When a homotopy equivalence of the closed unit ball in the Euclidean n- dimensional space is injective?

Question

I have a compact connected metric space $X$ of dimension $n$ which is homotopically equivalent to the closed unit ball $D^n$ in the n-dimensional Euclidean space. I am wondering if there is an homotopy equivalence $h$ such that either $h$ or its homotopy inverse is injective.

Thank you.

Answer 1

I dont think this is atall true...since $D^n$ is contractible so any map is actually homotopic to constant map...so if you casider $X= D^n$ then for given any map $f,g: D^n \rightarrow D^n$ $fog$ and $gof$ is homotopic to constant map and thus homotopic to identity map...