Singular cohomology

Remarks

A continuous map $f \colon X \to Y$ induces $f^* \colon H^n(Y; G) \to H^n(X; G)$, so $H^n(-; G)$ is a contravariant functor; compare the Hom functor. Evaluation $\langle \varphi, c \rangle = \varphi(c)$ of a cochain on a chain gives the Kronecker pairing $H^n(X; G) \times H_n(X) \to G$. The universal coefficient theorem gives a split exact sequence $0 \to \operatorname{Ext}(H_{n-1}(X), G) \to H^n(X; G) \to \Hom(H_n(X), G) \to 0$, with Ext the Ext functor and $H_n$ singular homology. When $G = \Z$ one writes $H^n(X)$.

Example: a $0$-cochain $\varphi$ is a function from points of $X$ to $G$, and $(\delta \varphi)(\sigma) = \varphi(\sigma(1)) - \varphi(\sigma(0))$ for a path $\sigma$. So $\varphi$ is a cocycle exactly when it is constant on each path component. Hence $H^0(X; G)$ is the group of functions from the set of path components of $X$ to $G$.

Non-example: on $X = [0, 1]$ the $0$-cochain $\varphi(x) = x$ with $G = \R$ is not a cocycle, because $\delta\varphi$ takes the value $1$ on the path $t \mapsto t$.