1.10 Homotopy invariance

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.

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