 Cap-product support lemma for a two-set cover

A valid support statement for the cap product uses a relative class. Let $M$ be a closed, oriented $n$-dimensional smooth manifold, and let $W\subset M$ be a compact codimension-zero submanifold with boundary. If $\alpha \in H^k(M;R)$ restricts to zero on $M\setminus \mathrm{int}W$, thenIndeed, the pair long exact sequence in cohomology lifts $\alpha$ to $H^k(M,M\setminus \mathrm{int}W;R)$. Excision, using a boundary collar, identifies this with $H^k(W,\partial W;R)$. Poincare-Lefschetz duality gives a class in $H_{n-k}(W;R)$ whose image is the desired cap product.

Vanishing restrictions on both sets of an open cover do not force this cap product to vanish. Cover a circle by two proper open arcs. Its generator $\alpha \in H^1(S^1;\mathbb{Z})$ restricts to zero on each arc, but $[S^1]⌢\alpha$ generates $H_0(S^1;\mathbb{Z})$ and is nonzero. The valid vanishing cup product from an open cover instead says that $\alpha ⌣\beta =0$ when $\alpha$ vanishes on one covering set and $\beta$ on the other.