The induced fundamental group homomorphism of a composition of continuous maps is the same as the composition of their induced homomorphisms

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Statement

Let [ilmath](X,\mathcal{ J })[/ilmath], [ilmath](Y,\mathcal{ K })[/ilmath] and [ilmath](Z,\mathcal{ H })[/ilmath] be topological spaces, let [ilmath]p\in X[/ilmath] be any fixed point (to act as a base point for the fundamental group [ilmath]\pi_1(X,p)[/ilmath]) and let [ilmath]\varphi:X\rightarrow Y[/ilmath] and [ilmath]\psi:Y\rightarrow Z[/ilmath] be continuous maps. Then:

• [ilmath](\psi\circ\varphi)_*\eq(\psi_*\circ\varphi_*)[/ilmath]
  • where [ilmath]\varphi_*[/ilmath] denotes the fundamental group homomorphism, [ilmath]\varphi_*:\pi_1(X,p)\rightarrow\pi_1(Y,\varphi(p))[/ilmath], induced by [ilmath]\varphi[/ilmath] - and "" for the others

Note that both of these maps have the form [ilmath]\big(:\pi_1(X,p)\rightarrow\pi_1(Z,\psi(\varphi(p))\big)[/ilmath]

Proof

This is a usual [ilmath]\text{LHS}\eq\text{RHS} [/ilmath] proof:

• Let [ilmath][f]\in\pi_1(X,p)[/ilmath] be given.
  • First we simplify them:
    1. [ilmath](\psi\circ\varphi)_*([f])[/ilmath]

       [ilmath]\eq[(\psi\circ\varphi)\circ f][/ilmath]

       [ilmath]\eq[\psi\circ\varphi\circ f][/ilmath]

    2. [ilmath](\psi_*\circ\varphi_*)([f])[/ilmath]

       [ilmath]\eq\psi_*(\varphi_*([f]))[/ilmath]

       [ilmath]\eq\psi_*([\varphi\circ f])[/ilmath]

       [ilmath]\eq[\psi\circ(\varphi\circ f)][/ilmath]

       [ilmath]\eq[\psi\circ\varphi\circ f][/ilmath]

  • We observe these are equal
• Since [ilmath][f]\in\pi_1(X,p)[/ilmath] was arbitrary (and the maps are well-defined for [ilmath][f][/ilmath] by the properties of the fundamental group homomorphism induced by a continuous map

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