Lecture 9: Paths, Homotopy of Paths, and the Fundamental Group
← Back to Course HomeKSM3E02: Algebraic Topology I — Fall 2026 | Tuesday, September 8, 2026 (12:00–1:00 PM) | Instructor: Pranav Haridas
Having developed the foundations of CW complexes, cell attachments, and homotopy equivalence in Lectures 1–8, we now turn to our first major algebraic invariant: the fundamental group $\pi_1(X, x_0)$. Introduced by Henri Poincaré in 1895, the fundamental group captures 1-dimensional "holes" and loops in a topological space by assigning to each pointed space a group whose elements are homotopy classes of loops based at $x_0$.
In this lecture, we rigorously define paths, path homotopy (homotopy relative to endpoints), path composition, and verify in detail that path homotopy classes of loops form a group under concatenation.
1. Paths and Homotopy Relative to Endpoints
Let $X$ be a topological space, and let $I = [0,1]$ denote the standard closed unit interval equipped with the subspace topology from $\mathbf{R}$.
Definition 1 (Paths and Loops).
- A path in $X$ from $x_0$ to $x_1$ is a continuous map $\gamma: I \to X$ such that $\gamma(0) = x_0$ (the initial point) and $\gamma(1) = x_1$ (the terminal point).
- If $\gamma(0) = \gamma(1) = x_0$, we say that $\gamma$ is a loop based at $x_0$. The point $x_0$ is called the basepoint.
To form an algebraic structure, we cannot simply compose arbitrary continuous curves directly; rather, we consider paths up to a deformation that leaves their endpoints fixed.
Definition 2 (Path Homotopy).
Let $\gamma_0, \gamma_1: I \to X$ be two paths in $X$ having the same initial point $x_0$ and the same terminal point $x_1$. A path homotopy between $\gamma_0$ and $\gamma_1$ is a continuous map $H: I \times I \to X$ satisfying:
- $H(s, 0) = \gamma_0(s)$ and $H(s, 1) = \gamma_1(s)$ for all $s \in I$,
- $H(0, t) = x_0$ for all $t \in I$ (initial endpoint remains fixed),
- $H(1, t) = x_1$ for all $t \in I$ (terminal endpoint remains fixed).
If such a map $H$ exists, we say that $\gamma_0$ and $\gamma_1$ are path homotopic, denoted $\gamma_0 \simeq_p \gamma_1$ (or simply $\gamma_0 \simeq \gamma_1$ when the context is clear).
Proposition 1 (Equivalence Relation).
Path homotopy $\simeq_p$ is an equivalence relation on the set of paths in $X$ with fixed initial point $x_0$ and terminal point $x_1$.
Proof
Reflexivity: For any path $\gamma$, the constant homotopy $H(s, t) = \gamma(s)$ is continuous, fixes endpoints, and shows $\gamma \simeq_p \gamma$.
Symmetry: If $H(s, t)$ is a path homotopy from $\gamma_0$ to $\gamma_1$, then $H'(s, t) = H(s, 1-t)$ is continuous, satisfies $H'(s, 0) = \gamma_1(s)$, $H'(s, 1) = \gamma_0(s)$, and leaves the endpoints fixed. Thus $\gamma_1 \simeq_p \gamma_0$.
Transitivity: If $H_1$ is a path homotopy from $\gamma_0$ to $\gamma_1$ and $H_2$ is a path homotopy from $\gamma_1$ to $\gamma_2$, define $H: I \times I \to X$ by: $$ H(s, t) = \begin{cases} H_1(s, 2t), & 0 \le t \le 1/2 \\ H_2(s, 2t-1), & 1/2 \le t \le 1 \end{cases} $$ By the Pasting Lemma, since $H_1(s, 1) = \gamma_1(s) = H_2(s, 0)$, $H$ is continuous, and it clearly preserves the endpoints for all $t$. Hence $\gamma_0 \simeq_p \gamma_2$.
The equivalence class of a path $\gamma$ under path homotopy is denoted by $[\gamma]$.
2. Path Composition and Reparametrization
Given two paths where the first ends where the second begins, we can traverse them in succession at twice the speed.
Definition 3 (Path Concatenation / Product).
Let $f, g: I \to X$ be paths such that $f(1) = g(0)$. The product path $f * g: I \to X$ is defined by:
Since $f(2 \cdot \frac{1}{2}) = f(1) = g(0) = g(2 \cdot \frac{1}{2} - 1)$, the Pasting Lemma guarantees that $f * g$ is continuous, and is a path from $f(0)$ to $g(1)$.
Lemma 1 (Well-Definedness on Homotopy Classes).
If $f_0 \simeq_p f_1$ and $g_0 \simeq_p g_1$ with $f_0(1) = g_0(0)$, then $f_0 * g_0 \simeq_p f_1 * g_1$.
Proof
Let $F: I \times I \to X$ be a path homotopy from $f_0$ to $f_1$, and let $G: I \times I \to X$ be a path homotopy from $g_0$ to $g_1$. Define $H: I \times I \to X$ by:
$$ H(s, t) = \begin{cases} F(2s, t), & 0 \le s \le 1/2 \\ G(2s-1, t), & 1/2 \le s \le 1 \end{cases} $$At $s = 1/2$, $F(1, t) = f_0(1) = g_0(0) = G(0, t)$ for all $t$. By the Pasting Lemma, $H$ is continuous. Furthermore, $H(0, t) = F(0, t) = f_0(0)$ and $H(1, t) = G(1, t) = g_0(1)$, so $H$ fixes both endpoints throughout. Hence $f_0 * g_0 \simeq_p f_1 * g_1$.
Consequently, we define the product on path homotopy classes by $[f] * [g] = [f * g]$.
3. The Group Structure of $\pi_1(X, x_0)$
Let $c_{x_0}: I \to X$ denote the constant path $c_{x_0}(s) = x_0$, and for any path $f$, let $\bar{f}(s) = f(1-s)$ denote the reverse path.
Theorem 1 (Fundamental Properties of Path Product).
The operation $*$ on path homotopy classes satisfies:
- Associativity: If $f * (g * h)$ is defined, then $([f] * [g]) * [h] = [f] * ([g] * [h])$.
- Identity: If $f$ is a path from $x_0$ to $x_1$, then $[c_{x_0}] * [f] = [f] = [f] * [c_{x_1}]$.
- Inverse: For any path $f$ from $x_0$ to $x_1$, $[f] * [\bar{f}] = [c_{x_0}]$ and $[\bar{f}] * [f] = [c_{x_1}]$.
Proof Details (Reparametrizations)
General Reparametrization Lemma: If $\phi: I \to I$ is a continuous map such that $\phi(0) = 0$ and $\phi(1) = 1$, then for any path $f$, $f \circ \phi \simeq_p f$. The homotopy is simply $H(s, t) = f((1-t)\phi(s) + ts)$.
Associativity: The paths $(f * g) * h$ and $f * (g * h)$ are reparametrizations of one another:
$$ ((f * g) * h)(s) = \begin{cases} f(4s), & 0 \le s \le 1/4 \\ g(4s-1), & 1/4 \le s \le 1/2 \\ h(2s-1), & 1/2 \le s \le 1 \end{cases} \quad\text{and}\quad (f * (g * h))(s) = \begin{cases} f(2s), & 0 \le s \le 1/2 \\ g(4s-2), & 1/2 \le s \le 3/4 \\ h(4s-3), & 3/4 \le s \le 1 \end{cases} $$The piecewise linear map taking $[0, 1/4] \to [0, 1/2]$, $[1/4, 1/2] \to [1/2, 3/4]$, and $[1/2, 1] \to [3/4, 1]$ gives the desired boundary-preserving homotopy.
Identity: The path $c_{x_0} * f$ stays at $x_0$ for $s \in [0, 1/2]$ and then traverses $f(2s-1)$. Contracting the idle interval $[0, 1/2]$ down to $0$ via a linear homotopy gives $c_{x_0} * f \simeq_p f$.
Inverse: For $f * \bar{f}$, we travel along $f$ and then retrace backwards. At stage $t \in [0, 1]$, define a path that traverses $f$ up to $f(1-t)$ and then immediately reverses back along $f$. At $t=0$, this is $f * \bar{f}$; at $t=1$, this is the constant path $c_{x_0}$.
Definition 4 (The Fundamental Group).
Let $X$ be a topological space and $x_0 \in X$. The set of path homotopy classes of loops based at $x_0$, equipped with the operation $[f][g] = [f * g]$, is a group called the fundamental group of $X$ with basepoint $x_0$, denoted:
4. Convex Spaces and Simply Connected Spaces
Example 1 (Convex Subsets of $\mathbf{R}^n$).
Let $C \subseteq \mathbf{R}^n$ be a convex set, and $x_0 \in C$. Any loop $f: I \to C$ based at $x_0$ is path homotopic to the constant loop $c_{x_0}$ via the straight-line homotopy:
Since $C$ is convex, $H(s, t) \in C$ for all $(s, t) \in I \times I$. Moreover, $H(0, t) = (1-t)x_0 + tx_0 = x_0$ and $H(1, t) = x_0$, so the basepoint remains fixed. Therefore, every loop is homotopic to the constant loop, which implies:
Definition 5 (Simply Connected Space).
A topological space $X$ is said to be simply connected if:
- $X$ is path-connected, and
- $\pi_1(X, x_0)$ is the trivial group for some (and hence any) basepoint $x_0 \in X$.