Simply connected spaces

Overview and basic definition

Definition: Simply connected

A space $X$ is simply connected if it is path-connected and if its fundamental group ${\pi }_1(X,x_0)$ is the trivial (one-element) group for some $x_0\in X$, and hence for all $x_0\in X$. The fact that the fundamental group is trivial is often denoted ${\pi }_1(X,x_0)=0$.

Intuitively, this means that every closed curve in $X$ can be continuously shrunk to a point in $X$, i.e., any loop with base point $x_0$ is homotopic relative to $x_0$ to the constant loop.

We have the following key facts about simply connected spaces:

• Any two paths having the same initial and final points are path-homotopic, i.e., any two paths are homotopic rel endpoints.
• Equivalently, a space is simply connected if and only if there is a unique path-homotopy class between any two points.
• If $p:\tilde{X}\to X$ is a covering space, the lifting correspondence $\varphi :{\pi }_1(X,x_0)\to p^{-1}(x_0)$ is a bijection; further, since $p$ is both continuous and open, then $p$ is a homeomorphism.

Relevant theorems:

• (Theorem) The fundamental group of the n-sphere is trivial in higher dimensions

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Properties of simply connected spaces

Lemma (Munkres 52.3): In a simply connected space, any two paths with the same initial and final points are path-homotopic.

Proof from Topology. Uses:

• (Proof pattern) Middle-man trick (but for the identity!)

Suppose $f,g$ are paths in $X$ from some point $x_0$ to $x_1$. Then $f\ast \bar{g}$ is a loop based at $x_0$, so by simply-connectedness, $f\ast g{\simeq }_p{1}_{x_0}$. Since path-homotopy is an equivalence relation, this implies

$$f{\simeq }_{p}f\ast \bar{g}\ast g{\simeq }_p{1}_{x_0}\ast g{\simeq }_{p}g.$$