fundamental group

Definition

The fundamental group of a topological space $X$ based at $x\in X$ is

${\pi }_1(X,x)=P(x,x)/\sim =\{\text{loops at x}\}/\text{homotopy}$

It is the homotopy equivalence classes of loops. We consider the group multiplication to be composition of loops i.e.

$[\gamma ]\cdot [\delta ]=[\gamma \circ \delta ]$

Proof this is a group

The fundamental group follows the group axioms as shown below:

Group operation

Composition of paths is defined as $(\delta \cdot \gamma )(s)=\{\begin{array}{ll}\gamma (2s)& 0\le s\le \frac{1}{2}\\ \delta (2s-1)& \frac{1}{2}\le s\le 1\end{array}$

Inversion

Inversion is defined as

$$\begin{array}{rl}P(x,y)& ⟶P(x,y)\\ \gamma (s)& ⟼{\gamma }^{-1}(s)\end{array}$$

where ${\gamma }^{-1}(s)=\gamma (1-s)$.

Identity element

There exists a constant path $C_x\in P(x,x)$ such that $C_x(t)=x$ for all $t\in I$.

Well defined for equivalence classes

• Composition of paths descends to homotopy classes

$[\delta ]\cdot [\gamma ]=[\delta \cdot \gamma ]$

• Constant paths are half identity up to homotopy For a path $\gamma$ from $x\to y$

$[\gamma ]\cdot [C_x]=[\gamma ]$

and

$[C_y]\cdot [\gamma ]=[\gamma ]$

• Inversion is defined in homotopy classes $[{\gamma }^{-1}]\cdot [\gamma ]=[C_x]$

Pushforward of continuous maps

Let $f:X\to Y$ be a continuous map. Then the pushforward

$$\begin{array}{rl}{f}_{\ast }:{\pi }_1(X,x)& ⟶{\pi }_1(Y,f(x))\\ [\gamma ]& ⟶[f\circ \gamma ]\end{array}$$

is a group homomorphism.

Thus, the fundamental group depends (up to isomorphism) only on $X$ and the path components of $X$.

Homotopy equivalence

Let $X$, $Y$ be (path connected) topological spaces that are homotopy equivalent. Then ${\pi }_1(X)\simeq {\pi }_1(Y)$.

Examples

${\pi }_1({\mathbb{R}}^n)\simeq 0$ ${\pi }_1(S^1)\simeq \mathbb{Z}$ Can be proved with Lebesgue number lemma.