1.10 Homotopy invariance

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

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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

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.

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