Homological Algebra

Abstract

The topologist Witold Hurewicz observed in 1935 that, if X is an aspherical path-connected polyhedron, then the homotopy type of X is determined by the fundamental group -π ¹ X of X. Thus, in particular, the homology groups of X are functions of π = π ¹ X. The Swiss topologist Heinz Hopf realized that this must mean that there was a purely algebraic procedure for passing from the group π to the homology groups of X, which we may now think of as the homology groups of π. Hopf then proceeded to invent such an algebraic procedure, guided by his own experience working on the homology theory of topological spaces, and bearing in mind his seminal work on the influence of the fundamental group of a path-connected topological space on its second homology group. Hopf, working in the early 1940’s, considered a free resolution of ℤ, the additive group of integers regarded as a trivial π-module; that is, an exact sequence of π-modules and π-module homomorphisms

$$ \cdots \mathop \to \limits^\partial {F_n}\mathop \to \limits^\partial {F_{n - 1}}\mathop \to \limits^\partial \cdots \mathop \to \limits^\partial {F_0}\mathop \to \limits^\varepsilon $$

((1))

where each F ⁿ is free. One then tensors (1) with a π-module B, i.e., one takes the tensor product over π of (1) with B, to obtain a chain-complex of abelian groups

$$ \cdots \mathop \to \limits^{\partial \otimes 1} {F_n}{ \otimes _\pi }B\mathop \to \limits^{\partial \otimes 1} {F_{n - 1}}{ \otimes _\pi }B\mathop \to \limits^{\partial \otimes 1} \cdots \mathop \to \limits^{\partial \otimes 1} {F_0}{ \otimes _\pi }B$$

((2))

and calculates the homology groups of this chain-complex.