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 \(x\).

  • (3.13) This map is well-defined: if \(\gamma_s\) is a homotopy then \(\delta^{-1}\cdot\gamma_s\cdot\delta\) is a homotopy from \(F_\delta([\gamma_0])\) to \(F_\delta([\gamma_1])\), so \(F_\delta([\gamma])\) doesn't depend on the choice of \(\gamma\) within its homotopy class.
  • (4.39) \(F_\delta\) is a homomorphism: given two loops \(\alpha,\beta\) based at \(y\), we have

    \begin{align*} F_\delta(\beta\cdot\alpha)&=\delta^{-1}\cdot\beta\cdot\delta\cdot\delta^{-1}\cdot\alpha\cdot\delta\\ &=\delta^{-1}\cdot\beta\cdot\alpha\cdot\delta\\ &=F_\delta(\beta\cdot\alpha),\\ F_\delta(1)&=\delta^{-1}\delta=1, \end{align*}

    as required.

  • (5.52) \(F_\delta\) is invertible: we have \(F_{\delta^{-1}}(F_\delta(\gamma))=\delta\cdot\delta^{-1}\cdot\gamma\cdot\delta\cdot\delta^{-1}=\gamma\), so \(F_{\delta^{-1}}\) is an inverse for \(F_\delta\).

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:

  • (11.13) If \(\pi_1(X,x)\) is abelian then two loops are based homotopic if and only if they are freely homotopic (conjugation does nothing in an abelian group).
  • (12.13) In many geometric examples, the homotopies we construct often move the basepoint (for example, in the proof of the fundamental theorem of algebra).

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 \(X\)?


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