I am caught up in a minor detail on a qualifying exam problem I am doing:
Show that there are no injective continuous functions $f: \Bbb R^n \to \Bbb R^2$ , $n>2$ with $f(0)=0$.
So far I have considered the punctured spaces at the origin, and so the problem changes to:
Show that there are no continuous injections from $\Bbb {R^n}-\{0\} \to \Bbb {R^2}-\{0\}$ for $n>2$.
To do this I want to look at the induced homomorphism between fundamental groups and say that it is nontrivial since the original function is injective. Can I say this? If so, why?
A continuous injective map doesn't necessarily induce an injective homomorphism on fundamental groups. Consider the embedding $S^1 \hookrightarrow \Bbb R^2$ for a counterexample.
What you need is the Borsuk–Ulam theorem. $f$ restricts to a continuous map $S^2 \to \Bbb R^2$, and this map cannot be injective by the theorem.