Algebraic Topology · Problem 3.1.4

Cohomology — Allen Hatcher

Mathematics — Topology · homology-groups-hom-chain-complex

Problem

What happens if one defines homology groups

h_n(X; G)

as the homology groups of the chain complex

\cdots \to \text{Hom}(G, C_n(X)) \to \text{Hom}(G, C_{n-1}(X)) \to \cdots

? More specifically, what are the groups

h_n(X; G)

when

G = \mathbb{Z}

\mathbb{Z}_m

, and

\mathbb{Q}