Try to prove the commutativity of the diagram in Hatcher Theorem 3.44. Sadly, I am stuck on the commutativity of the following diagram.
Let $M$ be a closed orientable manifold of dimension $d$, i.e $M$ is compact and has no boundary. Let $K$ be a compact subspace of $M$, why the following diagram commutes.
where $i:(M,\varnothing) \to (M, M\setminus K)$ is the inclusion map, $[M]$ is the fundamental class for $M$ and $\mu_K = i_\ast([M])$.
Notice for excisive pair $(A,B)$ in any topological space $X$ we have $$H^k(X,A) \otimes H_d(X,A\cup B) \to H_{d-k}(X,B), x\otimes y \mapsto x\cap y.$$ Now consider the inclusion map $i: (M, \varnothing, \varnothing) \to (M, \varnothing, M\setminus K)$ between triad, then by naturality of cap product we have following commutative diagram
(Hope this is correct.....)