1.04 Examples and simply-connectedness

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



  1. (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).
  2. (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.
  1. 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).
  2. 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

  1. 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?
  2. What about the unit circle \(S^1=\{(x,y)\in\mathbf{R}^2\ :\ x^2+y^2=1\}\) in two dimensions?


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.