1.05 Basepoint dependence

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


Change of basepoint

(0.00) By now, we have seen how to associate to a space \(X\) and a basepoint \(x\) a group \(\pi_1(X,x)\) of homotopy classes of loops in \(X\) based at \(x\). You might wonder what happens if we pick a different basepoint \(y\in X\).

(0.30) Given a path \(\delta\colon[0,1]\to X\) with \(\delta(0)=x\) and \(\delta(1)=y\) we obtain an isomorphism \(F_\delta\colon\pi_1(X,y)\to\pi_1(X,y)\).
(1.37) Given a loop \(\gamma\) in \(X\) based at \(y\), we define \(F_\delta([\gamma])\) to be the homotopy class of loops \([\delta^{-1}\cdot\gamma\cdot\delta]\) based at \(x\).

Free homotopy and conjugation

(7.20) Given all of this, we can now say what happens if we have a free homotopy \(\gamma_s\) connecting two loops \(\gamma_0,\gamma_1\) based at \(x\). Let \(\delta(t)=\gamma_t(0)\) be the loop traced out by the basepoint of the loop \(\gamma_s\) along the free homotopy. From the theorem on changing basepoints, \[\gamma_0=\delta^{-1}\gamma_1\delta.\] Therefore \(\gamma_0\) is conjugate to \(\gamma_1\) in \(\pi_1(X,x)\). Different loops \(\delta\) will give different conjugates.

(10.40) This implies that free homotopy classes of loops based at \(x\) are conjugacy classes in \(\pi_1(X,x)\). This is very useful:

Pre-class questions

  1. Suppose that \(X\) is a topological space and \(x\in X\) is a basepoint with \(\pi_1(X,x)\cong S_3\), where \(S_3\) is the group of permutations of three objects. How many free homotopy classes of loops are there in \(X\)?


Comments, corrections and contributions are very welcome; please drop me an email at j.d.evans at lancaster.ac.uk if you have something to share.

CC-BY-SA 4.0 Jonny Evans.