Relative singular cochain complex

For a subspace $A\subseteq X$ and an abelian group $G$, the relative singular cochains are $C^n(X,A;G)={\mathrm{Hom}}_{\mathbb{Z}}(C_n(X,A;\mathbb{Z}),G)(n\ge 0),$ and zero for $n<0$. The domain is the quotient chain group of Relative singular chain complex. The relative coboundary is positive precomposition with its induced boundary: $\delta \phi =\phi \bar{\mathrm{\partial }}$. The well-defined relative chain differential of Boundary on relative chains squares to zero, so precomposition twice also gives zero.

Precomposing with the quotient $q:C_n(X;\mathbb{Z})\to C_n(X,A;\mathbb{Z})$ identifies $C^n(X,A;G)$ with the subgroup of $C^n(X;G)$ vanishing on $C_n(A;\mathbb{Z})$. Indeed $\phi q$ vanishes there; conversely a homomorphism $\psi$ vanishing there defines $\bar{\psi }([c])=\psi (c)$, independently of the representative because its possible difference lies in $C_n(A;\mathbb{Z})$. These inverse maps are additive and intertwine coboundaries by $q\mathrm{\partial }=\bar{\mathrm{\partial }}q$.

In the simplex-function description of Singular cochain complex with coefficients, these are exactly functions zero on simplices whose image lies in $A$. Their faces also lie in $A$, so their coboundaries still vanish there. The quotient integer chain group is free on the complementary set of simplices: remove the coefficients on simplices in $A$ from each finite formal chain; the remaining coefficients uniquely determine its relative class. This identifies the quotient with a free group without a choice of representatives. It does not assert that extension by zero commutes with the differential.

Define $H^n(X,A;G)=\ker (\delta :C^n(X,A;G)\to C^{n+1}(X,A;G))\mathrm{/}\mathrm{im}(\delta :C^{n-1}(X,A;G)\to C^n(X,A;G))\mathrm{.}$ The image is inside the kernel by the relative square-zero calculation. As in Singular cohomology with coefficients, this is an abelian quotient, zero in negative degrees; degree zero has no incoming coboundaries. For $A=\mathrm{\varnothing }$ this is the absolute complex and cohomology, whereas $A=X$ gives zero in every degree. Empty $X$ and zero $G$ are included. All identifications are explicit and require no AC.