An Introduction to the Fundamental Group

Hold a rubber band against a flat tabletop and slide it around. No matter how you stretch or twist it, you can always shrink it down to a single point — the table has nothing to catch on. Now stand a coffee mug on the table and loop the band around its handle. Suddenly you are stuck: to shrink the loop to a point you would have to pull the rubber through the handle, and the handle is in the way. The band has discovered a hole.

This is the whole idea of algebraic topology in one gesture. Point-set topology — open sets, continuity, homeomorphism, connectedness — gives us a precise language for "shape", but it is hard to compute with. How do you prove, rather than merely feel, that a disc and a doughnut are genuinely different shapes? The trick, due to Poincaré, is to attach to every space an algebraic gadget — a group — that records which loops can be contracted and which get snagged on holes. Homeomorphic spaces get isomorphic groups, so if the groups differ, the spaces cannot possibly be the same. Topology becomes algebra.

That gadget is the fundamental group, written \pi_1(X, x_0). This page builds it from scratch: loops, homotopy, and the group law — then the flagship fact that the circle's fundamental group is the integers, and the payoff of using it as an invariant. It is the gateway from point-set topology into the algebraic world.

Step 1: loops based at a point

Fix a space X and a distinguished point x_0 \in X, the basepoint. A loop based at x_0 is a continuous map

\gamma : [0, 1] \to X \quad \text{with} \quad \gamma(0) = \gamma(1) = x_0.

Read [0,1] as a stopwatch: at time 0 you are standing at x_0, you wander continuously through X, and at time 1 you are back home at x_0. The map being continuous is exactly the promise that you never teleport — the path is drawn "without lifting the pen". A loop is a round trip.

The very simplest loop is the constant loop c(t) = x_0 for all t — the trip where you never leave home. Keep it in mind; it will turn out to be the identity element.

Step 2: homotopy — deforming one loop into another

Two loops \gamma and \delta based at the same point are homotopic (rel endpoints) if one can be continuously deformed into the other while keeping both ends pinned at x_0 the entire time. Formally, there is a continuous homotopy

H : [0,1] \times [0,1] \to X, \qquad H(s, 0) = \gamma(s), \quad H(s, 1) = \delta(s),

with H(0, u) = H(1, u) = x_0 for every u. The second coordinate u is a "morphing dial": at u = 0 you see \gamma, at u = 1 you see \delta, and in between a continuous movie of intermediate loops, each still a genuine loop at x_0. The slogan: no scissors, no gluing — only stretching and sliding. You may not tear the loop or lift it off the space; you may only deform it.

Homotopy is an equivalence relation on loops (it is reflexive, symmetric — run the movie backwards — and transitive — splice two movies together). So the loops at x_0 fall into homotopy classes: two loops share a class exactly when you can wobble one onto the other. We write [\gamma] for the class of \gamma. From now on we mostly stop caring about individual loops and care about their classes — the class is the thing that "remembers a hole".

A loop homotopic to the constant loop is called null-homotopic: it can be reeled all the way in to x_0. In the picture below, the little loop is null-homotopic; the loop around the hole is not.

Step 3: concatenation, and the birth of a group

Now we make the classes multiply. Given two loops \gamma and \delta based at x_0, their concatenation \gamma \cdot \delta is the loop "do \gamma first, then \delta" — run each at double speed so the round trip still fits in one unit of time:

(\gamma \cdot \delta)(t) = \begin{cases} \gamma(2t), & 0 \le t \le \tfrac{1}{2}, \\ \delta(2t - 1), & \tfrac{1}{2} \le t \le 1. \end{cases}

Because both loops pass through x_0 at the handover time t = \tfrac{1}{2}, the join is continuous — it is again a loop at x_0. This descends to classes: define [\gamma]\,[\delta] = [\gamma \cdot \delta] (one checks the result does not depend on which representatives you pick). The set of homotopy classes of loops at x_0, with concatenation as the operation, is a group — this is the group \pi_1(X, x_0), the fundamental group. Let us check the four axioms, each holding up to homotopy:

Closure: the concatenation of two loops at x_0 is again a loop at x_0.
Identity: the constant loop c is the identity — \gamma \cdot c and c \cdot \gamma are both homotopic to \gamma (waiting at home for the second half of the trip can be reparametrised away).
Inverses: the reverse loop \bar{\gamma}(t) = \gamma(1 - t) — the same journey walked backwards — satisfies \gamma \cdot \bar{\gamma} \simeq c: going out and retracing your steps contracts to standing still.
Associativity: (\gamma \cdot \delta) \cdot \varepsilon \simeq \gamma \cdot (\delta \cdot \varepsilon). The two only differ in the timing of the handovers, and a homotopy simply reparametrises time.

Notice how essential homotopy is: on the nose, \gamma \cdot c \ne \gamma (they have different speed schedules), and concatenation is not literally associative either. Everything works only after we pass to homotopy classes — which is precisely why we built classes first. The group lives on the [\,\cdot\,], not on the raw loops.

Yes. The 1 counts the dimension of the thing you are throwing at the space. \pi_1 maps in the 1-dimensional circle (a loop); \pi_2(X, x_0) maps in the 2-sphere and detects two-dimensional "voids"; in general \pi_n uses the n-sphere. There is even a \pi_0, the set of path-components — the 0-dimensional probe just asks "can you walk from here to there?", which is why \pi_0 is trivial exactly when the space is path-connected. The whole family is called the homotopy groups. They form a tower of ever-subtler invariants; \pi_1 is the first, most computable, and most intuitive rung, and unlike its higher cousins it need not be abelian.

Step 4: simply connected spaces — when every loop contracts

The dullest possible fundamental group is the trivial group \{e\}, containing only the identity. When \pi_1(X, x_0) = \{e\} it means every loop is null-homotopic: there are no holes to snag on, and any loop reels all the way in to a point. A space that is path-connected and has trivial fundamental group is called simply connected.

The plane \mathbb{R}^2 (indeed any \mathbb{R}^n) is simply connected: slide any loop straight home along the radial lines, H(s, u) = (1 - u)\,\gamma(s) + u\,x_0. This "straight-line homotopy" works in any convex set — a disc, a ball, a cube — because the segment from each loop point to x_0 stays inside.
The sphere S^2 (the surface of a ball) is simply connected too, which surprises people. A loop on the globe, even one that circles the equator, can be slid up and over a pole and squeezed to a point — there is no puncture to catch it. A sphere has an enclosed cavity, but that is a \pi_2 phenomenon, not a \pi_1 one.

The opposite is the circle S^1, the annulus, and the punctured plane \mathbb{R}^2 \setminus \{0\}: each has a hole, and a loop that encircles it is stubbornly non-contractible. Those are the interesting cases — so let us compute one.

The fundamental group is intuitive, which makes it easy to over-trust the intuition. Three snares:

"π₁ depends on the basepoint, so it is not really an invariant of the space." For a path-connected space it does not — moving the basepoint from x_0 to x_1 along a path h gives an isomorphism \pi_1(X, x_0) \cong \pi_1(X, x_1) (conjugate every loop by h). So people write \pi_1(X) and drop the basepoint. It only genuinely matters when the space is disconnected — you cannot walk between the pieces.
"Connected means simply connected." No. The annulus is (path-)connected — you can walk between any two points — yet it is not simply connected, because a loop around its hole will not contract. "Connected" is about walking; "simply connected" adds that every loop also contracts. Two different questions.
"The sphere S^2 has a hole, so its π₁ is nontrivial." The 2-sphere is simply connected: \pi_1(S^2) = \{e\}. Its "hole" is a hollow interior, invisible to loops, which slide over the surface and contract. Do not confuse the hole a circle has (a missing point, which traps loops) with the cavity a sphere encloses (which traps a \pi_2 membrane, not a loop).

The flagship computation: π₁(S¹) ≅ ℤ

Take the circle S^1 with basepoint at angle 0. Every loop, followed around, sweeps out a net number of full turns — clockwise counts negative, counter-clockwise positive. This integer is the winding number (or degree) of the loop. The central theorem of the subject says the winding number is a complete invariant of a loop up to homotopy:

\pi_1(S^1, 1) \cong \mathbb{Z}, the additive group of integers.
The isomorphism sends the class [\gamma] to its winding number n \in \mathbb{Z}.
Concatenating loops adds their winding numbers: wind m times, then n times, and the result winds m + n times.

Why can't the once-around loop be undone? Intuitively, to unwind it you would have to slide part of the loop across the missing centre of the circle — and there is nothing there to slide across. The hole is a topological wall. The rigorous proof lifts each loop to the real line via the "wrapping" map p(t) = (\cos 2\pi t, \sin 2\pi t): a loop upstairs starts at 0 and ends at some integer n, and that integer — which cannot jump continuously — is the winding number. Because winding numbers add under concatenation, the map \pi_1(S^1) \to \mathbb{Z} is a group isomorphism.

This one computation is the engine of the whole theory. It is the first genuinely non-trivial fundamental group, and — as we will see — it single-handedly separates the disc from the annulus and powers two famous theorems.

The Fundamental Theorem of Algebra. Every non-constant complex polynomial has a root. Sketch: if p(z) had no root, then as z travels a large circle, p(z) traces a loop whose winding number is the degree d > 0; but shrinking the circle to a point continuously deforms that loop to a constant (winding 0). An integer cannot equal both d and 0 — contradiction. Roots must exist.

Brouwer's Fixed-Point Theorem (2D). Every continuous map of the disc to itself has a fixed point. If some map moved every point, you could push each point away from its image out to the boundary, giving a continuous retraction of the disc onto its boundary circle. But that would force \pi_1(D^2) = \{e\} to map onto \pi_1(S^1) = \mathbb{Z} compatibly with the identity on the circle — a group homomorphism from the trivial group cannot hit all of \mathbb{Z}. Impossible, so a fixed point exists. Algebra forbidding a geometric move.

π₁ is a topological invariant — the payoff

Here is why all this machinery earns its keep. A homeomorphism f : X \to Y carries loops to loops and homotopies to homotopies, so it induces an isomorphism of groups:

X \cong Y \quad \Longrightarrow \quad \pi_1(X, x_0) \cong \pi_1\big(Y, f(x_0)\big).

Contrapositive — and this is the whole point — if the fundamental groups differ, the spaces cannot be homeomorphic. A hard geometric non-existence ("there is no continuous invertible map between them") reduces to an easy algebraic inequality ("these two groups are not isomorphic").

The disc is not the annulus. The disc D^2 is convex, so \pi_1(D^2) = \{e\}; the annulus deformation-retracts onto its core circle, so \pi_1(\text{annulus}) \cong \mathbb{Z}. Since \{e\} \not\cong \mathbb{Z}, no homeomorphism can exist. You felt this with the rubber band at the top of the page; now it is a theorem.
A line is not a circle, a plane is not a punctured plane, and so on — each is a \pi_1 difference caught red-handed.
The torus has \pi_1 \cong \mathbb{Z} \times \mathbb{Z}. A doughnut surface has two independent kinds of loop — one going through the hole and one going around the tube — and neither can be deformed into the other or contracted. Each contributes an independent copy of \mathbb{Z} counting how many times you wrap that way, and (because you can slide the two loops past each other) the two commute. So a torus is provably different from a sphere (\{e\}) and from a two-holed surface (a non-abelian group).

That is the arc of this page: loops become classes, classes become a group, and the group becomes a fingerprint of shape that algebra can read off. From here the subject opens up — covering spaces, the van Kampen theorem for gluing spaces together, and the higher homotopy groups — but every one of them stands on the little rubber band you just learned to shrink.