Right now I have some problem about the exercise in the chapter of Massey's book, this chapter talks about several important facts about the $Jordan \space Brouwer \space Theorem$,i.e.,$Let \space A \space be \space a \space subset \space of \space S^n \space homeomorphic \space to \space S^{n-1}, then\space S^n -A \space has \space two \space components, and \space the \space boundary\space of \space these\space components \space are \space A$
The questions are: 1. Show that no proper subset of $S^m$ can be homeomorphic to $S^n,n>m$
2.Prove that any continuous map $f:S^n \to R^n$ cannot be $1-1$
I guess I have to make some calculation about some homology groups, but which space? Or maybe there are other solutions to this problems?
You do not need any homology computation.
About the first question: since $S^n$ must be a proper subset of $S^m$ you have an injection in $S^m \setminus \{ pt\}$ this is homeomorphic to $\mathbb{R}^m$ via the usual stereographic projection. Then you can insert $\mathbb{R^m}$ into $\mathbb{R^n}$ and you finally get an embedding $S^n \hookrightarrow\mathbb{R}^n$. There is corollary of Jordan-Brouwer saying that if you have an embedding $M \hookrightarrow \mathbb{R}^n$ with $M$ topological manifold of dimension $n$ then the map is open. Can you end the proof?
For the second question just apply again this corollary. Or alternatively compose $f$ with the one point compactification of $\mathbb{R}^n$. Are you familiar with this concept and what result you get for $\mathbb{R}^n$?