Regarding a cochain φ∈C1(X;G) as a function from paths in X to G, show that if φ is a cocycle, then
(a) φ(fg)=φ(f)+φ(g),
(b) φ takes the value 0 on constant paths,
(c) φ(f)=φ(g) if f≃g,
(d) φ is a coboundary iff φ(f) depends only on the endpoints of f, for all f.
[In particular, (a) and (c) give a map H1(X;G)→Hom(π1(X),G), which the universal coefficient theorem says is an isomorphism if X is path-connected.]