A continuous map induces a homomorphism between 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, $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$ (which may be the same) a continuous function, $f:X\rightarrow Y$ induces a group homomorphism between the fundamental groups of $X$ and $Y$.

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

Proof

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