# 1.04 Examples and simply-connectedness

## Video

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

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