Fundamental groups

Overview

The fundamental group ${\pi }_1(X,x)$ of a topological space $X$ at a point $x\in X$ is the quotient space of loops $f:I\to X$ based at $x$ after identifying loops that are homotopic relative to the endpoints $\partial I=\{0,1\}$.

Two fundamental groups are isomorphic if they are connected by a path. In particular, if $\alpha$ is such a path between two basepoints $x_0,x_1$, then we can define an isomorphism $\hat{\alpha }$ between fundamental groups called the change-of-basepoint map.

A space is simply connected if it is path-connected, hence any two fundamental groups are isomorphic, and its fundamental group is trivial.

We have the following methods for “computing” the fundamental group of a space:

• (Theorem) van Kampen
• (Theorem) A group acting on a simply connected space is isomorphic to the fundamental group of its orbits

Relevant theorems:

• (Theorem) The fundamental group of the circle is isomorphic to the additive group of integers
• (Theorem) The fundamental group of the n-sphere is trivial in higher dimensions

Related notes: Induced homomorphism between fundamental groups, Simply connected spaces

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Basic definition

Definition: Fundamental group of a space

Let $X$ be any space any $x_0\in X$ be any point. A loop based at $x_0$ is a path in $X$ that starts and ends at $X_0$.

The fundamental group of $X$ relative to $x_0$ is the set ${\pi }_1(X,x_0)$ of path-homotopy classes of loops in $X$ based at $x_0$. The group operation is path concatenation defined by

$$[f]\cdot [g]=[f\ast g],$$

inverses $[f{]}^{-1}=[\bar{f}]$ are reverse paths, and the identity is $e=[1_{x_0}]$.

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Change-of-basepoint isomorphisms

Theorem (52.1): Points connected by a path have isomorphic fundamental groups

Let $X$ be any space and $x_0,x_1\in X$ be any points, and suppose $\alpha$ is a path from $x_0$ to $x_1$. Then we define a corresponding map change-of-basepoint map between fundamental groups

$$\hat{\alpha }:{\pi }_1(X,x_0)\to {\pi }_1(X,x_1)$$

by

$$\hat{\alpha }([f])=[\bar{\alpha }]\ast [f]\ast [\alpha ].$$

The map $\hat{\alpha }$ is a group isomorphism.

🔺Review. Why is $\hat{\alpha }$ well-defined?

Theorem: Properties of change-of-basepoint isomorphisms

Let $\alpha$ be a path between two points $x_0,x_1\in X$, and let $\hat{\alpha }:{\pi }_1(X,x_0)\to {\pi }_1(X,x_1)$ be the corresponding change-of-basepoint isomorphism.

• (i) If $\beta$ is a path from $x_1$ to $x_2$, then $\widehat{(\beta \ast \alpha )}=\hat{\alpha }\circ \hat{\beta }$;
• (ii) wip Compatibility with induced homomorphisms between fundamental groups:

Theorem ( Topology HW 9.5): The fundamental group is abelian if and only if any two change-of-basepoint maps are equal

Let $X$ be path-connected and $x_0,x_1\in X$. Then ${\pi }_1(X,x_0)$ is abelian if and only if for any paths $\alpha ,\beta$ from $x_0$ to $x_1$, the corresponding isomorphisms satisfy $\hat{\alpha }=\hat{\beta }$ as functions ${\pi }_1(X,x_0)\to {\pi }_1(X,x_1)$.

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Examples

The fundamental group of Euclidean space is a single point

Let $c(t)=x$ be the constant loop, which gives a special element $[c]\in {\pi }_1(X,x)$. Then for any loop $f$ in ${\mathbb{R}}^n$, the straight line homotopy between $f$ and $c$ is a homotopy relative to $\partial I$.