# 1.10 Homotopy invariance

## Video

Below the video you will find accompanying notes and some pre-class questions.

## Notes

### Homotopy invariance of the fundamental group

(0.00) In this section, we will prove that homotopy equivalent spaces have isomorphic fundamental groups.

(0.20) If $$F\colon X\to X$$ is a continuous map which is homotopic to the identity map $$id_X\colon X\to X$$, then the induced map $$F_*\colon\pi_1(X,x)\to\pi_1(X,F(x))$$ is an isomorphism.

(1.26) Let $$F_t\colon X\to X$$ be a homotopy from $$F_0=F$$ to $$F_1=id_X$$ and let $$\delta$$ be the path traced out by the basepoint $$x$$ under this homotopy, that is $\delta(t)=F_t(x).$ Now, for $$[\gamma]\in\pi_1(X,x)$$, we have $F_*[\gamma]=[F\circ\gamma]$ and $$F\circ\gamma$$ is freely homotopic to $$\gamma$$ via the free homotopy $$F_s\circ\gamma$$. This is a free (rather than based) homotopy because the basepoint of $$F_s\circ\gamma$$ is $$\delta(s)$$. Using our results on basepoint dependence, this implies that $[F\circ\gamma]=[\delta\cdot\gamma\cdot\delta^{-1}].$ and that the map $$[\gamma]\to[\delta\cdot\gamma\cdot\delta^{-1}]$$ is an isomorphism $$\pi_1(X,x)\to\pi_1(X,F(x))$$. Since $$F_*[\gamma]=[F\circ\gamma]$$, this tells us that $$F_*$$ is an isomorphism.

(6.38) If $$F\colon X\to Y$$ and $$G\colon Y\to X$$ are homotopy inverses then $$F_*\colon\pi_1(X,x)\to\pi_1(Y,F(x))$$ and $$G_*\colon\pi_1(Y,y)\to\pi_1(X,G(y))$$ are isomorphisms.

(7.36) The composition $$F\circ G\colon Y\to Y$$ is homotopic to $$id_Y$$. By the previous lemma, $$(F\circ G)_*\colon\pi_1(Y,y)\to\pi_1(Y,F(G(y)))$$ is an isomorphism. By functoriality, we have $$(F\circ G)_*=F_*\circ G_*$$, so $$F_*\circ G_*$$ is an isomorphism. This implies that $$G_*$$ is injective (otherwise $$F_*\circ G_*$$ would fail to be injective) and $$F_*$$ is surjective (otherwise $$F_*\circ G_*$$ would fail to be surjective).

(9.58) By arguing the same way about the composition $$G\circ F$$, we get that $$G_*$$ is surjective and $$F_*$$ is injective. This implies that both $$F_*$$ and $$G_*$$ are bijections, and hence isomorphisms.

## Pre-class questions

1. The punctured plane $$\mathbf{C}\setminus\{0\}$$ is homotopy equivalent to the (1-dimensional) unit circle $$S^1$$. Find 1-dimensional'' topological spaces homotopy equivalent to $$\mathbf{C}\setminus\mu_n$$ where $$\mu_n$$ is the set of $$n$$th roots of unity (a doodle, rather than a proof, will suffice). We will be able to use this to compute $$\pi_1(\mathbf{C}\setminus\mu_n)$$ once we have proved Van Kampen's theorem.