Can we prove, if $f:\mathbb{D}^2\rightarrow\mathbb{D}^2$ is a homeomorphism then $f(S^1)=S^1$ and $f(\textrm{int}(\mathbb{D}^2))=\textrm{int}(\mathbb{D}^2)$, using fixed point theorem? I have already solved this problem using fundamental groups, but can we solve this problem using fixed point theorem?
Asked
Active
Viewed 82 times
4
-
possible not, because fixed point theorem doesn't hold for many spaces, for which the first statement does, the intuitive reason of this statement is the fact "if $f$ is a homeomorphism, then $x$ and $f(x)$ must have homeomorphic neighborhoods" – Andrey Ryabichev May 20 '16 at 10:58
1 Answers
0
I can't see any obvious way to do it using the fixed point theorem - the natural way is to consider what happens when you remove a point from the boundary $S^1$, which I suspect is what you mean by "using fundamental groups".
Josh Hunt
- 1,463
-
Yea, that approach I know, my query is, is it possible to kill this problem using fixed point theorem? – maths_freak_007 May 20 '16 at 08:03
-
As in my answer, I can't see any way to do so using the fixed point theorem. – Josh Hunt May 20 '16 at 08:05