Homology calculation using Mayer Vietoris sequences

Question

So suppose we have $p \in S^n$ and suppose that $X$ is a polyhedron.

I want to show that $$H_q(X \times S^n, X \times p) \cong H_{q-n}(X)$$

I was given the hint to start out by writing $S^n$ as the union of upper and lower hemispheres, and to proceed by induction on n.

Can anyone offer some insight?

Answer 1

Hint: Cover $S^n$ by two hemispheres $A$ and $B$ such that $p\in A\cap B$, and consider the Mayer-Vietoris sequence for covering $(X\times S^n,X\times p)$ by $(X\times A, X\times p)$ and $(X\times B,X\times p)$.

More details are hidden below.

The Mayer-Vietoris sequence in question looks like $$H_q(X\times A,X\times p)\oplus H_q(X\times B,X\times p)\to H_q(X\times S^n, X\times q)\to H_{q-1}(X\times (A\cap B), X\times p)\to H_{q-1}(X\times A,X\times p)\oplus H_{q-1}(X\times B,X\times p).$$ Since $A$ and $B$ are contractible, the first and last terms are $0$, giving an isomorphism $H_q(X\times S^n, X\times q)\to H_{q-1}(X\times (A\cap B), X\times p)$. Since $A\cap B\cong S^{n-1}$, you can then apply the induction hypothesis to $H_{q-1}(X\times (A\cap B), X\times p)$.