A question on Brouwer Fixed point theorem for two homeomorph discs

Question

Brouwer Fixed point theorem states that

Theorem: any continuous map from a unit two dimensional disc $D^2$ into itself has a fixed point.

Q: Is it possible to use this theorem for two homeomorph discs $D^2$ and $D'^2$ (with different boundary)? and in this case what is the meaninig of fixed point?

Answer 1

You cannot really formulate such a theorem at all.

Fixed points are for self-maps (i.e. $f:X \to X$ for some $X$).

The FPP (fixed point property) i.e. "every continuous self-map of $X$ has a fixed point" is a topological property. So any space $Y$ homeomeomorphic to $X$ has it when $X$ has it and vice versa.

This is proved easily by "map transportation": if $f: Y \to Y$ is a continuous self map and $h: X \to Y$ is a homeomorphism, then $h^{-1} \circ f \circ h$ is a continuous self map of $X$, so has a fixed point $p \in X$ and then $h(p)$ is one for $f$ (all this is simple to check). So your $D'^2$ does have the FPP. Maybe that is what you really want?

Answer 2

It doesn't make sense to talk about fixed points for a map $f : D^2 \to D'^2$ with differing domain and codomain. So no, you can't really use the theorem, because I don't even see how you could state it.