Kronecker evaluation pairing

Fix a space $X$, abelian group $G$ and integer $n\ge 0$. For a singular cocycle $\phi \in Z^n(X;G)$ and an integral singular cycle $c\in Z_n(X;\mathbb{Z})$, define their evaluation to be $\phi (c)\in G$, using Singular cohomology with coefficients and The singular chain complex and singular homology. If $c={\sum }_{\sigma }{a}_{\sigma }\sigma$ is its finite formal expansion, this value is ${\sum }_{\sigma }{a}_{\sigma }\phi (\sigma )$.

The Kronecker pairing on quotient classes is the rule $⟨[\phi ],[c]⟩=\phi (c),H^n(X;G)\times H_n(X;\mathbb{Z})\to G\mathrm{.}$ Its descent through both quotients, biadditivity and naturality are justified by the following lemma The kronecker pairing is independent of cocycle and cycle representatives ↗, before any use of the quotient pairing. It is not an assertion that evaluation gives an isomorphism to the full dual of homology.

For negative $n$, the two groups are zero and the pairing is the zero map. Empty $X$ and zero $G$ also give zero evaluation. When $G$ is an $R$-module, the cohomology variable is $R$-linear and the integral homology variable is additive; an $R$-bilinear pairing instead uses chains and homology over $R$ with $R$-linear cochains. No representative-selection function is included in the definition.