# 1.05 Basepoint dependence

## Video

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

## 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

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 $$X$$?