Regarding a cochain \(\varphi \in C^1(X; G)\) as a function from paths in \(X\) to \(G\), show that if \(\varphi\) is a cocycle, then
- \(\varphi(f \cdot g) = \varphi(f) + \varphi(g)\),
- \(\varphi\) takes the value 0 on constant paths,
- \(\varphi(f) = \varphi(g)\) if \(f \simeq g\),
- \(\varphi\) is a coboundary iff \(\varphi(f)\) depends only on the endpoints of \(f\), for all \(f\).
[In particular, (a) and (c) give a map \(H^1(X; G) \to \operatorname{Hom}(\pi_1(X), G)\), which the universal coefficient theorem says is an isomorphism if \(X\) is path-connected.]
Solution
1. A \(1\)-cochain \(\varphi\) assigns to each path \(f\) a value \(\varphi(f) \in G\), and \(\delta\varphi = 0\) (cocycle condition) means \(\varphi(\partial\sigma) = 0\) for every \(2\)-simplex \(\sigma\).
definition of cocycle.
2. (a) For paths \(f, g\) with \(f(1) = g(0)\), the concatenation \(f \cdot g\) bounds a \(2\)-simplex (the triangle with edges \(f\), \(g\), and \(f \cdot g\)), so \(\varphi(f \cdot g) - \varphi(f) - \varphi(g) = \varphi(\partial\sigma) = 0\).
1 (the cocycle vanishes on the boundary of the triangle).
3. Hence \(\varphi(f \cdot g) = \varphi(f) + \varphi(g)\).
2.
4. (b) A constant path \(c\) satisfies \(c \cdot c = c\), so \(\varphi(c) = \varphi(c \cdot c) = \varphi(c) + \varphi(c)\), forcing \(\varphi(c) = 0\).
3.
5. (c) If \(f \simeq g\) (rel endpoints), then \(f\) and \(g\) differ by the boundary of a \(2\)-chain (a homotopy gives a \(2\)-chain whose boundary is \(f - g\)), so \(\varphi(f) - \varphi(g) = \varphi(\partial(\text{homotopy})) = 0\).
1.
6. Hence \(\varphi(f) = \varphi(g)\).
5.
7. (d) (\(\Rightarrow\)) If \(\varphi = \delta\psi\) is a coboundary, then \(\varphi(f) = \psi(f(1)) - \psi(f(0))\) depends only on the endpoints.
definition of coboundary.
8. (\(\Leftarrow\)) If \(\varphi(f)\) depends only on the endpoints, define \(\psi(x) = \varphi(f_x)\) for any path \(f_x\) from a fixed basepoint to \(x\); then \(\varphi(f) = \psi(f(1)) - \psi(f(0)) = \delta\psi(f)\), so \(\varphi\) is a coboundary.
7, reversed.
9. Q.E.D.
3, 4, 6, <1>7–<1>8.