Winding numbers and circle degree

Try

A circle map can stretch and reverse angular motion without moving points at constant speed. For f(exp(2πit))=exp(2πi(2t+(1/4)sin(2πt))), the real lifted angle changes by two turns between t=0 and t=1. The oscillating term changes intermediate behavior but not the final displacement. The degree records that integer.

Predict

Predict whether f must fix the circle point 1 to have a degree. It need not. A rotation has degree one even though its image of 1 changes. Predict whether a free homotopy can change degree by moving this basepoint image. Both ends of the lifted parameter interval may move, but their integer difference cannot vary continuously.

Experiment

Compare lifted-angle formulas on one turn.

Formula θ(t)	Endpoint displacement	Degree
t+1/3	1	1
−2t	−2	−2
(1/4)sin(2πt)	0	0

For each formula verify that the endpoint images agree after exponentiation, so it defines a circle map. A real formula whose endpoint difference is not an integer would fail this descent condition.

Understand

For a continuous f:S¹→S¹, lift the path t↦f(exp(2πit)) to a real θ(t). Its endpoint difference d=θ(1)−θ(0) is an integer because the endpoint images agree. Changing the lift adds a constant integer by path uniqueness, so d is well defined. This is degree f. For a based map it is the integer multiplying the generator under f_*.

Degree is invariant under unrestricted homotopy. Lift the associated parameter square. Its right and left boundary lifts project to the same point at each deformation time, so their difference is an integer. This difference is continuous, hence constant on the interval. Unlike endpoint homotopy, neither boundary value must itself remain fixed.

Equal degree is also sufficient for homotopy. If θ and ψ are real lifts with the same endpoint displacement d, interpolate them linearly. For every s, the resulting endpoints still differ by d, so exponentiation identifies the two ends of the parameter interval. The continuous map on the square therefore descends through its quotient to a homotopy on S¹×[0,1]. The quotient identification is legitimate because the compact interval-to-circle quotient, multiplied by a compact interval, is a quotient map. This proves the full classification of circle maps by degree.

For composition, induced homomorphisms on π₁ compose, and basepoint changes act by conjugation. Since the circle group is abelian, those conjugations do nothing. Thus degree(g∘f)=degree(g)degree(f). Directly, an extended real lift satisfies θ(t+1)=θ(t)+d, so passing through a second lifted map multiplies the integer changes.

For a loop γ avoiding a point a in the plane, normalize it to (γ−a)/||γ−a|| on the circle. Its degree is the winding number about a. This definition works for continuous loops without derivatives; the familiar integral formula needs additional regularity and a separate derivation.

Apply

A circle-valued map on the boundary of a disk can extend across the disk only if its degree is zero: radial contraction inside the disk makes its boundary map null homotopic. Conversely, the equal-degree proof supplies a homotopy to a constant, which fills the disk by collapsing the constant end to the center. Explain why the identity boundary map fails the necessary condition.

Continue

Degree is a homotopy invariant of maps, while winding is its application to a loop around a missing point. Neither is a count of visible crossings in an arbitrary sampled plot. Next, degree and functoriality rule out a disk retraction and force a fixed point for every continuous self-map of the closed disk.

Continue to No retraction and the disk fixed-point theorem.