A continuous map induces a homomorphism between fundamental groups

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• A continuous map induces a homomorphism on fundamental groups

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Note: there is an important precursor theorem: The relation of path-homotopy is preserved under composition with continuous maps.

Statement

Given two topological spaces, [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] (which may be the same) a continuous function, [ilmath]f:X\rightarrow Y[/ilmath] induces a group homomorphism between the fundamental groups of [ilmath]X[/ilmath] and [ilmath]Y[/ilmath].

• We denote this induced homomorphism, [ilmath]f_*:\pi_1(X,p)\rightarrow\pi_1(Y,f(p))[/ilmath] and it is given by [ilmath]f_*:[g]\mapsto[f\circ g][/ilmath]

Proof

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