The Fundamental Group

The fundamental group $\pi_1(X, x_0)$ is the first and most intuitive algebraic invariant of a topological space. It captures the "holes" in $X$ by studying loops up to continuous deformation, and its algebraic structure provides powerful tools for distinguishing spaces.

Definition

Definition9.7Loop

A loop in $X$ based at $x_0$ is a continuous map $\gamma: [0, 1] \to X$ with $\gamma(0) = \gamma(1) = x_0$ . The point $x_0$ is called the basepoint.

Definition9.8Fundamental Group

The fundamental group of $X$ at the basepoint $x_0$ , denoted $\pi_1(X, x_0)$ , is the set of path-homotopy classes of loops based at $x_0$ : $\pi_1(X, x_0) = \{[\gamma] : \gamma \text{ is a loop based at } x_0\}$ with the group operation given by concatenation of loops: $[\gamma_1] \cdot [\gamma_2] = [\gamma_1 * \gamma_2]$ where $(\gamma_1 * \gamma_2)(t) = \begin{cases} \gamma_1(2t) & t \in [0, 1/2] \\ \gamma_2(2t - 1) & t \in [1/2, 1] \end{cases}$ .

Group Structure

Theorem9.2$\pi_1(X, x_0)$ is a Group

Under the concatenation operation, $\pi_1(X, x_0)$ is a group with:

Identity: The constant loop $e(t) = x_0$ for all $t$ .
Inverse: $[\gamma]^{-1} = [\bar{\gamma}]$ where $\bar{\gamma}(t) = \gamma(1 - t)$ (the reverse path).
Associativity: $[\gamma_1] \cdot ([\gamma_2] \cdot [\gamma_3]) = ([\gamma_1] \cdot [\gamma_2]) \cdot [\gamma_3]$ .

Proof

Identity: We need $[\gamma * e] = [\gamma]$ . The homotopy is: $H(t, s) = \begin{cases} \gamma\left(\frac{2t}{1+s}\right) & t \in [0, (1+s)/2] \\ x_0 & t \in [(1+s)/2, 1] \end{cases}$ which continuously "speeds up" $\gamma$ and shrinks the constant portion. Similarly for $[e * \gamma] = [\gamma]$ .

Inverse: We need $[\gamma * \bar{\gamma}] = [e]$ . The homotopy: $H(t, s) = \begin{cases} \gamma(2t) & t \in [0, s/2] \\ \gamma(s) & t \in [s/2, 1 - s/2] \\ \gamma(2 - 2t) & t \in [1 - s/2, 1] \end{cases}$ deforms $\gamma * \bar{\gamma}$ to the constant loop by "pushing" the turnaround point back to the basepoint.

Associativity: The homotopy reparametrizes the three-fold concatenation: $(\gamma_1 * \gamma_2) * \gamma_3 \simeq \gamma_1 * (\gamma_2 * \gamma_3) \quad (\text{rel } \{0, 1\}).$ The two sides differ only in how the interval $[0, 1]$ is partitioned into three pieces; a linear reparametrization provides the homotopy.

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Dependence on Basepoint

Theorem9.3Change of Basepoint

If $X$ is path-connected, then $\pi_1(X, x_0) \cong \pi_1(X, x_1)$ for any $x_0, x_1 \in X$ . The isomorphism depends on the choice of a path from $x_0$ to $x_1$ .

Proof

Let $\alpha$ be a path from $x_0$ to $x_1$ . Define $\hat{\alpha}: \pi_1(X, x_0) \to \pi_1(X, x_1)$ by $\hat{\alpha}([\gamma]) = [\bar{\alpha} * \gamma * \alpha]$ . This is a group homomorphism with inverse $\hat{\bar{\alpha}}$ , hence an isomorphism.

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RemarkNotation for Path-Connected Spaces

When $X$ is path-connected, the fundamental group is independent of basepoint up to isomorphism (but not canonical isomorphism). We sometimes write $\pi_1(X)$ without specifying the basepoint.

Examples

ExampleFundamental Groups

$\pi_1(\mathbb{R}^n) = 0$ : $\mathbb{R}^n$ is contractible, so every loop is null-homotopic.
$\pi_1(S^1) \cong \mathbb{Z}$ : Generated by the loop $\gamma(t) = e^{2\pi i t}$ winding once around the circle. The integer $n$ corresponds to winding $n$ times. (Proof: see the theorems section.)
$\pi_1(T^2) \cong \mathbb{Z} \times \mathbb{Z}$ : The torus has two independent loops (one around each $S^1$ factor).
$\pi_1(S^n) = 0$ for $n \geq 2$ : Higher-dimensional spheres are simply connected.
$\pi_1(\mathbb{R}P^2) \cong \mathbb{Z}/2\mathbb{Z}$ : Real projective space has a loop (the image of a semicircle under $S^2 \to \mathbb{R}P^2$ ) that is not null-homotopic but whose square is.
$\pi_1(\text{figure-eight}) \cong F_2$ : The free group on two generators.

Simply Connected Spaces

Definition9.9Simply Connected

A topological space $X$ is simply connected if it is path-connected and $\pi_1(X, x_0) = 0$ (the trivial group) for some (hence every) basepoint.

Equivalently, $X$ is simply connected if it is path-connected and every loop in $X$ is null-homotopic.

ExampleSimply Connected Spaces

$\mathbb{R}^n$ for all $n \geq 1$ .
$S^n$ for $n \geq 2$ .
Any convex subset of $\mathbb{R}^n$ .
Any contractible space.
The universal cover of any space.

RemarkFunctoriality

The fundamental group is a functor $\pi_1: \mathbf{Top}_* \to \mathbf{Grp}$ from pointed topological spaces to groups. A continuous map $f: (X, x_0) \to (Y, y_0)$ induces a group homomorphism $f_*: \pi_1(X, x_0) \to \pi_1(Y, y_0)$ by $f_*([\gamma]) = [f \circ \gamma]$ . This is well-defined because homotopies are preserved: if $H: \gamma_0 \simeq \gamma_1$ , then $f \circ H: f \circ \gamma_0 \simeq f \circ \gamma_1$ .