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

From Maths

## 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^{[1]}:

- [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]

## References