The Fundamental Group of a Topological Space at a Point

Definition: Let $X$ be a topological space. A path $\alpha : I \to X$ is a Loop at $x$ if $\alpha(0) = x$ and $\alpha(1) = x$.

Recall the material on the following pages:

• Products of Paths Relative to {0, 1} in a Topological Space

• Associativity of Products of Paths in a Topological Space

• Constant Paths in a Topological Space

• Inverse Paths in a Topological Spaces

In particular, if $\alpha, \beta, \gamma : I \to X$ are loops at $x$ then the operation;

(1)

\begin{align} \quad [\alpha][\beta] = [\alpha \beta] \end{align}

Is well defined, and furthermore:

• 1) $[\alpha]([\beta][\gamma]) = ([\alpha][\beta])[\gamma]$ (Associative Property).

• 2) $[\alpha][c_x] = [\alpha]$ and $[c_x][\alpha] = [\alpha]$ (The Existence of an Identity).

• 3) $[\alpha][\alpha^{-1}] = [c_x]$ and $[\alpha^{-1}][\alpha] = [c_x]$ (The Existence of Inverses).

So the set of homotopy classes $[\alpha]$ with the property that $\alpha(0) = x = \alpha(1)$ form a group. This group is given a special name.

Definition: Let $X$ be a topological space and let $x \in X$. The Fundamental Group of $X$ at $x$ is $\pi_1(X, x) = \{ [\alpha] : \alpha(0) = x = \alpha(1) \}$ with the operation of homotopy class multiplication given for all $\alpha, \beta \in \pi_1(X, x)$ by $[\alpha][\beta] = [\alpha \beta]$.

The simplest example of determining a fundamental group is with the topological space $X = \{ x \}$. This topological space contains only one element, and so $\pi_1(X, x)$ contains all of the homotopy classes of loops that start at $x$. But the only such loop is the constant look $[c_x]$. So:

(2)

\begin{align} \quad \pi_1(\{ x \}, x) = \{ [c_x] \} \end{align}

Theorem 1 (The Fundamental Group of the Circle): Let $S^1 = \{ (x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1 \}$ denote the unit circle. Then $\pi_1(S^1, x) \cong \mathbb{Z}$, that is, the fundamental group of the circle is infinite cyclic.

We do not yet have the tools to prove that the Fundamental group of the circle is isomorphic to the group of integers. In fact, proving this result is very long and cumbersome!