A description/proof of that fundamental group homomorphism induced by composition of continuous maps is composition of fundamental group homomorphisms induced by maps
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.
Target Context
- The reader will have a description and a proof of the proposition that for any continuous map from any 1st topological space into any 2nd topological space and any continuous map from the 2nd topological space into any 3rd topological space, the fundamental group homomorphism induced by the composition of the maps is the composition of the fundamental group homomorphisms induced by the maps.
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, \(T_1, T_2, T_3\), and any continuous maps, \(f_1: T_1 \to T_2\) and \(f_2: T_2 \to T_3\), the fundamental group homomorphism induced by \(f_2 \circ f_1\), \((f_2 \circ f_1)_*\) is the composition of the fundamental group homomorphisms induced by \(f_1\) and \(f_2\), \({f_1}_*\) and \({f_2}_*\), which is \((f_2 \circ f_1)_* = {f_2}_* \circ {f_1}_*\).
2: Proof
For any continuous loop path, \(f: I \to T_1\), and its homotopic equivalence class, \([f]\), \((f_2 \circ f_1)_* ([f]) = {f_2}_* \circ {f_1}_* ([f])\)? \((f_2 \circ f_1)_* ([f]) = [f_2 \circ f_1 (f)] = {f_2}_* ([f_1 (f)]) = {f_2}_* ({f_1}_* ([f]))\). So, yes.
