Where we are: from homotopy of maps to loops with a basepoint
Guide 1 of this rung gave you homotopy of maps and homotopy equivalence: two maps are the same up to continuous deformation, two spaces are the same if maps go back and forth that are inverse up to homotopy, and a contractible space is one homotopy equivalent to a point. That is a beautiful equivalence relation — but on its own it gives no numbers, no algebra, nothing you can compute with. This guide manufactures the first genuine algebraic invariant out of exactly those raw materials, by restricting attention to a very special class of maps: loops.
An honesty note up front, the same posture as last guide: this is graduate algebraic topology. We assume you are comfortable with point-set topological spaces, continuity, homeomorphisms, path-connectedness, and basic group theory — words like quotient group, normal subgroup, and free group are used, not re-derived. Vol I's survey of covering spaces is the one prerequisite we will lean on hard; if that feels distant, skim it before the π₁(S¹) computation, where it does all the work.
Building the group: loops, concatenation, and why a basepoint is needed
Fix a space X and a chosen point x_0 in it, the basepoint. A loop at x_0 is a continuous path gamma: [0,1] -> X with gamma(0) = gamma(1) = x_0 — it leaves home and returns. Two loops are equivalent if there is a homotopy of loops between them: a continuous deformation that holds both endpoints pinned at x_0 the whole time (this is a homotopy rel endpoints, stricter than free homotopy). Picture a rubber band tacked to a pin at x_0, free to be slid and stretched anywhere in X but never unpinned. The equivalence classes are the elements of our group-to-be.
Now the group operation: concatenation. Given loops gamma and delta, run gamma at double speed on [0, 1/2], then delta at double speed on [1/2, 1]; call the result gamma followed by delta. The constant loop sitting at x_0 is the identity, and the reverse loop gamma-bar (same path traversed backwards) is the inverse — concatenating a loop with its reverse can be deformed, rel endpoints, back to the constant loop. None of these laws hold on the nose: gamma followed by (delta followed by epsilon) is not literally the same parametrized path as (gamma followed by delta) followed by epsilon. They hold only up to homotopy — which is exactly why we passed to equivalence classes first. On classes, associativity, identity, and inverses are all genuine.
The resulting group is the fundamental group pi_1(X, x_0). The basepoint is not decoration: concatenation only makes sense when the second loop starts where the first ends, which forces every loop to share one home. The reassuring fact is that for a path-connected X the choice barely matters — a path from x_0 to x_1 induces an isomorphism pi_1(X, x_0) ≅ pi_1(X, x_1) by conjugation. So we usually drop the basepoint and write pi_1(X), keeping in mind the isomorphism is non-canonical (it depends on the chosen path up to homotopy), which is why basepoints never fully vanish from the theory.
First sanity checks: functoriality, and what makes a space simply connected
Two facts make pi_1 useful rather than merely defined. First, it is functorial: a continuous basepoint-preserving map f: X -> Y sends a loop at x_0 to a loop at f(x_0), and respects concatenation up to homotopy, so it induces a group homomorphism f_*: pi_1(X) -> pi_1(Y); identity maps go to identity homomorphisms and composites to composites. Second, a homotopy equivalence induces an isomorphism on pi_1. That second fact is the whole point: pi_1 cannot tell apart spaces that are homotopy equivalent, so it is an invariant of homotopy type, not just of homeomorphism type. A disk and a point have the same pi_1 (trivial); they are not homeomorphic but they are homotopy equivalent.
A space with trivial fundamental group is called simply connected: every loop contracts to the basepoint. Any contractible space is simply connected — R^n, any convex set, the disk D^2 all have pi_1 = 0. But the converse fails, and this is a classic trap to flag now: the sphere S^2 is simply connected (any loop on a globe slides off to a point) yet it is not contractible. Simple connectivity is only the vanishing of pi_1; contractibility demands all higher homotopy vanish too. Do not conflate the two — S^2 is the standard counterexample.
The signature computation: π₁(S¹) = ℤ via the covering R -> S¹
Now the theorem that justifies the whole apparatus: pi_1(S¹) is isomorphic to the integers ℤ. The intuition is the winding number — a loop on the circle wraps around some net number of times, counterclockwise counting positive — and the claim is that this integer is a complete homotopy invariant of the loop, and that concatenation adds the winding numbers. To make this rigorous we use the covering map p: R -> S¹ given by p(t) = (cos 2 pi t, sin 2 pi t), which wraps the real line around the circle like an infinite helix projected down, with the integers all sitting above the basepoint 1.
The engine is two lifting properties of any covering space, which the next guide will state in full generality but which we use here for this one map. Path lifting: given a path in S¹ starting at the basepoint and a chosen point upstairs over it, there is a unique path in R that starts at that point and projects down to the given one. Homotopy lifting: a homotopy of paths downstairs lifts uniquely to a homotopy upstairs, once the starting lift is fixed. R is simply connected (it is contractible), and that is exactly the leverage — upstairs there is no room for loops to be subtle, so all the information collapses to the endpoint of the lift.
- Take any loop gamma in S¹ based at 1. Lift it to a unique path gamma-tilde in R starting at 0. Since gamma ends at the basepoint, gamma-tilde must end at some point lying over 1 — that is, at an integer n. Define the degree of gamma to be that integer n.
- Show degree is well-defined on homotopy classes. If two loops are homotopic rel basepoint, lift the homotopy; its endpoint stays over 1 throughout, and an integer-valued continuous function of the deformation parameter is constant. So homotopic loops have equal degree, giving a well-defined map pi_1(S¹) -> ℤ.
- Show it is a homomorphism. The lift of a concatenation gamma followed by delta is the lift of gamma followed by a shifted copy of the lift of delta; the endpoints add, so degree(gamma · delta) = degree(gamma) + degree(delta).
- Show it is a bijection. Surjective: the loop omega_n(t) = p(nt) winds n times and has degree n, hitting every integer. Injective: if degree is 0, the lift is a loop in R from 0 to 0; since R is contractible that lift contracts rel endpoints, and projecting down contracts gamma — so the kernel is trivial.
That is the complete proof, and it is worth pausing on what powered it: the entire computation rode on R being simply connected. The covering R -> S¹ is in fact the universal cover of the circle, and the appearance of ℤ is no accident — ℤ is the group of deck transformations of this cover, the integer translations t -> t + n that permute the fiber over the basepoint. This is the first whisper of the next guide's central theme: pi_1 of the base equals the deck group of its universal cover. Hold that thought.
Reading the answer: degree, applications, and honest limits
Why does this one computation matter so much? Because it is the seed crystal everything else grows from. Once you know pi_1(S¹) = ℤ, the no-retraction lemma follows (there is no continuous retraction of the disk onto its boundary circle, since it would force the identity ℤ -> ℤ to factor through pi_1(D^2) = 0), and from that the Brouwer fixed-point theorem in dimension two falls out immediately. The fundamental theorem of algebra also yields to a degree argument on a large circle. A single computed group powers a chain of theorems that look, at first glance, to have nothing to do with loops.
A few more computations now come cheaply, each by the same homotopy-equivalence or covering logic. The annulus, the Möbius band, and the punctured plane all deformation-retract onto a circle, so each has pi_1 = ℤ. The n-torus has pi_1 = ℤ^n, computed via its universal cover R^n -> T^n. The figure-eight, as noted, has the free group F_2 — that one needs the Seifert-van Kampen theorem of guide 4, the tool that computes pi_1 of a space glued from pieces with known fundamental groups. Each of these is one of the small, concrete examples worth more than a general theorem.
Two convention warnings before you read further. First, sign: counterclockwise is positive degree here, but a minority of texts orient the other way, flipping the isomorphism pi_1(S¹) ≅ ℤ by a sign — harmless, but check before comparing formulas. Second, basepoint bookkeeping varies: some authors suppress x_0 entirely and quietly assume path-connectedness, others carry it throughout for the pointed-space machinery of fibrations. Pick a source, note its conventions, and stay loyal; most apparent contradictions between books on this rung are conventions colliding, not mathematics disagreeing.