Poincare–Lefschetz duality with local coefficients

If $A=\mathrm{\varnothing }$, $M=N$ is compact, [F1] identifies $z$ with the compact twisted fundamental class, and both displayed maps are [F3]. This includes compact zero-manifolds. Hence suppose $A\mathrm{\ne }\mathrm{\varnothing }$, so $n\ge 1$. Choose a collar and put $C_d=c(A\times [0,d))$, $K_d=M\setminus C_d$ for $0<d<1$. Every compact subset $L$ of $N$ lies in some $K_d$. Indeed, on a fixed closed collar segment the height coordinate has compact image on $L$, and that image omits zero because $L\cap A=\mathrm{\varnothing }$; hence its positive part has a positive minimum. Points outside the segment already lie in every sufficiently small core. Choosing $d$ below that minimum gives $L\subseteq K_d$.

The collar retraction $C_d\simeq A$, the natural cohomology pair sequences, and the five lemma give $H^k(M,C_d;\mathcal{L})\cong H^k(M,A;\mathcal{L})$. Removing the closed boundary inside $C_d$, local-coefficient excision gives $H^k(M,C_d;\mathcal{L})\cong H^k(N,N\setminus K_d;\mathcal{L}{\mathrm{\mid }}_N)$. Both isomorphisms commute with decreasing $d$, because they are induced by inclusions, restrictions, and the collar homotopies with their coefficient-transport comparisons. Cofinality from step 1.1 and the common-support criterion [F6] therefore give an isomorphism $J:H^k(M,A;\mathcal{L})\stackrel{\sim }{\to }{H}_c^k(N;\mathcal{L}{\mathrm{\mid }}_N)$.

Under $J$, cap with $z$ is cap with the interior support class followed by inclusion $N\hookrightarrow M$. Indeed [F1] constructed $z$ so that its image in $H_n(M,C_d;{\mathcal{O}}_M^R)$ is the excision image of $[N{]}_{K_d}^{\mathrm{t}\mathrm{w}}$. Represent a class by a cocycle vanishing on $C_d$ and represent the equality of these two relative classes by a boundary plus a chain in $C_d$. The cap boundary identity in [F2] turns the boundary term into a target boundary, and the $C_d$ term caps to zero. Thus the two cap classes agree. Twisted duality on $N$ is an isomorphism by [F3], and inclusion $N\hookrightarrow M$ is a homotopy equivalence with the coefficient comparison in [F4]. Together with step 2.1 this proves the second displayed isomorphism.

For the first displayed map, compare the five-term cohomology window $H^{k-1}(A;\mathcal{L})\to H^k(M,A;\mathcal{L})\to H^k(M;\mathcal{L})\to H^k(A;\mathcal{L})\to H^{k+1}(M,A;\mathcal{L})$ with the homology window $H_{n-k}(A;\mathcal{P})\to H_{n-k}(M;\mathcal{P})\to H_{n-k}(M,A;\mathcal{P})\to H_{n-k-1}(A;\mathcal{P})\to H_{n-k-1}(M;\mathcal{P})$. Use vertically, in order, twisted duality on $A$, the second isomorphism just proved, the desired first cap map, twisted duality on $A$, and the next-degree second isomorphism. The boundary identification in [F1] identifies $\mathcal{P}{\mathrm{\mid }}_A$ with ${\mathcal{O}}_A^R{\otimes }_R\mathcal{L}{\mathrm{\mid }}_A$. Both rows are exact by [F5].

The middle inclusion and quotient squares commute because they use the same cap chains before and after passage to the relevant quotient. For a degree-$(k-1)$ boundary cocycle $a$, choose an extension $\tilde{a}$ to $M$. The positive cohomology connector is represented by $\delta \tilde{a}$, and the cap boundary formula gives $\delta \tilde{a}\cap z=\tilde{a}\cap \mathrm{\partial }z+(-1{)}^k\mathrm{\partial }(\tilde{a}\cap z)$. Since $\mathrm{\partial }z=[A{]}^{\mathrm{t}\mathrm{w}}$, this proves the first connector square; the same calculation one degree later proves the last. For the homology connector square, a degree-$k$ cocycle $b$ gives $\mathrm{\partial }(b\cap z)=(-1{)}^{k}b{\mathrm{\mid }}_A\cap \mathrm{\partial }z$, so multiplying that homology connector by $(-1{)}^k$ makes the square commute. Multiplication by this unit preserves exactness. Thus [F5]'s five lemma applies to the ladder in step 4.1 and proves the first displayed map is an isomorphism.

Empty $M$ and the zero ring or zero system give the unique isomorphisms of zero groups. The endpoint degrees $k=0,n$ occur in the same exact windows; negative chain and cochain degrees vanish. Disconnected manifolds split componentwise, and compactness makes only finitely many components occur. The boundary may be empty or disconnected. The collar choice changes ${\mathcal{O}}_M^R$ only by the natural isomorphism specified in its definition, under which the class and cap maps correspond by their local characterization. The sole AC use is inherited from boundaryless twisted duality [F3], namely its countable coordinate-neighborhood selection; collar cores, individual representatives, and finite exact windows add no choice. The published oriented Poincare--Lefschetz theorem is recovered after an orientation trivializes ${\mathcal{O}}_M^R$.