APPENDIX B PROOFS OF THE PROPOSITIONS IN §9.B

The proofs of Propositions 9.8–11 depend on the following statement found at the bottom of page 296 of [D, §VIII.7.12] which is elevated here to the status of a proposition. To state it, as in Chapter 9, let X denote an m-dimensional manifold oriented over a PID R and give every open submanifold of X the induced orientation. Also, coefficients of homology and cohomology modules are taken in some R-module G (unless specified otherwise) and the coefficient module is suppressed from the notation. Let K be a closed subset of X and set Ω_{c}^{K} to be the family of closed subsets of K that are cobounded in K. Note that Ω_{c}^{K} is cofinal in Ω^{K}, the family of locally compact subsets of K that are cobounded in K directed by containment. Thus, Ȟ_{c}^{∗}(K) = \underrightarrow{lim} Ȟ^{∗}(K, A) where A ∈ Ω_{c}^{K}. Suppose x̌_{c} ∈ Ȟ_{c}^{i}(K). By the definition of a direct limit, choose A ∈ Ω_{c}^{K} so that there exists x̌ ∈ Ȟ^{i}(K, A) whose image under the canonical map to the direct limit maps to x̌_{c}. As Ȟ^{∗}(X, A) is itself the direct limit of the singular cohomology modules of neighborhood pairs of (K, A), choose an open neighborhood pair (U, W) of (K, A) so that there exists y ∈ H^{i}(U, W) whose image under the canonical map to the direct limit is x̌. Let e_{1}: (U, U \ K) ↪ (X, X \ K) be the excision.

B.1 Proposition. In H_{m−i}(U, U \ K),

x̌_{c} ⌢ o^{U} = y ⌢ o_{K\W}^{U} ;

hence, in H_{m−i}(X, X \ K),

x̌_{c} ⌢ o^{X} = e_{1 ∗}(y ⌢ o_{K\W}^{U}) .

Proof. Let C := cl (K \ A), note that C is compact by choice of A, and observe that the following equalities between sets hold:

(1) (U \ K) ∪ W = U \ (K \ W);

(2) (U \ K) ∪ (W \ A) = (U \ A) \ (K \ W);

(3) (U \ (K ∩ C)) ∪ (W \ (A ∩ C)) = (U \ (A ∩ C)) \ (K \ W). Too, observe that the left-hand side of each equality is the union of open sets and that there are inclusions of pairs

(4) (U \ A, (U \ A) \ (K \ W)) ⊂ (U, U \ (K \ W));

(5) (U \ A, (U \ A) \ (K \ W)) ⊂ (U \ (A ∩ C), (U \ (A ∩ C)) \ (K \ W)).

The proposition then follows from an analysis of the diagram in Figure 9 where the arrows will be defined as the analysis proceeds.

\begin{aligned}H_{c}^{i}(K) & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & _{⌢ o^{X}} & Ȟ^{i}(K, A) & ^{i} & _{e_{5}^{∗}} & _{≃} & _{⌢ o^{U}} & _{⌢ o^{X}} & H^{i}(U, W) & ^{i′} & _{i′′} & _{≃} & _{≃} & _{⌢ o_{K\W}^{U}} & Ȟ^{i}(K ∩ C, A ∩ C) & _{⌢ o^{U}} & _{≃} & H^{i}(U \ (A ∩ C), W \ (A ∩ C)) & _{⌢ o_{K\W}^{U\(A∩C)}} & _{e_{4}^{∗}} & _{≃} & H_{m−i}((U \ A) ∪ (U \ C), (U \ K) ∪ (U \ C)) & H^{i}(U \ A, W \ A) & _{⌢ o_{K\W}^{U\A}} & H_{m−i}(U \ A, U \ K) & _{e_{2 ∗}} & _{≃} & _{j′} & _{e_{3 ∗}} & _{≃} & H_{m−i}(U, U \ K) & _{e_{1 ∗}} & _{≃} & H_{m−i}(X \ A, X \ K) & _{j} & H_{m−i}(X, X \ K) \\ & & Figure 9 \end{aligned}

The set equalities (1), (2), and (3) and the fact that the left-hand side of each is the union of open sets yield that in Figure 9 the cap product homomorphisms ⌢ o_{K\W}^{U}, ⌢ o_{K\W}^{U\A}, and ⌢ o_{K\W}^{U\(A∩C)} determined by fundamental classes along the compact K \W are well-defined. Of course, the arrows labeled with ⌢ o^{X} or ⌢ o^{U} are Poincaré duality isomorphisms.

In Figure 9, the remaining arrows are defined and the diagram is shown commutative as follows. All unlabeled arrows are inclusion induced. All arrows labeled with an e_{j}^{∗} or an e_{j ∗} for some integer j are excision isomorphisms. The arrow labeled i is the canonical map into the direct limit as are the arrows labeled i′ and i′′ in triangle 1. It follows that triangle 1 commutes. The vertical arrow in quadrilateral 2, labeled ⌢ o^{U}, is a relative Poincaré-Alexander duality isomorphism whence quadrilateral 2 commutes by the very definition of ⌢ o^{U}. Parallelogram 3 commutes by naturality of cap products since o_{K\W}^{U\A} has image o_{K\W}^{U\(A∩C)} under the homomorphism induced by the inclusion in (5). Similarly, quadrilateral 5 commutes since o_{K\W}^{U\A} has image o_{K\W}^{U} under the homomorphism induced by the inclusion in (4). Triangle 4 and rectangle 6 commute because all arrows in them are inclusion induced. Subdiagram 7 commutes because the curvilinear arrow of the subdiagram is by definition the composite of the vertical arrows in the subdiagram; similarly, by definition of its curvilinear arrow, subdiagram 8 commutes. Because H_{∗}(U, U \K) = \underrightarrow{lim} H_{∗}(U \A, U \K) taken over A ∈ Ω_{c}^{K} (see [D, §VII.7.12, p. 296]) the inclusion induced homomorphism j′ can be regarded as the canonical map into the direct limit, and analogous remarks apply when U is replaced by X and j′ by j. Thus, passage to the direct limit over A ∈ Ω_{c}^{K} yields for Z := U or Z := X that the curvilinear arrow marked ⌢ o^{Z} passes to the Poincaré duality isomorphism

(B.1)  ⌢ o^{Z}: Ȟ_{c}^{i}(K) → H_{m−i}(Z, Z \ K) .

With Z := X, the last sentence immediately implies that

(B.2)  ⌢ o^{X} ^{∘} i = j^{∘} ⌢ o^{X}

where on the left ⌢ o^{X} denotes the dashed curvilinear arrow of subdiagram 9 but on the right it denotes the solid curvilinear arrow so marked in subdiagrams 8 and 9. Thus, equality (B.2) states that subdiagram 9 is commutative. Analogously, one sees that with Z := U

(B.3)  ⌢ o^{U} ^{∘} i = j′^{∘} ⌢ o^{U}

where on the left ⌢ o^{U} is the arrow of (B.1) which does not appear in Figure 9, but on the right, ⌢ o^{U} is the curvilinear arrow so marked in subdiagram 7.

The proposition then follows easily from equality (B.3) and a trivial chase on the diagram in Figure 9 starting at y ∈ H^{i}(U, W). □

B.2 The Proof of Proposition 9.8. Suppose x̌_{c} ∈ Ȟ_{c}^{i}(V). Then Proposition B.1 yields that for some open subset W of V, W is cobounded in V and where K := V \ W, K is compact and for some y ∈ H^{i}(V, W) = H^{i}(V, V \ K)

x̌_{c} ⌢ o^{V} = y ⌢ o_{K}^{V} .

Next, note that the diagram

Ȟ_{c}^{i}(V) _{⌢ o^{V}} H^{i}(V, V \ K)^{i_{∗}^{V}}  _{⌢ o_{K}^{V}}  H^{i}(X, X \ K)^{e_{K ∗}}_{≃}  _{⌢ o_{K}^{X}}  ^{i_{∗}^{X}}  Ȟ_{c}^{i}(X)_{⌢ o^{X}} H_{m−i}(V)  H_{m−i}(X)

is commutative where i_{∗}^{V} and i_{∗}^{X} are the canonical maps into the direct limits: The triangles of morphisms at left and right are commutative because the oblique arrow in each triangle is obtained from the vertical arrow in the triangle by passage to the direct limit over the family of vertical maps obtained by replacing K with any compact subset of V (respectively X) in the triangle on the left (right). Also, as o_{K}^{X} is the image of o_{K}^{V} under the excision isomorphism induced by the inclusion of pairs (V, V \ K) ⊂ (X, X \ K), it follows that the middle rectangle commutes by naturality of singular cap products. Since x̌_{c} was an arbitrary element of Ȟ_{c}^{i}(V), the proposition follows by chasing the class y around the diagram because by definition of the top horizontal arrow in the diagram in the statement of the proposition the image of x̌_{c} under that horizontal arrow is i_{∗}^{X} ^{∘} e_{K ∗}^{−1} y. □

B.3 The Proof of Proposition 9.9. First, note that the hypotheses imply

V \ K_{1} = (V \ L_{1}) \ (K_{1} \ L_{1}) ⊂ (X \ L_{2}) \ (K_{2} \ L_{2}) = X \ K_{2}

whence there is an excision (V \L_{1}, V \K_{1}) ⊂ (X\L_{2}, X\K_{2}). Thus, the bottom horizontal arrow of the diagram in the statement of the proposition is well-defined and is an isomorphism because the other three sides of the diagram are isomorphisms. Too, the reader can if desired easily verify that (V \ L_{1}) ∪ (X \ K_{2}) = X \ L_{2} and that (V \ L_{1}) ∩ (X \ K_{2}) = V \ K_{1} which together imply that the excision yields an isomorphism on homology as V \ L_{1} and X \ K_{2} are open in X \ L_{2}.

Next, set K := K_{1} \ L_{1} = K_{2} \ L_{2}. Note that K is closed relative to both V \ L_{1} and X \ L_{2}. Thus, as in the preamble to Proposition B.1, for any x̌_{c} ∈ Ȟ_{c}^{i}(K) there exists A ∈ Ω_{c}^{K} and an open (relative to both V \ L_{1} and X \ L_{2}) neighborhood pair (U, W) of (K, A) so that for some x̌ ∈ Ȟ^{i}(K, A) and for some y ∈ H^{i}(U, W), under the canonical maps to the direct limits y has image x̌ and x̌ has image x̌_{c}. By Proposition B.1,

x̌_{c} ⌢ o^{U} = y ⌢ o_{K\W}^{U} ∈ H_{m−i}(U, U \ K) .

Commutativity of the triangle of excision induced isomorphisms

H_{m−i}(U, U \ K)_{e_{2 ∗}}_{≃}  _{e_{1 ∗}} _{≃} H_{m−i}(V \ L_{1}, V \ K_{1})  _{e_{3 ∗}}_{≃}  H_{m−i}(X \ L_{2}, X \ K_{2})

yields by the second part of Proposition B.1 that

x̌_{c} ⌢ o^{X\L_{2}} = e_{1 ∗}(y ⌢ o_{K\W}^{U}) = e_{3 ∗} ^{∘} e_{2 ∗}(y ⌢ o_{K\W}^{U}) = e_{3 ∗}(x̌_{c} ⌢ o^{V \L_{1}}) . □

B.4 The Proof of Proposition 9.10. Suppose x̌_{ℓ} ∈ Ȟ_{c}(K_{ℓ}). Then Proposition B.1 yields that for some open pair (V_{ℓ}, W_{ℓ}) in M_{ℓ} where W_{ℓ} contains a cobounded subset A_{ℓ} of K_{ℓ},

(B.4)  x̌_{ℓ} ⌢ o_{ℓ} = y_{ℓ} ⌢ o_{K_{ℓ}\W_{ℓ}}

where y_{ℓ} ∈ H^{m_{ℓ}−i_{ℓ}}(V_{ℓ}, W_{ℓ}) and o_{K_{ℓ}\W_{ℓ}} is the fundamental class along the compact set K_{ℓ} \ W_{ℓ}. Then by the standard identity of singular theory relating cap and cross products (see [D, §VII.12.17]) (B.5) (y_{1} ⌢ o_{K_{1}\W_{1}}) × (y_{2} ⌢ o_{K_{2}\W_{2}}) = (−1)^{m_{1}(m_{2}−i_{2})}(y_{1} × y_{2}) ⌢ (o_{K_{1}\W_{1}} × o_{K_{2}\W_{2}}) .

Because

o_{K_{1}\W_{1}} × o_{K_{2}\W_{2}} = (o_{1} × o_{2})_{(K_{1}\W_{1})×(K_{2}\W_{2})}^{M_{1}×M_{2}} ,

i.e., because the product on the left side is the generator of H_{2m}(X × X, X × X \ (K_{1} \W_{1} ×K_{2} \W_{2})) corresponding to the product orientation and because x̌_{1} ×x̌_{2} is represented in Ȟ((K_{1}, A_{1}) × (K_{2}, A_{2})) by the canonical image of y_{1} × y_{2}, the result follows from the equalities (B.4) and (B.5) and Proposition B.1. □

B.5 The Proof of Proposition 9.11. Let x̌ ∈ Ȟ^{i}(K, L) and let y̌_{c} ∈ Ȟ_{c}^{j}(K). As in the preamble to Proposition B.1, for some A ∈ Ω_{c}^{K}, for some open neighborhood pair (U, ω) of (K, A), some z ∈ H^{j}(U, ω) has image y̌_{c} under the canonical map to the direct limit. By the properties of direct systems, we may without loss of generality assume that for some W, (U, W) is an open neighborhood pair of (K, L) and some x ∈ H^{i}(U, W) has image x̌ under the canonical map to the direct limit. The proposition will follow from a diagram chase on the diagrams in Figures 10 and 11 once those diagrams are defined and shown commutative.

Toward that end, note that K \ ω and K \ (W ∪ ω) are compact and observe that the following equalities between sets hold:

(1) U \ (K \ ω) = (U \ K) ∪ ω;

(2) U \ (K \ (W ∪ ω)) = (U \ K) ∪ W ∪ ω;

(3) (U \ L) \ (K \ (W ∪ ω)) = (U \ K) ∪ (W \ L) ∪ (ω \ L). Observe too that there are inclusions of open pairs

(4) (U, (U \ K) ∪ ω) ⊂ (U, (U \ K) ∪ W ∪ ω);

(5) (U \ L, (U \ K) ∪ (W \ L) ∪ (ω \ L)) ⊂ (U, (U \ K) ∪ W ∪ ω).

Also, as in earlier proofs, a homomorphism labeled by e_{j ∗} or e_{∗}^{j} for some integer j is an excision isomorphism, and one labeled by i_{j ∗} or i_{∗}^{j} for some integer j is inclusion induced.

Then the diagram in Figure 10 is well-defined and commutes: It follows from equalities (1), (2), and (3) and that the right-hand side of each is expressed as the union of open sets that each of the factor cap product homomorphisms in the diagram in Figure 10 labeled by ⌢ o_{K\W}^{U}, ⌢ o_{K\(W ∪ω)}^{U}, or ⌢ o_{K\(W ∪ω)}^{U\L} is well-defined. It follows that the triangle of homomorphisms at the top left of the diagram in Figure 10 commutes by naturality of singular cap products because o_{K\(W ∪ω)}^{U} is the image of o_{K\ω}^{U} under the homomorphism induced by the inclusion

H^{i}(U, W) ⊗ H^{j}(U, ω) H^{i}(U, W) ⊗ H_{m−j}(U, U \ K) H^{i}(U, W) ⊗ H_{m−j}(X, X \ K)  ^{e_{2}^{∗} ⊗ i_{3}^{∗}}_{1^{∗} ⊗ ⌢ o_{K\(W ∪ω)}^{U}}_{1^{∗} ⊗ ⌢ o_{K\W}^{U}}  H^{i}(U \ L, W \ L) ⊗ H^{j}(U \ L, ω \ L)_{1^{∗} ⊗ ⌢ o_{K\(W ∪ω)}^{U\L}} H^{i}(U, W) ⊗ H_{m−j}(U, (U \ K) ∪ W)_{1^{∗} ⊗ e_{3 ∗}}_{≃}  _{e_{2}^{∗} ⊗ e_{4 ∗}^{−1}}_{≃}_{1^{∗} ⊗ i_{1 ∗}}_{1^{∗} ⊗ e_{1 ∗}} _{≃}  H^{i}(U \ L, W \ L) ⊗ H_{m−j}(U \ L, (W \ L) ∪ (U \ K)) H^{i}(U, W) ⊗ H_{m−j}(X, (X \ K) ∪ W) _{e_{2}^{∗} ⊗ e_{5 ∗}^{−1}}_{≃}_{1^{∗} ⊗ i_{2 ∗}}

Figure 10

H^{i}(U, W) ⊗ H^{j}(U, ω)_{e_{2}^{∗} ⊗ i_{3}^{∗}}  ^{⌣}  H^{i+j}(U, W ∪ ω)_{e_{6}^{∗}}  ^{ǐ_{(K,L)}}  Ȟ_{c}^{i+j}(K, L)_{≃} H^{i}(U \ L, W \ L) ⊗ H^{j}(U \ L, ω \ L)_{1^{∗} ⊗ ⌢ o_{K\(W ∪ω)}^{U\L}}  _{⌣}  H^{i+j}(U \ L, (W ∪ ω) \ L)_{⌢ o_{K\(W ∪ω)}^{U\L}} _{ǐ_{K\L}}  Ȟ_{c}^{i+j}(K \ L)_{⌢ o^{X\L}} H^{i}(U \ L, W \ L) ⊗ H_{m−j}(U \ L, (U \ K) ∪ (W \ L)) _{⌢} H_{m−(i+j)}(U \ L, U \ K) _{e_{7 ∗}}_{≃} H_{m−(i+j)}(X \ L, X \ K)

Figure 11 of (4). Similarly, the rectangle of homomorphisms at the top right of the diagram in Figure 10 commutes because o_{K\(W ∪ω)}^{U} is the image of o_{K\(W ∪ω)}^{U\L} under the homomorphism induced by the inclusion of (5).

Also, the diagram in Figure 11 is well-defined and commutes: For the same reasons as in the diagram in Figure 10, in the diagram in Figure 11, the factor cap product homomorphisms labeled by ⌢ o_{K\(W ∪ω)}^{U\L} are well-defined. It follows that the lower left rectangle commutes as a consequence of the standard associative law of cup and cap products in singular theory; see [D, §VII.12.7]. The upper left rectangle commutes by naturality of cup products. The horizontal arrows ǐ_{(K,L)}^{∗} and ǐ_{K\L}^{∗} are the canonical maps to the direct limits. It follows that the upper right rectangle commutes because the natural isomorphism that is the right edge is obtained by passage to the limit from e_{6}^{∗} and analogous excisions obtained by replacing W ∪ ω with V where (U, V) is a neighborhood pair of (K, L) and K \ V is bounded. Finally, the rectangle in the lower right corner commutes as a consequence of the proof of Proposition B.1, in particular the commutativity of the diagram in Figure 9: For the composite e_{7 ∗}^{∘} ⌢ o_{K\(W ∪ω)}^{U\L} corresponds to the composite of the two arrows in the right edge of the diagram in Figure 9, i_{K\L}^{∗} corresponds to the composite of the arrows in the top edge of that diagram, and ⌢ o^{X\L} corresponds to the dashed curvilinear arrow in that diagram.

To begin the diagram chase, note that the diagrams in Figures 10 and 11 can be joined by identifying the top horizontal arrow of the diagram in Figure 10 with the top vertical arrow in the left edge of the diagram in Figure 11 and also by identifying the rightmost vertical arrow of the diagram in Figure 10 with the bottom vertical arrow in the left edge of the diagram in Figure 11. Next, note that x ⊗ z has image x̌ ⌣ y̌_{c} under the composite of the arrows in the top row of Figure 11 by definition of the cup product on Ȟ^{∗}(K, L) ⊗ Ȟ_{c}^{∗}(K). It follows that the image of x⊗z under the composition of the arrows in the top row and right edge of Figure 11 is (x̌ ⌣ y̌_{c}) ⌢ o^{X} where here ⌢ o^{X} is just the composite of the arrows in the right edge. On the other hand, Proposition B.1 yields that the image of x ⊗ z under the composite obtained by starting at the top left hand corner of the diagram in Figure 10 and traveling down along its left edge to H^{i}(U, W) ⊗ H_{m−j}(X, X \ K) is x ⊗ (y̌_{c} ⌢ o^{X}). The commutativity of both diagrams shows that the image of x ⊗ (y̌_{c} ⌢ o^{X}) under the composite homomorphism C^{(U,W)} obtained by first following the horizontal and oblique arrows in the bottom edge of the diagram in Figure 10 and then following the two bottom horizontal arrows in the diagram in Figure 11 is also (x̌ ⌣ y̌_{c}) ⌢ o^{X}. Because the Čech cap product of elements in Ȟ^{∗}(K, L)⊗H_{∗}(X, X \K) is obtained by passing to the direct limit from composites of the form C^{(U,W)}, it follows that

x̌ ⌢ (y̌_{c} ⌢ o^{X}) = (x̌ ⌣ y̌_{c}) ⌢ o^{X}

as desired. □