1.05 Basepoint dependence

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

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Notes

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

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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Previous video: 1.04 Examples and simply-connectedness.
Next video: 1.06 Fundamental theorem of algebra: reprise.
Index of all lectures.