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.