416: Fundamental Group Homomorphism Induced by Homeomorphism Is 'Groups - Group Homomorphisms' Isomorphism

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A description/proof of that fundamental group homomorphism induced by homeomorphism is 'groups - group homomorphisms' isomorphism

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Topics

About: topological space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

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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.

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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.

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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.

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Main Body

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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 $f_1$, ${f_1}_{\ast }$, is a 'groups - group homomorphisms' isomorphism.

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2: Proof

For any continuous loop path on $T_1$, $f:I\to T_1$, and its homotopic equivalence class, $[f]$, ${f_1}_{\ast }([f])=[f_1\circ f]$. For any continuous loop paths on $T_1$, $f,f^{\prime }$, such that $[f]\ne [f^{\prime }]$, $[f_1\circ f]\ne [f_1\circ f^{\prime }]$? Let us suppose that $[f_1\circ f]=[f_1\circ f^{\prime }]$. As $f_1$ is a homeomorphism, there is the continuous inverse, ${f_1}^{-1}$. $[{f_1}^{-1}\circ f_1\circ f]=[f]=[{f_1}^{-1}\circ f_1\circ f^{\prime }]=[f^{\prime }]$, a contradiction. So, ${f_1}_{\ast }$ 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 ${f_1}_{\ast }([{f_1}^{-1}\circ f^{″}])=[f_1\circ {f_1}^{-1}\circ f^{″}]=[f^{″}]$, so, ${f_1}_{\ast }$ is surjective, and so, is bijective.

So, ${f_1}_{\ast }$ is a 'groups - group homomorphisms' isomorphism, by the proposition that any bijective group homomorphism is a 'groups - groups homomorphisms' isomorphism.

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References

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