Computational Homology used to verify that two spaces are homeomorphic The following two facts seem really intriguing and I am trying to figure out how to use computations of homology to deduce them:
(a) The boundary of an ($n+1$)-simplex is homeomorphic to $S^n$.
(b) An $n$-dimensional convex body is a compact convex set in $\mathbb{R}^n$. Show that any two $n$-dimensional convex bodies are homeomorphic.  
Any input to help me think about these exercises would be greatly appreciated.
 A: You can find proofs of both facts in Ch.1 section 16 of Bredon's book Topology and Geometry.  Here are the two relevant statements.
16.3 Proposition. Let $C\subset \mathbb R^n$ be a compact convex body with $0\in int(C)$. Then the function $f\colon\partial C\to S^{n-1}$ given by $f(x)=x/||x||$ is a homeomorphism.
This is easy to verify now that you know what the map is!
16.4 Theorem. A compact convex body $C$ in $\mathbb R^n$ with nonempty interior is homeomorphic to the closed $n$ ball, and $\partial C\cong S^{n-1}$.
To prove this, assume by translating if necessary that $0\in int(C)$. Define $k\colon D^n\to C$ by $k(x)=||x||f^{-1}(x/||x||)$ for $x\neq 0$ and $k(0)=0$, where $f$ is as above. Now check this is a homeomorphism.
A: Homological computations won't help you in proving that two spaces are homeomorphic. Having isomorphic homology groups is (usually) a much weaker condition than being homeomorphic.
In the examples you are interested in, it is possible to write down explicit homeomorphisms. My advise would be to try and find these yourself.
