1.04 Examples and simply-connectedness

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

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Notes

Examples

(0.21) We saw in the previous video that all loops based at $0$ in $\mathbf{R}^n$ are based homotopic to the constant loop, so $\pi_1(\mathbf{R}^n,0)\cong\{1\}$ (i.e. it is the trivial group).
(0.54) Let $S^1$ denote the unit circle in $\mathbf{C}$ . The fundamental group $\pi_1(S^1,1)$ is isomorphic to the integers $\mathbf{Z}$ : the homotopy class of a loop is determined by the number of times it winds around the circle. We will prove this later: for now, you will need to take it on trust.

Simply-connected spaces

(2.15) A path-connected space

$X$ is called simply-connected if

$\pi_1(X,x)=\{1\}$ .

We will see later that the fundamental group is independent (up to isomorphism) of the basepoint when

$X$ is path-connected, so the choice of

$x$ in this definition does not matter.

(3.28) If

$X$ is a simply-connected space and

$x,y\in X$ then there is a unique homotopy class of paths from

$x$ to

$y$ .

Here, a homotopy of paths from

$x$ to

$y$ means a map

$H\colon[0,1]\times[0,1]\to X$ such that

$H(s,0)=x$ and

$H(s,1)=y$ for all

$s\in[0,1]$ .

(5.50) Suppose we have two paths

$\alpha,\beta$ from

$x$ to

$y$ . Because

$\pi_1(X,x)=\{1\}$ , the loop

$\beta^{-1}\cdot\alpha$ (based at

$x$ ) is homotopic to the constant map

$\epsilon_x$ at

$x$ . Now

$\alpha\simeq\beta\cdot\beta^{-1}\cdot\alpha\simeq\beta\cdot\epsilon_x\simeq\beta.$

Fundamental group of the 2-sphere

(7.38) Let $S^2=\{(x,y,z)\in\mathbf{R}^3\:\ x^2+y^2+z^2=1\}$ be the unit sphere in $\mathbf{R}^3$ ; since points on the sphere can be specified by two coordinates (latitude and longitude), we say that the sphere is 2-dimensional. Let $N,S$ be the North and South poles respectively.

The fundamental group

$\pi_1(S^2,S)$ is trivial (the 2-sphere is simply-connected).

(8.20) Let

$\gamma\colon[0,1]\to S^2$ be a loop.

If $\gamma(t)\neq N$ for all $t\in[0,1]$ then $\gamma$ is contractible. This is because $S^2\setminus\{N\}$ is homeomorphic to the plane via stereographic projection and every loop in the plane is contractible (as we saw here).
If $\gamma$ passes through the North pole then we can find a homotopic loop which misses the North pole (which then implies that $\gamma$ is nullhomotopic, by the first point).

(11.50) To prove this second point, let

$U$ be a neighbourhood of the North pole and let

$V$ be a neighbourhood of

$S^2\setminus U$ . Because

$\gamma$ is continuous, the preimages

$\gamma^{-1}(U)$ and

$\gamma^{-1}(V)$ consist of a collection of connected open intervals (open in the subspace topology on

$[0,1]$ , so

$[0,\epsilon)$ and

$(1-\epsilon,1]$ count as open) which cover the interval

$[0,1]$ . Because the interval

$[0,1]$ is compact, this admits a finite subcover. We can therefore find a finite sequence of times

$0=t_0\leq t_1\leq \cdots\leq t_n=1$ such that

$\gamma_i:=\gamma_{[t_i,t_{i+1}]}$ has image contained in either

$U$ or in

$V$ .

(16.20) Whenever $\gamma_i$ has image in $U$ , we will replace the subpath $\gamma_i$ with a homotopic path disjoint from $N$ (the subpaths which are contained in $V$ automatically miss $N$ ). To that end, pick* any path $\delta_i$ in $U\setminus\{N\}$ from $\gamma(t_i)$ to $\gamma_{t_{i-1}}$ .

(18.40) By the lemma above, $\gamma_i\simeq\delta_i$ (i.e. these paths are homotopic in $U$ with fixed endpoints) since the disc $U$ is simply-connected. Therefore, replacing each $\gamma_i$ with $\delta_i$ we get a homotopic path which avoids the North pole.

(20.00) *The reason we can find $\delta_i$ is because $U\setminus\{N\}$ is path-connected (see here).

Pre-class questions

What about the unit sphere $S^n=\{(x_0,\ldots,x_n)\in\mathbf{R}^{n+1}\ :\ \sum_{k=0}^n x_k^2=1\}$ in higher dimensions? Is it simply-connected?
What about the unit circle $S^1=\{(x,y)\in\mathbf{R}^2\ :\ x^2+y^2=1\}$ in two dimensions?

Navigation

Previous video: 1.03 Concatenation and the fundamental group.
Next video: 1.05 Basepoint dependence.
Index of all lectures.