Singular and cellular local chain complexes

Let $\mathcal{L}$ be a left $R$-module local system on an arbitrary space $X$. For a singular simplex $\sigma :{\mathrm{\Delta }}^n\to X$, write $v_i=\sigma (e_i)$ and ${\gamma }_{01}^{\sigma }(t)=\sigma (1-t,t,0,\dots ,0)$.

The intrinsic singular local chain group is $C_n^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(X;\mathcal{L})={⨁}_{\sigma :{\mathrm{\Delta }}^n\to X}{\mathcal{L}}_{v_0}\mathrm{.}$ Write an element of the $\sigma$-summand as $m\sigma$. Its boundary is $\mathrm{\partial }(m\sigma )=T_{{\gamma }_{01}^{\sigma }}(m)(\sigma {\delta }_0)+{\sum }_{i=1}^n(-1{)}^{i}m(\sigma {\delta }_i),$ and the boundary of a zero-simplex is zero. The exceptional zeroth face transports its coefficient from the old first vertex $v_0$ to the new first vertex $v_1$; all other faces keep $v_0$.

The intrinsic singular local cochain group is the product $C_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}^n(X;\mathcal{L})={\prod }_{\sigma :{\mathrm{\Delta }}^n\to X}{\mathcal{L}}_{v_0},$ so a cochain $\phi$ assigns $\phi (\sigma )\in {\mathcal{L}}_{v_0}$ to every simplex, with no finite-support requirement. Its positive coboundary, consistent with Singular cochain complex with coefficients, is $(\delta \phi )(\sigma )=T_{{\gamma }_{01}^{\sigma }}^{-1}(\phi (\sigma {\delta }_0))+{\sum }_{i=1}^{n+1}(-1{)}^i\phi (\sigma {\delta }_i)\mathrm{.}$ Now the exceptional face value is transported back from $v_1$ to $v_0$.

For $A\subseteq X$, restrict $\mathcal{L}$ along the inclusion and set $C_{\ast }^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(X,A;\mathcal{L})=C_{\ast }^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(X;\mathcal{L})\mathrm{/}{C}_{\ast }^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(A;\mathcal{L}),C_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}^{\ast }(X,A;\mathcal{L})=\ker (C_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}^{\ast }(X;\mathcal{L})\to C_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}^{\ast }(A;\mathcal{L}))\mathrm{.}$

For a connected CW complex with chosen basepoint $x$, universal cover $\tilde{X}$, $\pi ={\pi }_1(X,x)$, and base fiber $M={\mathcal{L}}_x$ carrying $gm=T_{\bar{g}}m$, the equivalent universal-cover models are $C_{\ast }^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(\tilde{X};R){\otimes }_{R[\pi ]}M,{\mathrm{Hom}}_{R[\pi ]}({}_{R[\pi ]}{C}_{\ast }^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(\tilde{X};R),M)\mathrm{.}$ The first tensor product is the balanced construction of The tensor product $M{\otimes }_{R}N$ from the additive group underlying the free $\mathbb{Z}$-module on $M\times N$, elementary tensors, and finite tensor sums using the right action of Right action on universal-cover chains. For the second, convert that right chain module to a left one by $g\cdot c=c\cdot g^{-1}\mathrm{.}$ Thus an equivariant cochain has the correctly typed rule $\phi (c\cdot g)=g^{-1}\phi (c),$ not a module-Hom between one right and one left module.

The intrinsic/tensor identification sends $\tilde{\sigma }\otimes m$ to the simplex $\sigma =p\tilde{\sigma }$ with coefficient obtained by transporting $m$ along the path represented by $\tilde{\sigma }(e_0)$. If $\tilde{\sigma }$ is replaced by $\tilde{\sigma }\cdot g$, that path is changed by the loop $g^{-1}$ and the coefficient becomes the transport of $gm$, exactly the balanced relation. The analogous statement for cochains proves the displayed equivariance rule. These maps respect the two boundary formulas term by term.

The cellular local chain and cochain groups of a connected CW complex are $C_{\ast }^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(X;\mathcal{L})=C_{\ast }^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(\tilde{X};R){\otimes }_{R[\pi ]}M,C_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}^{\ast }(X;\mathcal{L})={\mathrm{Hom}}_{R[\pi ]}({}_{R[\pi ]}{C}_{\ast }^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(\tilde{X};R),M)\mathrm{.}$ For a CW pair, quotient by the lifted subcomplex before tensoring and use the corresponding relative cellular complex before applying equivariant Hom. For a disconnected CW complex, take the direct sum of chain complexes and the degreewise product of cochain complexes over its components. Universal-cover coordinates of this componentwise description require a supplied basepoint in each component; the intrinsic complexes require no simultaneous basepoint choice and remain the definition when no such family is supplied.

The next lemma proves square-zero and independence of supplied lift bases. Empty spaces and negative degrees give zero groups, the zero local system gives the zero complexes, and constant systems reduce to the ordinary formulas because every transport is the identity.