A description/proof of that fundamental group homomorphism induced by homotopy equivalence is 'groups - group homomorphisms' isomorphism
Topics
About: topological space
About: group
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 homotopy equivalence.
- The reader knows a definition of %category name% isomorphism.
- The reader admits the proposition that for any 2 homotopic maps, any point on the domain, and the fundamental group homomorphisms induced by the maps, the 2nd homomorphism is the composition of the canonical 'groups - group homomorphisms' isomorphism between the codomains of the homomorphisms after the 1st homomorphism.
- The reader admits the proposition that any bijective group homomorphism is a 'groups - group homomorphisms' isomorphism.
Target Context
- The reader will have a description and a proof of the proposition that the fundamental group homomorphism induced by any homotopy equivalence 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, \(T_1, T_2\), any homotopy equivalence, \(f: T_1 \to T_2\), and any point, \(p \in T_1\), the fundamental group homomorphism induced by \(f\), \(f_*: \pi_1 (T_1, p) \to \pi_1 (T_2, f (p))\), is a 'groups - group homomorphisms' isomorphism.
2: Proof
There is a continuous map, \(f': T_2 \to T_1\), such that \(f' \circ f \simeq 1_{T_1}\) and \(f \circ f' \simeq 1_{T_2}\).
There is a homotopy, \(F: T_1 \times I \to T_1\), such that \(F (p', 0) = f' \circ f (p')\) and \(F (p', 1) = 1_{T_1} (p') = p'\). \((f' \circ f)_*: \pi_1 (T_1, p) \to \pi_1 (T_1, f' \circ f (p))\) and \((1_{T_1})_*: \pi_1 (T_1, p) \to \pi_1 (T_1, p)\). There is the canonical 'groups - group homomorphisms' isomorphism, \(\phi: \pi_1 (T_1, f' \circ f (p)) \to \pi_1 (T_1, p)\), \([g] \mapsto [\gamma^{-1}] [g] [\gamma]\), where \(\gamma: I \to T_1\), \(t \mapsto F (p, 1 - t)\), such that \((1_{T_1})_* = \phi \circ (f' \circ f)_*\), by the proposition that for any 2 homotopic maps, any point on the domain, and the fundamental group homomorphisms induced by the maps, the 2nd homomorphism is the composition of the canonical 'groups - group homomorphisms' isomorphism between the codomains of the homomorphisms after the 1st homomorphism.
As \((1_{T_1})_*\) is a bijection and \(\phi\) is a bijection, \((f' \circ f)_*\) is a bijection. \((f' \circ f)_* = f'_* \circ f_*\), and as the left hand side is a bijection, \(f_*\) is an injection and \(f'_*\) is a surjection.
By the likewise argument for \(f \circ f' \simeq 1_{T_2}\), \(f'_*\) is an injection and \(f_*\) is a surjection.
So, \(f_*\) is a bijection.
So, \(f_*\) is a 'groups - group homomorphisms' isomorphism, by the proposition that any bijective group homomorphism is a 'groups - group homomorphisms' isomorphism.
