Homeomorphic topological spaces have isomorphic fundamental groups

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Statement

Let $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$ be homeomorphic topological spaces, let $p\in X$ be given (this will be the base point of the fundamental group $\pi_1(X,p)$) and let $\varphi:X\rightarrow Y$ be that homeomorphism. Then: :

• $\pi_1(X,p)\cong\pi_1(Y,\varphi(p))$ - where $\cong$ denotes group isomorphism here, but can also be used to denote topological isomorphism (AKA: homeomorphism)

That is to say:

• $\big(X\cong_\varphi Y)\implies(\pi_1(X,p)\cong_{\varphi_*}\pi_1(Y,\varphi(p))\big)$

Proof

The idea is to recall that the definition of a categorical isomorphism means that the map composed with its inverse is the identity of the codomain and the inverse composed with the map is the identity on the domain.

Noting that a homeomorphism is an instance of this we see that both compositions are identity maps, which as we know from the induced fundamental group homomorphism of the identity map is the identity map of the fundamental group and using:

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

Shows that the induced maps are categorical isomorphisms too.

Using:

• A map is a group isomorphism if and only if it satisfies the properties of a categorical isomorphism and the result follows.

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Good to do formally