1.04 Examples and simply-connectedness

Video

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

Notes

Examples

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?

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