415: Fundamental Group Homomorphism Induced by Composition of Continuous Maps Is Composition of Fundamental Group Homomorphisms Induced by Maps|T.B.P.

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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 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.

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}}_{*}

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.

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