Fully relative Poincaré–Lefschetz duality

If $C\mathrm{\ne }\mathrm{\varnothing }$, use [F6] to glue its two collars into a map $c:C\times (-1,1)\to D$, with negative height on $A$ and positive height on $B$. This is a homeomorphism onto an open neighborhood of $C$. Indeed it is bijective onto the union of the two collar images, with their only intersection at $C$. The two halves are closed in the source and their images closed relative to the union, since $A,B$ are closed in $D$; the two continuous inverse maps therefore paste continuously. The image is open: each complement of a collar image is closed in its compact piece, hence closed in $D$, and their union is the complement of the glued image. Fix $0<ϵ<1$ and set $U=A\cup c(C\times [0,ϵ))$ and $V=B\cup c(C\times (-ϵ,0])$. They are open in $D$: the complement of $U$ is the closed collar complement in $B$, and similarly for $V$. They cover $D$ and have intersection $W=c(C\times (-ϵ,ϵ))$. If $C=\mathrm{\varnothing }$, the disjoint compact pieces are open in $D$; set $U=A,V=B,W=\mathrm{\varnothing }$.

The space $U$ deformation retracts onto $A$ by replacing positive seam height $t$ with $(1-s)t$ and fixing $A$. This is continuous at height zero by the collar coordinates, and outside the attached strip it is the identity. Likewise $V$ retracts onto $B$. Denote the latter retraction by $r$. If $C\mathrm{\ne }\mathrm{\varnothing }$, set $C_{ϵ}=c(C\times [0,ϵ))\subset B$. The retraction and its homotopy send $W$ into $W$, with image $C_{ϵ}$ at the end, so $(V,W)$ retracts onto $(B,C_{ϵ})$. The inclusion $C\to C_{ϵ}$ is also a deformation retract. For $C=\mathrm{\varnothing }$, take $C_{ϵ}=\mathrm{\varnothing }$ and all these maps are identities. By [F7], the pair sequences [F5] and [F4], the inclusions induce isomorphisms on relative homology and cohomology for $(M,A)\to (M,U)$, $(M,B)\to (M,V)$, $(B,C)\to (B,C_{ϵ})$ and $(B,C_{ϵ})\to (V,W)$. The maps of pair sequences commute directly on inclusion/quotient chains and restriction cochains; their connectors commute by lifting the same representative. Thus the five-lemma applications have their required naturality.

There is a short exact cochain sequence $0\to C^{\ast }(M,D;R)\to C^{\ast }(M,U;R)\to C^{\ast }(D,U;R)\to 0.$ The last map is restriction. A cochain on $D$ vanishing on $U$ extends by zero on the other simplices of $M$ and still vanishes on $U$, proving surjectivity in every degree. Its kernel is precisely the cochains vanishing on $D$. The resulting cohomology sequence is exact, with connector $[\gamma ]\mapsto [\delta \tilde{\gamma }]$: two extensions differ by a cochain vanishing on $D$, so they give the same relative class; changing $\gamma$ by a coboundary and extending its primitive also changes that class by zero. Exactness at the middle term follows by subtracting an extended primitive when the restriction is a coboundary. At the right term, a connector that bounds allows subtracting the relative primitive from an extension to make it closed. At the left term, a relative cocycle bounding in $C^{\ast }(M,U)$ is the connector of that primitive's restriction. These are all three positions, including the initial zero-degree injection because negative cochains vanish. Thus no unproved triple-sequence theorem is needed.

Put $Q=C_{\ast }(M;R)\mathrm{/}(C_{\ast }(U;R)+C_{\ast }(V;R))$. Since $U,V$ are open in their union $D$, [F2] gives the canonical chain equivalence $q:Q\to C_{\ast }(M,D;R)$ and its dual equivalence. Let $\widehat{z}=q_{\ast }^{-1}z$. The relative cap of [F2] gives $S_p:H^p(M,U;R)\to H_{n-p}(M,V;R)$ by $a\mapsto a\cap \widehat{z}$. Transport this map through the isomorphisms of step 2.1 to define the claimed map on $(M,A)$ and $(M,B)$. This is a compatible neighborhood replacement, rather than an assertion that arbitrary closed pieces form an open triad. If smaller positive collar widths are used with the same collars, inclusions commute with $q$ and with the cap formula, so the transported map is unchanged. For two choices of collars, there are common smaller neighborhoods of this form: the compact sets $A,B$ have open neighborhoods equal to the intersections of the respective $U$'s and $V$'s in $D$, and compactness of $C$ puts sufficiently thin strips of the first collar inside these neighborhoods. This last assertion follows from finitely many product neighborhoods of points of $C\times \{0\}$, taking the minimum of their finitely many positive widths. The same comparison through these inclusions shows independence of the collars.

Restriction gives an isomorphism $H^k(D,U)\to H^k(V,W)$ by [F3]: excise the closed set $D\setminus V$, which is contained in the open $U$. Together with step 2.1 this identifies $H^k(D,U)$ with $H^k(B,C)$. The analogous inclusion on relative homology is an isomorphism by the same excision. The class of $[D]$ in $H_{n-1}(D,U)$ corresponds under this homology comparison to a class $b_V$ in $H_{n-1}(V,W)$. We prove that its image $d=r_{\ast }{b}_V$ in $H_{n-1}(B,C_{ϵ})$ is the image of $[B,C]$. Put $N=B\setminus C$ and $K=B\setminus C_{ϵ}$. The collar makes $C_{ϵ}$ open in $B$, so $K$ is compact and lies in the boundaryless manifold $N$. Excision of the closed set $C$ inside $C_{ϵ}$ identifies $H_{n-1}(B,C_{ϵ})$ with $H_{n-1}(N,N\setminus K)$. At every $x\in K$, the retraction is the identity near $x$ and the preceding excision comparison is induced by inclusion, so $d$ restricts to the prescribed local generator inherited from $[D]$. By the pointwise injectivity and realization in [F11], $d$ is the unique orientation class supported on $K$. The image of $[B,C]$ has the same description: [F8] gives its prescribed generator at every point of $N$, hence at every point of $K$, and the same excision square transports those values. Therefore the two classes in $H_{n-1}(B,C_{ϵ})$ agree. If $C$ is empty, $K=B$ and the identical argument compares the absolute classes.

It follows that $E_k:H^k(D,U)⟶H_{n-1-k}(V),[\gamma ]⟼[(\gamma {\mathrm{\mid }}_V)\cap b_V]$ is an isomorphism. More precisely, first pass by step 3.2 to $H^k(V,W)$, then restrict to $H^k(B,C_{ϵ})$ and to $H^k(B,C)$. Step 2.1 proves that these are isomorphisms; the last class caps with $[B,C]$ by [F1] to give an isomorphism to $H_{n-1-k}(B)$. Include $B$ into $V$, an isomorphism on homology by step 2.1. This composite equals $E_k$. Indeed $r$ and inclusion are inverse on the relative groups, $r_{\ast }{b}_V$ is the image of $[B,C]$ by step 3.2, and the relative version of the literal chain naturality identity [F9] identifies the two cap outputs after $r_{\ast }$. Since $r_{\ast }$ is an isomorphism on absolute homology, the outputs in $V$ agree. This also proves that the definition of $E_k$ is independent of all representatives.

For use in the exact diagram, representatives can be chosen compatibly as follows. Start with a relative cycle $v$ for $z$, so $b=\mathrm{\partial }v$ is a cycle in $D$ for $[D]$, by [F8]. Use the operator $E$ of [F2] for the open cover $U,V$ of $D$, and replace $v$ by $v-Eb$. Its boundary is $Pb=b-\mathrm{\partial }Eb$, since $\mathrm{\partial }b=0$, and lies in $C_{\ast }(U)+C_{\ast }(V)$. Thus it represents $\widehat{z}$ in $Q$. Write $Pb=b_U+b_V$ by assigning each simplex in both members to $U$ and each remaining simplex to its member. This is a specified linear splitting on the small simplex basis. Since $\mathrm{\partial }{b}_U=-\mathrm{\partial }{b}_V$ and the intersection of these simplex subcomplexes is $C_{\ast }(W)$, $b_V$ is a relative cycle of $(V,W)$ and represents precisely the excision class in step 3.2. In the rest of the proof use this adjusted $v$, so $\mathrm{\partial }v=b_U+b_V$.

Use step 2.2's exact five-term window $H^{p-1}(D,U)\stackrel{\delta }{\to }{H}^p(M,D)\to H^p(M,U)\to H^p(D,U)\stackrel{\delta }{\to }{H}^{p+1}(M,D)$ above the pair homology window $H_{n-p}(V)\to H_{n-p}(M)\to H_{n-p}(M,V)\stackrel{(-1{)}^p\mathrm{\partial }}{\to }{H}_{n-p-1}(V)\to H_{n-p-1}(M)\mathrm{.}$ The five vertical maps are $E_{p-1},T_p,S_p,E_p,T_{p+1}$, where $T_k$ is cap with $z$ from [F1] for $M$. The first, second, fourth and fifth are isomorphisms by [F1] and step 4.1. Both rows are exact; the unit multiplying the homology connector leaves its kernel and image unchanged.

All four squares commute using the adjusted cycle of step 4.2. A degree-$(p-1)$ cocycle $\gamma$ on $(D,U)$ extends to a cochain $\tilde{\gamma }$ on $(M,U)$ as in step 2.2. The cap with $b_U$ is zero, because it vanishes on all simplices in $U$. Thus [F9] gives $\delta \tilde{\gamma }\cap v=\tilde{\gamma }\cap b_V+(-1{)}^p\mathrm{\partial }(\tilde{\gamma }\cap v),$ which proves $T_p\delta =i_{\ast }{E}_{p-1}$. Replacing $p$ by $p+1$ proves the last square. The second square compares the same cap chain in $M$ and modulo $V$, so commutes literally. For a cocycle $\phi$ on $(M,U)$, [F9] gives $\mathrm{\partial }(\phi \cap v)=(-1{)}^p\phi \cap b_V$, again because the $b_U$ term vanishes. Hence $(-1{)}^p\mathrm{\partial }{S}_p=E_p\mathrm{r}\mathrm{e}\mathrm{s}$, the third square with precisely the chosen sign. For a negative source cochain degree these are the zero identities; all cap expressions of negative output degree are zero with the same boundary formula.

The five lemma [F4] applied to steps 5.1–6.1 proves that $S_p$ is an isomorphism. Transporting through step 2.1 proves the claimed isomorphism of step 3.1. Empty pieces were handled in the Given paragraph; a disjoint decomposition with $C=\mathrm{\varnothing }$ uses literal open pieces throughout. For the zero ring all groups and classes are zero. In dimension one the boundary pieces are zero-manifolds with empty common boundary and [F1] on each point is multiplication by its orientation unit. Empty $M$ and dimension zero reduce to the ordinary empty-boundary case. All integer degrees, in particular $p=0,n$, have already been included in the exact-window computation; ordinary degree-zero groups are used. Degenerate simplices stay in their declared subspaces under face restriction and the small-chain operators. The only AC use is [F10]'s inherited atlas and local UCT choices in [F1]. Two supplied collars, finitely many compact-neighborhood widths, the explicit small-chain operator and extension by zero require no further AC or selection of component orientations.