Basic Algebraic Topology

Chapter 4 Homology Groups

Recall the homology version of Cauchy’s theorem in complex analysis of 1-variable ([Shastri, 2009] p.214):

Theorem 4.0.1 Let Ω be an open subset in $\mathbb{C}$ and γ be a cycle in Ω. Then the following conditions on γ are equivalent:

1. (a) ${\int }_{\gamma }fdz=0$ for all holomorphic functions f on Ω.
2. (b) The winding number of γ around a, $\eta (\gamma ,a)=0$ for all a ∈ $\mathbb{C}$ \Ω.

A cycle γ satisfying the condition (a) or equivalently, (b), was called null-homologous.

As indicated in the introduction to Chapter 1, the homology and cohomology theories have their roots in complex analysis. In the integral calculus of several variables, especially in Stokes’ theorem, you may witness a neat interaction between homology and cohomology theories. However, it was Poincaré ...