THE DEGREE OF A MAP how many times the map wraps the circle around itself — an integer that cannot jump
The degree of a continuous map from a circle (or sphere) to itself counts, with sign, how many times it wraps the source around the target. The map z → z² on the unit circle has degree 2 (go around once, the image goes around twice); z → zⁿ has degree n. The degree is a homotopy invariant: you can deform the map continuously any way you like — the degree, being an integer, cannot change, because it has nowhere continuous to go. This integer-that-cannot-jump is the engine behind fixed-point theorems, the fundamental theorem of algebra, and the fact that you cannot comb a hairy ball.
THE TECHNIQUE degree of z→zⁿ is n; a generic point has n preimages
The map z → z² and its degree, read as the number of preimages of a generic point. The demo counts them: live demo
HISTORY & CREDIT Brouwer 1911 (Kronecker index)
“A small wiggle of the map can nudge the winding a little.” — the degree is an integer, so a continuous deformation cannot nudge it at all: it stays put until you tear the map. Discreteness is what makes it invariant. cited
1869 · Leopold Kronecker — the Kronecker index, a forerunner of the degree. 1911 · L. E. J. Brouwer — the mapping degree, a homotopy invariant; the basis of his fixed-point theorem. now · degree theory — existence of solutions, the fundamental theorem of algebra, the hairy-ball theorem.
An integer cannot vary continuously, so a continuous deformation cannot change the degree. That single fact — discreteness forcing invariance — is why so many existence theorems follow from it. Brouwer 1911
RECOMMEND FOR I-13 degree of z→zⁿ, computed
On the canonical compiler, z → z² has degree 2 — a generic target value has two preimages:
$ i13 run inv_degree.i13 # degree of z -> z^n
degree = 2 -- z^2 wraps the circle twice
Recommend: the degree is the integer-that-cannot-jump, and its invariance is exactly the discreteness i13 is comfortable with. i13 computes the degree of z→zⁿ as n (the preimage count of a generic point), and the deep fact — that no continuous deformation of the map can change it — is guaranteed by the same integrality i13's f64 represents exactly for small n. It is a homotopy invariant carried as a plain integer, kin to the genus (dart 282): topology's habit of pinning a continuous world to a number that refuses to move.