Hatcher-Algebraic-Topology · Problem 3.1.5

Cohomology

Mathematics — Graph Theory · cocycle-path-properties

Problem

Regarding a cochain φ∈C1(X;G)\varphi \in C^1(X; G) as a function from paths in XX to GG, show that if φ\varphi is a cocycle, then (a) φ(fg)=φ(f)+φ(g)\varphi(fg) = \varphi(f) + \varphi(g), (b) φ\varphi takes the value 00 on constant paths, (c) φ(f)=φ(g)\varphi(f) = \varphi(g) if f≃gf \simeq g, (d) φ\varphi is a coboundary iff φ(f)\varphi(f) depends only on the endpoints of ff, for all ff. [In particular, (a) and (c) give a map H1(X;G)→Hom(π1(X),G)H^1(X; G) \to \mathrm{Hom}(\pi_1(X), G), which the universal coefficient theorem says is an isomorphism if XX is path-connected.]

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