology II

Winter 2003

This course is an introduction to homology theory. We will start with simplicial and singular homology theory: axioms, homological algebra, homology with coefficients, Mayer-Vietoris sequence, degrees of maps, Euler characteristic. Next, we will introduce CW-complexes and study cellular homology, as well as homology with coefficients. We will illustrate these techniques with many geometrical examples (surfaces, projective spaces, grassmanians, lens spaces, products, etc), and derive various applications (Invariance of Domain, Jordan Curve Theorem, Borsuk-Ulam Theorem, Brouwer and Lefschetz-Hopf Fixed Point Theorems, etc).

There are several software packages that facilitate the computation of homology groups. To start with, we will use the Simplicial Homology package, which runs under the GAP (Groups, Algorithms and Programming) system for computational discrete algebra. Other useful packages are Macaulay 2, a software system for research in Algebraic Geometry and Commutative Algebra, and Mathematica, a very comprehensive symbolic programming system. An advanced program is Kenzo, designed specifically for Computational Algebraic Topology.