Compute the fundamental group of the circle

Try

A loop on the circle may travel around several times, reverse direction or pause. A continuous lifted angle on the real line records all of this without a discontinuous jump at a chosen branch cut. Starting the lift at zero, its endpoint must be an integer. That integer will classify the loop completely.

Predict

Predict whether a finite list of sampled circle points determines the winding number. A loop can insert extra full turns between samples, so samples alone do not certify it. In the live experiment, the integer winding is part of an exact supplied parameterization. The display illustrates lifting; the proof below applies to every continuous based loop.

Experiment

Live experiment: a closed loop can have an open lift

The covering map is p(z)=zᵈ on the unit circle. The base loop is α(t)=exp(2πiwt), where w is its signed number of turns and 0≤t≤1. Start the lift at sheet j, the point exp(2πij/d). Predict its ending sheet before stepping to t=1.

Cover degree d12345Base winding w-3-2-10123Starting sheet j012Path time t00.250.50.751Reset lift

Both circles use the same fixed scale. Base point: (1, 0); lifted point: (1, 0). The points alone do not record completed turns: base angle is 0 turns and lifted angle is 0 turns.

At t=1 the lift ends at sheet 1. It is not closed because 3 does not divide 1.

The exact lift is α̃(t)=exp(2πi(j+wt)/d). Raising it to d gives exp(2πij)exp(2πiwt)=α(t), and its initial value is the prescribed sheet. Uniqueness follows from the covering path-lifting theorem. A loop returns to its starting sheet exactly when w is a multiple of d; this is the subgroup dℤ in π₁(S¹)=ℤ.

This finite selector samples an explicitly defined continuous path. Winding is supplied by the formula, not inferred from five samples: different loops can pass through the same sampled points. Changing d resets j and t so the starting sheet remains valid. Settings are temporary and do not award mastery.

Use the live covering experiment to change the number of sheets and the supplied winding, then predict whether the lifted path closes before inspecting its endpoint. The static example below is separate from the widget settings.

For α(t)=exp(2πi(4t−t²)), its real lift beginning at zero is 4t−t².

t	Lifted angle in turns
0	0
1/2	7/4
1	3

The projected path is a loop because its final angle is an integer. Its class will be three, although its speed is not constant.

Understand

Let p:R→S¹ be p(u)=exp(2πiu). Every loop α based at 1 has a unique lift α̃ starting at zero. Since p(α̃(1))=1, the endpoint α̃(1) is an integer. Define W([α]) to be this integer.

Homotopy lifting makes W well defined: an endpoint homotopy of circle loops lifts to a square whose right edge stays in the integer fiber. That edge is constant, so its integer endpoint cannot change during the homotopy.

Every integer n occurs, represented by α_n(t)=exp(2πint). To prove completeness, take two loops whose lifts start at zero and end at the same integer n. Their real lifts are endpoint homotopic by straight-line interpolation: (1−s)α̃(t)+sβ̃(t). At t=0 this is zero, and at t=1 it is n. Projecting gives an endpoint homotopy on the circle. Thus equal integers imply equal loop classes; unequal integers cannot be homotopic by invariance.

For multiplication, lift α first. If its endpoint is n, lift β from n by adding n to β’s zero-start lift. The concatenated lift ends at n+W([β]). Therefore W([α][β])=W([α])+W([β]). This proves π₁(S¹,1) is isomorphic to the additive integers, not merely that an integer can be assigned to some selected examples.

A constant loop has integer zero, a once-around counterclockwise loop has one, and reversing a loop negates the integer. These signs use the chosen exponential orientation. The real line’s convexity supplies the crucial injectivity step: paths with common lifted endpoints can be deformed upstairs without obstruction.

Apply

The punctured plane strongly deformation retracts onto the circle, so its fundamental group is also Z. A loop that winds once around the missing point cannot contract while staying in that plane. This now proves the nonexistence that a failed straight-line interpolation alone could not establish. For several missing points, a single total winding number will no longer capture every loop relation.

Continue

The classification rests on three ingredients: existence of lifts, fixed lifted endpoints under homotopy, and straight-line deformations in R. It is independent of numerical sampling. Next, use the same lifted-angle mechanism to define degree for arbitrary circle maps and derive an obstruction to retracting a disk onto its boundary.

Continue to Winding numbers and circle degree.