By John Roe
The winding quantity is among the most elementary invariants in topology. It measures the variety of occasions a relocating aspect $P$ is going round a hard and fast element $Q$, only if $P$ travels on a course that by no means is going via $Q$ and that the ultimate place of $P$ is equal to its beginning place. this easy inspiration has far-reaching functions. The reader of this e-book will learn the way the winding quantity will help us convey that each polynomial equation has a root (the primary theorem of algebra), warrantly a good department of 3 items in house by means of a unmarried planar reduce (the ham sandwich theorem), clarify why each uncomplicated closed curve has an within and an outdoor (the Jordan curve theorem), relate calculus to curvature and the singularities of vector fields (the Hopf index theorem), permit one to subtract infinity from infinity and get a finite solution (Toeplitz operators), generalize to offer a basic and lovely perception into the topology of matrix teams (the Bott periodicity theorem). a lot of these matters and extra are constructed beginning in basic terms from arithmetic that's universal in final-year undergraduate classes. This publication is released in cooperation with arithmetic complicated learn Semesters.
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Extra resources for Winding Around: The Winding Number in Topology, Geometry, and Analysis
Sample text
O, Uj, ... ,vn E C. , it is the map t *-)> (nt — k)i'k+i + (k + 1 — nt)vk for k/n ≤ f ≤ (fc 4 1 )/??. ) It is a polygonal loop if VQ = vn . We*11 denote a polygonal path (or loop) by (wo, . . , vn). Suppose we have a polygonal loop (UQ, . . , vn) that does not pass through a point p. Choose a ray R from p to oo that is not parallel to any of the edges of the given loop and floes not pass through any of its vertices (we'll call this a. transverse ray with respect to the loop). For each edge e* = (u*, u*+i) of t he loop we can define an intersection number if e*.
The inequality shows that for each w = f{)(x)/ f\(x). ' — 1| < |UJ| -b 1. It is not hard to see that w 6 C satisfies this inequality iff w £ R“. Thus fo/fi never takes negative real values and the result follows from the previous lemma. 5. Let X he a compart metric space and /y, f\ maps from X to C \ {()}. The maps /y and f\ air homotopic if and only if f\j /(> is an exponential. Proof. 2). r) = fo(x) aiul h(lyx) = fi(x). ,r)| > e for all s £ [0, 1] and x £ X. 19) of hy there is > 0 such that |h(syx) — h(s\x)\ < e whenever s, s' £ [0, 1] with — \s s'\ < S.
3. 3. Computing (lie winding number of a polygonal loop. 5. The winding number of a polygonal loop around p is equal to the sum of its edge- intersection numbers with a transverse ray: i({vk. vk+1), R), wn((«o,.. Vn),p) ■ . k whew R is a transverse ray from p to infinity. ) Proof. , from the unbounded component inward to p, keeping track of the changes in the winding number. There is one small issue to deal with: our definition of a ’’polygonal loop" allows for different edges to overlap (for instance a polygonal loop with 6 edges that goes around the same triangle twice).