Compatibility of orientation classes

Question

Let $U$ be an oriented topological manifold of pure dimension $n$ and $K$ a compact of $U$. There is an orientation class $$or_{U,K} \in H_n(U, U \backslash K).$$ Let $V$ an open subset of $U$ such that $K \subset V \subset U.$ There is as well an orientation class $or_{V,K} \in H_n(V,V\backslash K).$ Now let us consider the canonical map $$i : H_n(V,V\backslash K) \to H_n(U,U\backslash K).$$ Do we have $$i(or_{V,K}) = or_{U,K}?$$

Answer 1

I believe the orientation class is characterized by the fact that it’s image under inclusion for each $p\in U$, $$H_n(U,U-K)\rightarrow H_n(U,U-p)$$ is sent to the chosen orientation. If they disagreed this couldn’t happen as the fact that manifolds are normal and excision implies that $$H_n(V,V-p)\rightarrow H_n(U,U-p)$$ is an isomorphism.