Local Coefficients, Twisted Homology, and Duality

Reversal respects endpoint-fixed path homotopy, is its own inverse on classes, and sends $[c_x]$ to itself. Hence ${\rho }_x$ is a canonical bijection that preserves the identity and inverses.

Reversing a concatenation gives $\overline{\alpha \ast \beta }=\bar{\beta }\ast \bar{\alpha }$ up to the standard endpoint-fixed reparametrization. By [F3], ${\rho }_x([\alpha ])\circ {\rho }_x([\beta ])=[\bar{\alpha }]\circ [\bar{\beta }]=[\bar{\beta }\ast \bar{\alpha }]={\rho }_x([\alpha ][\beta ])$. Thus ${\rho }_x$ is a homomorphism.

The bijective homomorphism in steps 1.1–1.2 is the asserted group isomorphism. Without reversal, [F3] gives $[\alpha ]\circ [\beta ]=[\beta \ast \alpha ]$, which proves the final anti-isomorphism warning as well.

A connected CW complex is path connected. The image of each characteristic disk is path connected, so it lies in one path component. Hence every path component $P$ intersects each closed cell either in the whole cell or not at all. Both $P$ and its complement therefore have closed intersection with every closed cell, and the weak-topology test in [F5] makes both sets closed. Thus $P$ is also open; connectedness leaves only one path component. Consequently the set of paths $x\to y$ is nonempty for every $y\in X$. Use [F4] once to choose such a path $p_y$, taking $p_x=c_x$.

For a local system $\mathcal{L}$, give ${\mathcal{L}}_x$ the action in the statement. If $a,b$ are loop classes, covariance gives $T_{ab}=T_b{T}_a$ and hence $T_{\overline{ab}}=T_{\bar{a}}{T}_{\bar{b}}$; therefore $(ab)m=a(bm)$. Constants act identically and reversals act inversely. The operators are $R$-linear, so [F3] extends this action uniquely to an $R[{\pi }_1(X,x)]$-module. Naturality makes the component ${\eta }_x$ of every coefficient morphism equivariant. This defines the evaluation functor $E$.

Conversely let $M$ be a left $R[{\pi }_1(X,x)]$-module. Define a local system $Q(M)$ with every fiber equal to the underlying $R$-module $M$. For a path $\gamma :y\to z$, put ${\mathrm{\ell }}_{\gamma }=[p_y\ast \gamma \ast {\bar{p}}_z]\in {\pi }_1(X,x)$ and $Q(M)([\gamma ])(m)={\mathrm{\ell }}_{\gamma }^{-1}\cdot m$. Endpoint-fixed homotopies do not change ${\mathrm{\ell }}_{\gamma }$. Constants give the identity. If $\gamma :y\to z$ and $\delta :z\to w$, cancellation of ${\bar{p}}_z\ast p_z$ gives ${\mathrm{\ell }}_{\gamma \ast \delta }={\mathrm{\ell }}_{\gamma }{\mathrm{\ell }}_{\delta }$, so $Q(M)(\gamma \ast \delta )=({\mathrm{\ell }}_{\gamma }{\mathrm{\ell }}_{\delta }{)}^{-1}\cdot -={\mathrm{\ell }}_{\delta }^{-1}\cdot ({\mathrm{\ell }}_{\gamma }^{-1}\cdot -)=Q(M)(\delta )Q(M)(\gamma )$. Thus $Q(M)$ is a covariant functor. A module map, used on every fiber, is a natural transformation by equivariance; hence $Q$ is a functor.

Since $p_x$ is constant, for a loop $a$ at $x$ the action obtained by evaluating $Q(M)$ is $a\cdot m=Q(M)(\bar{a})m=a\cdot m$. Hence $EQ$ is literally the identity on modules and their maps. For a local system $\mathcal{L}$, define ${\epsilon }_y:Q(E\mathcal{L}{)}_y={\mathcal{L}}_x\to {\mathcal{L}}_y$ by ${\epsilon }_y=T_{p_y}$. The action on $E\mathcal{L}$ and the definition of $Q$ give $Q(E\mathcal{L})(\gamma )=T_{p_y\ast \gamma \ast {\bar{p}}_z}$. Therefore $T_{p_z}Q(E\mathcal{L})(\gamma )=T_{\gamma }{T}_{p_y}$, so $\epsilon$ is a natural isomorphism $QE\mathcal{L}\to \mathcal{L}$. It is natural in $\mathcal{L}$ because coefficient morphisms commute with every $T_{p_y}$.

Steps 1.2–3.1 exhibit the required equivalence. A second chosen path family produces another $Q^{\mathrm{\prime }}$ and the component $[p_y\ast {\bar{p}}_y^{\mathrm{\prime }}{]}^{-1}$ acting on $M$ gives the natural isomorphism $Q(M{)}_y\to Q^{\mathrm{\prime }}(M{)}_y$; the same cancellation as in step 2.1 proves naturality. Thus path choices affect the displayed model but not its natural-isomorphism class. The only AC use was the point-indexed family in step 1.1; for a one-point space only the constant path is needed, while the empty case is excluded by the chosen basepoint.

Expand ${\mathrm{\partial }}^2(m\sigma )$. As in the ordinary simplicial cancellation, every codimension-two face occurs twice with opposite signs. If neither deletion removes the current first vertex, both coefficients remain $m$. If only the first vertex is removed, both occurrences use the same edge transport. In the remaining exceptional pair, one occurrence transports along $v_0{v}_1$ and then $v_1{v}_2$, while the other transports along $v_0{v}_2$; these paths are endpoint-fixed homotopic inside $\sigma ({\mathrm{\Delta }}^n)$, so functoriality of the local system makes the transports equal. Hence all paired terms cancel and ${\mathrm{\partial }}^2=0$. Reversing these transports gives the identical paired-face calculation for ${\delta }^2=0$.

Fix a basepoint and write a supplied old lifted oriented $n$-cell basis as $e_j$ with $\mathrm{\partial }{e}_j={\sum }_i{e}_i\cdot r_{ij}$. Any supplied new basis has $e_j^{\mathrm{\prime }}={ϵ}_j{e}_j\cdot a_j$, with ${ϵ}_j\in \{1,-1\}$ and $a_j\in \pi$. Then $\mathrm{\partial }{e}_j^{\mathrm{\prime }}={\sum }_i{e}_i^{\mathrm{\prime }}\cdot ({ϵ}_i{a}_i^{-1}{r}_{ij}{a}_j{ϵ}_j)$. Thus the new incidence matrix is obtained from the old one by the appropriate diagonal changes. In the tensor complex $e_j^{\mathrm{\prime }}\otimes m={ϵ}_j{e}_j\otimes a_{j}m$, so diagonal coefficient change is a chain isomorphism; precomposition by its inverse is the corresponding equivariant-cochain isomorphism.

In the universal-cover models, the singular and cellular boundaries already square to zero and are right $R[\pi ]$-linear. Therefore $(\mathrm{\partial }\otimes 1{)}^2=0$, while precomposition gives ${\delta }^2\phi =\phi {\mathrm{\partial }}^2=0$. The intrinsic/model identifications in [F1] intertwine the formulas, so this also verifies every component and relative quotient or kernel.

If the basepoint changes from $x$ to $x^{\mathrm{\prime }}$, a supplied path $q:x\to x^{\mathrm{\prime }}$ identifies the loop groups by $a\mapsto [\bar{q}\ast a\ast q]$ and the fibers by $T_q$. The local-system identity $T_q{T}_{\bar{a}}=T_{\overline{\stackrel{ˉ}{q}\ast a\ast q}}{T}_q$ intertwines the two left module actions. Lifting $q$ identifies the two pointed universal-cover models and their deck actions, so it yields chain isomorphisms on tensor and equivariant-Hom complexes. A different $q$ changes this comparison by the already accounted-for group action, hence by an isomorphic diagonal basis change rather than by a new homology theory.

Steps 1.1–2.2 prove square-zero and all asserted independence statements. They are conditional on supplied lift/orientation/path choices and do not select a set-indexed family, so no AC is used. Empty spaces, zero modules, zero ring, degree zero, absent cells, and degenerate singular simplices are included in the same zero or paired-face calculations.

Define $(f,\eta {)}_{\mathrm{\#}}(m\sigma )={\eta }_{\sigma (e_0)}(m)(f\sigma )$. On the exceptional zeroth face, [F2] says ${\eta }_{\sigma (e_1)}{T}_{\sigma [0,1]}^{\mathcal{L}}=T_{f\sigma [0,1]}^{\mathcal{K}}{\eta }_{\sigma (e_0)}$; all other faces use the same first vertex. Hence this is a chain map and carries the subcomplex on $A$ into that on $B$, so [F1] gives the asserted $f_{\ast }$.

For a cochain $\phi$ on $Y$, define $((f,\theta {)}^{\mathrm{\#}}\phi )(\sigma )={\theta }_{\sigma (e_0)}(\phi (f\sigma ))$. The same naturality square, inverted on the exceptional face, makes this commute with coboundary. It preserves the relative kernel because $f(A)\subseteq B$, and hence induces the asserted $f^{\ast }$. Taking $f=1_X$ proves covariance in a coefficient morphism for both theories.

Substitution in the two displayed chain-level formulas proves the identity laws. For composable maps $X\stackrel{f}{\to }Y\stackrel{g}{\to }Z$, the homology coefficient morphism is $\mathcal{L}\to f^{\ast }\mathcal{K}\to f^{\ast }{g}^{\ast }\mathcal{N}=(gf{)}^{\ast }\mathcal{N}$, and the cohomology coefficient morphism is the reverse composite; componentwise substitution proves the composition laws without a basepoint or lift choice.

For a homotopy $H$, define $({\tau }_H{)}_x=T_{t\mapsto H(x,t)}^{\mathcal{K}}$. A path square $(s,t)\mapsto H(\gamma (s),t)$ shows by its two boundary routes that these components satisfy the naturality equation, so ${\tau }_H$ is a coefficient isomorphism. Triangulate each prism ${\mathrm{\Delta }}^n\times I$ as in the ordinary prism operator and transport the coefficient from its initial first vertex along the corresponding prism edge. The usual oriented-prism cancellation is unchanged; the only new comparisons are transports along the two boundary routes of a triangular face, and those are equal because the face supplies an endpoint-fixed homotopy. Thus the resulting $P_H$ satisfies $(f_1,{\eta }_1{)}_{\mathrm{\#}}-(f_0,{\eta }_0{)}_{\mathrm{\#}}=\mathrm{\partial }{P}_H+P_H\mathrm{\partial }$ when ${\eta }_1={\tau }_H{\eta }_0$.

Precomposing a local cochain with the prism operator and applying the coefficient map in the reverse direction gives a cochain homotopy $(f_1,{\theta }_1{)}^{\mathrm{\#}}-(f_0,{\theta }_0{)}^{\mathrm{\#}}=P_H^{\ast }\delta +\delta P_H^{\ast }$ when ${\theta }_0={\theta }_1{\tau }_H$. Chain- or cochain-homotopic maps induce equal maps on (co)homology by applying the identity to cycles/cocycles and observing that the difference is a boundary/coboundary. This proves the homotopy clauses. Empty pairs, zero systems, degree zero, degenerate simplices, and constant homotopies obey the same formulas, and no AC is used.

Barycentric subdivision works for intrinsic local chains by transporting each coefficient from the original first vertex to the first vertex of each subsimplex along the straight segment inside the original simplex. The paired-face proof used for ordinary subdivision has only triangular path comparisons, which agree by local-system functoriality; the usual subdivision prism therefore gives $\mathrm{\partial }D+D\mathrm{\partial }=1-S$. For a finite chain, a sufficiently high subdivision is small relative to any excisive open cover. This reproduces the chain-homotopy and excision argument of [F2] with fibers tracked, without assuming the later general local-excision theorem.

Apply step 1.1 to the pair $(X^m\cup A,X^{m-1}\cup A)$. Excision separates the open $m$-cells outside $A$. On each cell, transport from one supplied point trivializes $\mathcal{L}$, and the relative pair is the disk-boundary pair; its chain contraction leaves one copy of that fiber in degree $m$ and zero in every other degree. Chains are finite, so the separated relative group is the direct sum over cells. In universal-cover coordinates this is exactly $C_m^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(\tilde{X},\tilde{A};R){\otimes }_{R[\pi ]}{\mathcal{L}}_x$. Thus consecutive skeletal relative local homology is concentrated in degree $m$, and the displayed intrinsic identification follows.

Write $Y_m=X^m\cup A$, with $Y_{-1}=A$, and $C_m=H_m(Y_m,Y_{m-1};\mathcal{L})$. The degreewise short exact chain sequence for a triple gives the usual connecting map $[c]\mapsto [\mathrm{\partial }c]$ and its exact homology sequence by a direct cycle-boundary chase. Step 2.1 and induction over the skeleta give $H_k(Y_m,A;\mathcal{L})=0$ for $k>m$. For fixed $n$, exactness for $(Y_n,Y_{n-1})$ therefore gives an injection $j_n:H_n(Y_n,A;\mathcal{L})\to C_n$ with image $\ker {\mathrm{\partial }}_n$. The quotient map $H_{n-1}(Y_{n-1},A;\mathcal{L})\to C_{n-1}$ is injective by the same vanishing one skeleton lower, including $n=0$ with $Y_{-1}=A$. Consequently $\ker d_n=\ker {\mathrm{\partial }}_n=j_n{H}_n(Y_n,A;\mathcal{L})$. Finally, exactness for $(Y_{n+1},Y_n)$ and $H_n(Y_{n+1},Y_n;\mathcal{L})=0$ give an exact sequence $C_{n+1}\to H_n(Y_n,A;\mathcal{L})\to H_n(Y_{n+1},A;\mathcal{L})\to 0$. Under $j_n$, the first image is exactly $\mathrm{im}{d}_{n+1}$, so taking the quotient proves $H_n(C_{\ast }^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}})\cong H_n(Y_{n+1},A;\mathcal{L})$.

Attaching cells of dimension greater than $n+1$ does not change $H_n$ because their consecutive relative local groups vanish in degrees $n$ and $n+1$. Every finite singular local cycle, and every finite chain bounding it, lies in some finite skeleton modulo $A$: [F3] places each compact simplex image in a finite CW subcomplex, and the finitely many resulting subcomplexes have a common finite maximum cell dimension. Hence passage through the increasing skeleta is respectively surjective and injective on the colimit, and step 3.1 gives the asserted comparison for arbitrary-dimensional and infinite CW pairs.

A cellular map preserves skeleta, the exceptional coefficient transport by naturality, and the connecting formula $[c]\mapsto [\mathrm{\partial }c]$. Therefore all identifications in steps 2.1–4.1 commute with the chain map from the supplied coefficient morphism. This proves naturality.

With supplied oriented lifts, write $\mathrm{\partial }{\tilde{e}}_j^n={\sum }_i{\tilde{e}}_i^{n-1}\cdot r_{ij}$, where $r_{ij}$ is the finite signed sum of deck elements determined by lifted attaching incidences. Tensoring sends the $j$th fiber element $m$ to the $i$th component $\sum r_{ij}m$. These are group-ring incidences, not ordinary integer degrees. The basis-change lemma [F1] handles altered lifts and orientations. Empty pairs, absent cells, degree zero, the zero ring/system, and disconnected complexes are included componentwise; no choice is made unless a global family of lifts is separately supplied.

The inclusion of local chains on $A$ is injective, and the quotient is the relative local chain group by [F1], so $0\to C_{\ast }(A;\mathcal{L}{\mathrm{\mid }}_A)\to C_{\ast }(X;\mathcal{L})\to C_{\ast }(X,A;\mathcal{L})\to 0$ is degreewise exact. The standard chase in [F3] uses only representatives and ${\mathrm{\partial }}^2=0$, so it gives the first long exact sequence with connector $[c]\mapsto [\mathrm{\partial }c]$.

Restriction of local cochains from $X$ to $A$ is surjective: extend a simplex function by zero on every simplex not contained in $A$. Its kernel is the relative cochain group, so $0\to C^{\ast }(X,A;\mathcal{L})\to C^{\ast }(X;\mathcal{L})\to C^{\ast }(A;\mathcal{L}{\mathrm{\mid }}_A)\to 0$ is exact. The cochain chase of [F3] gives the second sequence; the explicit zero extension makes the stated connector well defined, and a different extension differs by a relative cochain and changes $\delta \tilde{a}$ by a relative coboundary.

The maps in [F2] commute with the inclusions, quotient maps, restrictions, and differentials in the two short exact sequences. Applying them to the representative formulas for the connectors proves commutativity of every naturality square, with $\mathcal{L}\to f^{\ast }\mathcal{K}$ in homology and $f^{\ast }\mathcal{K}\to \mathcal{L}$ in cohomology.

Exactness at degree zero includes the initial zero group because negative chain and cochain degrees vanish. If $A=\mathrm{\varnothing }$, the relative complex is the absolute complex and the $A$ terms are zero; if $A=X$, the relative complex is zero. Empty $X$, zero coefficients, points, degenerate simplices, and disconnected spaces require no modification. The extension by zero is a displayed function, so no AC is used.

Define local barycentric subdivision $S$ on a simplex $\sigma$ by transporting its first-vertex coefficient along the affine segment in ${\mathrm{\Delta }}^n$ to each subdivided first vertex. Define the standard barycentric prism $T$ with the same transports. All face cancellations are literal except for routes across an affine triangle; those give the same local-system map because they are endpoint-fixed homotopic inside ${\mathrm{\Delta }}^n$. Thus $S$ is a chain map and $\mathrm{\partial }T+T\mathrm{\partial }=1-S$. Both operators preserve the image of each simplex, and $T$ preserves every small-chain subcomplex.

Put $B=X\setminus Z$. Since $\bar{Z}\subseteq \mathrm{int}A$, the interiors of $A$ and $B$ cover $X$. Apply Step 1.1 to the small complex generated by simplices in $A$ or $B$. Its operators preserve $C_{\ast }(A;\mathcal{L})$ because every term stays in the image of its original simplex, so they descend to the quotient by $C_{\ast }(A;\mathcal{L})$. The quotient of the small complex is canonically $C_{\ast }(B;\mathcal{L})\mathrm{/}{C}_{\ast }(A\cap B;\mathcal{L})$: its remaining basis consists exactly of simplices in $B$ but not $A$. Thus inclusion $(B,A\cap B)\to (X,A)$ is a chain-homotopy equivalence on relative local chains. Dualizing that actual equivalence gives the restriction equivalence on relative local cochains. Taking (co)homology proves both excision isomorphisms with the restricted ambient system.

Let $C_{\ast }^{U,V}(X;\mathcal{L})$ be the local subcomplex generated by simplices lying in $U$ or $V$. There is a degreewise exact sequence $0\to C_{\ast }(U\cap V;\mathcal{L})\stackrel{c\mapsto (c,-c)}{\to }{C}_{\ast }(U;\mathcal{L})\oplus C_{\ast }(V;\mathcal{L})\stackrel{(a,b)\mapsto a+b}{\to }{C}_{\ast }^{U,V}(X;\mathcal{L})\to 0$. Step 1.1 makes the last complex chain-homotopy equivalent to the full complex. The long exact sequence of [F2] gives the homology Mayer–Vietoris sequence with the signs displayed in [F4].

Dually, let $C_{U,V}^q(X;\mathcal{L})$ be the product of the first-vertex fibers over singular $q$-simplices whose images lie in $U$ or in $V$, with the intrinsic coboundary from [F1]. Compatible restrictions give $0\to C_{U,V}^{\ast }(X;\mathcal{L})\to C^{\ast }(U;\mathcal{L})\oplus C^{\ast }(V;\mathcal{L})\stackrel{(u,v)\mapsto u-v}{\to }{C}^{\ast }(U\cap V;\mathcal{L})\to 0$. Surjectivity of the last map follows by extending a simplex function on $U\cap V$ by zero on simplices of $U$ not contained in the intersection. The small-cochain complex computes full cohomology by step 1.1, and [F3] gives the cohomology sequence with the difference convention from [F4].

Subdivision, restriction, addition, difference, and the connector representative formulas commute with maps carrying the ordered cover into another ordered cover and with correctly directed coefficient morphisms. This proves naturality. Replacing an excisive cover by interiors gives the stated extension. Empty intersections, $U=X$, zero systems, degree zero, disconnected spaces, and degenerate simplices are covered by the same exact sequences. All subdivisions of a given finite chain are finite and explicitly defined, so no AC is used.

Apply the local-coefficient cohomological excision of [F5], using its simplexwise small-chain inverse, to the open-cell neighborhoods in the relative $m$-skeleton. A separated open $m$-cell has constant coefficients after transport from one point, and the relative disk-boundary cellular cochain complex has one copy of that fiber in degree $m$ and zero elsewhere. Hence $H^q(X^m\cup A,X^{m-1}\cup A;\mathcal{L})=0$ for $q\mathrm{\ne }m$, while the degree-$m$ group is the product of the dual cell-coordinate groups, precisely $C_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}^m(X,A;\mathcal{L})$.

In connected universal-cover coordinates, $g\cdot c=c\cdot g^{-1}$ makes the cellular boundary left $R[\pi ]$-linear and applying ${\mathrm{Hom}}_{R[\pi ]}(-,{\mathcal{L}}_x)$ gives the cellular coboundary by precomposition. Rewriting left equivariance at $c\cdot g=g^{-1}\cdot c$ gives $\phi (c\cdot g)=g^{-1}\phi (c)$, so every map is typed as claimed.

For the triple $X^{m-1}\cup A\subseteq X^m\cup A\subseteq X^{m+1}\cup A$, the connecting maps in [F3] compose to the cellular coboundary. Exactness and the concentration in step 1.1 give, by a direct kernel-image chase, $\ker d^n\mathrm{/}\mathrm{im}{d}^{n-1}\cong H^n(X^{n+1}\cup A,A;\mathcal{L})$. Attaching cells in dimensions above $n+1$ leaves this group unchanged because the two adjacent relative groups vanish.

For an infinite CW complex, use the telescope in [F4] and split it into alternating closed skeletal cylinders with overlapping half-cylinders. The local cochain Mayer--Vietoris sequence of [F5], applied to interiors of these cylinder neighborhoods, gives the exact sequence for this cover. The pieces are disconnected unions. By [A1], a family of componentwise cocycles is a coboundary in their product complex exactly when one may choose a primitive in every component; hence the cohomology of each piece is the product of the cohomologies of its skeletal components. Retraction of the pieces onto the skeleta then identifies the Mayer--Vietoris product map with $\mathrm{\Delta }:{\prod }_m{H}^q(X^m,A^m;\mathcal{L})\to {\prod }_m{H}^q(X^m,A^m;\mathcal{L})$, where $\mathrm{\Delta }((a_m){)}_m=a_m-i_m^{\ast }{a}_{m+1}$. Step 2.1 says that both the degree-$n$ and degree-$(n-1)$ inverse systems are eventually constant with isomorphism transition maps. For any eventually constant system, $\ker \mathrm{\Delta }$ is its stable value, while $\mathrm{\Delta }$ is onto: set the first stable-tail coordinate to zero, recurse forward there through the inverse transition maps, and then recurse through the finitely many earlier coordinates toward zero. Exactness therefore identifies $H^n$ of the telescope with the stable value $H^n(X^{n+1}\cup A,A;\mathcal{L})$. Pair homotopy invariance from [F4] identifies this with $H^n(X,A;\mathcal{L})$, with no inverse-limit remainder.

A cellular map preserves the skeletal triples and their connectors, and [F3] makes the restriction maps natural for a coefficient morphism $f^{\ast }\mathcal{K}\to \mathcal{L}$. The telescope splitting and the map $\mathrm{\Delta }$ are natural as well. Therefore the identifications in steps 1.1--3.1 commute with induced cochain maps. Empty pairs, no cells, zero systems/rings, degree zero, negative degrees, and disconnected products are all covered by the same componentwise exact chase. AC is used only in Step 3.1 to assemble componentwise primitives.