A Homology calculation using Mayer-Vietoris sequence

[PsychonautQQ]

Hey PF! This isn't for homework, just me messing around with some thoughts in caluclating various homology groups.

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

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?

 

[mathwonk]

before i think about this, do you have any response to my answer to your previous question?

 

[andrewkirk]

Hey PF! This isn't for homework, just me messing around with some thoughts in caluclating various homology groups.

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

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?

I've been meaning to have a go at this, but other things keep on getting in the way. A couple of questions first though: Is the polyhedron $X$ a 2D surface or a solid? The former is homeomorphic to the sphere $S^2$ and the latter to the solid ball in 3D space $B_1((0,0,0))$. Since both are simpler shapes than a polyhedron, it makes me wonder why they specify a polyhedron rather than a sphere or a solid ball. Could it be because they want part of the challenge to be to make the fairly elementary observation that a polyhedral surface is homeomorphic to $S^2$ otr that a solid polyhedron is homeomorphic to a solid ball?