A description/proof of that fundamental group homomorphism induced by homeomorphism is 'groups - group homomorphisms' isomorphism
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of fundamental group homomorphism induced by map.
- The reader knows a definition of %category name% isomorphism.
- The reader admits the proposition that any bijective group homomorphism is a 'groups - groups homomorphisms' isomorphism.
Target Context
- The reader will have a description and a proof of the proposition that the fundamental group homomorphism induced by any homeomorphism is a 'groups - group homomorphisms' isomorphism.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description
For any topological spaces, \(T_1, T_2\), and any homeomorphism, \(f_1: T_1 \to T_2\), the fundamental group homomorphism induced by \(f_1\), \({f_1}_*\), is a 'groups - group homomorphisms' isomorphism.
2: Proof
For any continuous loop path on \(T_1\), \(f: I \to T_1\), and its homotopic equivalence class, \([f]\), \({f_1}_* ([f]) = [f_1 \circ f]\). For any continuous loop paths on \(T_1\), \(f, f'\), such that \([f] \neq [f']\), \([f_1 \circ f] \neq [f_1 \circ f']\)? Let us suppose that \([f_1 \circ f] = [f_1 \circ f']\). As \(f_1\) is a homeomorphism, there is the continuous inverse, \({f_1}^{-1}\). \([{f_1}^{-1} \circ f_1 \circ f] = [f] = [{f_1}^{-1} \circ f_1 \circ f'] = [f']\), a contradiction. So, \({f_1}_*\) is injective. For any continuous loop path on \(T_2\), \(f'': I \to T_2\), and its homotopic equivalence class, \([f'']\), there is \([{f_1}^{-1} \circ f'']\), and \({f_1}_* ([{f_1}^{-1} \circ f'']) = [f_1 \circ {f_1}^{-1} \circ f''] = [f'']\), so, \({f_1}_*\) is surjective, and so, is bijective.
So, \({f_1}_*\) is a 'groups - group homomorphisms' isomorphism, by the proposition that any bijective group homomorphism is a 'groups - groups homomorphisms' isomorphism.
