Real singular cochain complex

For a topological space $X$, the real singular cochain complex is $C^k(X;\mathbb{R})={\mathrm{Hom}}_{\mathbb{R}}(C_k(X;\mathbb{R}),\mathbb{R}),{\delta }^k\phi =\phi \circ {\mathrm{\partial }}_{k+1}\mathrm{.}$ Here the chains and boundary are Real singular chain complex, so $C^k=0$ for $k<0$. Addition and scalar multiplication of cochains are pointwise. Precomposition with the real-linear boundary is real-linear. Explicitly, for $k\ge 0$, $({\delta }^k\phi )([\sigma ])={\sum }_{i=0}^{k+1}(-1{)}^i\phi ([\sigma \circ {\delta }_i])\mathrm{.}$ There is no extra degree sign. For every chain $c$, $({\delta }^{k+1}{\delta }^k\phi )(c)=\phi ({\mathrm{\partial }}_{k+1}{\mathrm{\partial }}_{k+2}c)=0$, so ${\delta }^2=0$. The same assertion in negative degrees follows from the zero source, including ${\delta }^{-1}=0$.

By Real singular cochains identify with functions on the supplied simplex basis, evaluation on the supplied basis identifies this with Singular cochain complex with coefficients for coefficient group $\mathbb{R}$: it is ${\mathrm{Hom}}_{\mathbb{Z}}(C_k(X;\mathbb{Z}),\mathbb{R})$, not ${\mathrm{Hom}}_{\mathbb{Z}}(C_k(X;\mathbb{R}),\mathbb{R})$. That lemma verifies the identification commutes with coboundary. A cochain may have arbitrary values on infinitely many simplices; it is not required to have finite support.

At degree zero a cochain is a function on points, and its coboundary on a path is the final value minus the initial value. For the empty space all cochains vanish. For a point the unnormalized cochain groups are $\mathbb{R}$ in each nonnegative degree, with ${\delta }^k=0$ for even $k$ and ${\delta }^k=\mathrm{id}$ for odd $k$. No choice is involved.