The Winding Number via Homotopy Theory

Polar coordinates arise naturally from the universal cover of the punctured plane.

The winding number of a closed curve $\gamma\colon\mathbb{S}^1\to\mathbb{R}^2$ about the origin is an integer $w(\gamma)$ assigned to the curve that, well, measures how many times it winds around the origin. In another note we considered one means of computing the winding number of a curve, using algebraic topology (specifically, homology and cohomology) to access information about the homotopy type of the space of maps of the circle into the punctured plane. This approach is limited to smooth curves as it requires complex analysis, and so for general topological problems we require a more general means of computing this number.

Another familiar definition of the winding number of a curve involves a choice of coordinates: if the closed curve $\gamma$ is represented as a $2\pi$-periodic function $\mathbb{R}\to\mathbb{R}^2$ missing $0$, then we define $w(\gamma)$ by writing $\gamma$ in polar coordinates $\gamma(t)=(r(t),\theta(t))$, and then computing

$w(\gamma):=\frac{\theta(2\pi)-\theta(0)}{2\pi}$

This definition raises its own host of questions: why is this invariant under perturbations of $\gamma$? Were in the world did polar coordinates come from? How important are the exact form of polar coordinates, or the $2\pi$ in the denominator to the overall picture? And finally - how can we possibly tell these two extremely different definitions give the same answers? Here again, the this simple-but-cryptic formula is powered by the engine of algebraic topology, finding a way to grab onto the slippery concept of “the curve $\gamma$ up to homotopy” with an explicit, computable invariant.

Topology of Maps $\mathbb{S}^1\to\mathbb{R}^2\smallsetminus \{0\}$

Topology is the art of blurring out the details of a problem that do not matter, while being careful not to blur so many details that the information you are interested in vanishes alongside them. In our case, we are interested in a map of the circle into the plane that misses the origin, so the set of objects we care about is the funtion space $C(\mathbb{S}^1,\mathbb{R}^2\smallsetminus 0)$ of continuous maps of the circle into the punctured plane.

The details we do not care about are things like speed at which $\gamma$ traverses its image, or even the or small continuous deformations of $\gamma$ (of course, with it still avoiding the origin). This type of imprecision is exactly captured by the notion of homotopy: two functions $f,g\colon X\to Y$ are called homotopic if there is a continuous map $H\colon I\times X\to Y$ where $H(0,-)=f$ and $H(1,-)=g$.

If two curves $\gamma, \eta\in C(\mathbb{S^1},\mathbb{R}^2\smallsetminus 0)$ are homotopic, then one can continuously wiggle $\gamma$ (via a homotopy $\gamma_t:=H(t,-)$) until it turns into $\eta$, all while avoiding the origin. We use this to define an equivalence relation on $C(\mathbb{S}^1,\mathbb{R}^2\smallsetminus 0)$ where $\gamma\sim\eta$ if and only if they are homotopic. Blurring our vision to disregard continuous deformation is then rigorously encoded by taking the quotient space of $C$ with respect to this equivalence relation. We denote this space by $[\mathbb{S}^1,\mathbb{R}^2\smallsetminus 0]=C(\mathbb{S}^1,\mathbb{R}^2\smallsetminus 0)/\sim$ and denote the homotopy class of a map $\gamma$ as $[\gamma]\in[\mathbb{S}^1,\mathbb{R}^2\smallsetminus 0]$.

This space represents a precise distillation of the information of a curve $\gamma$ in the punctured plane, up to continuous deformations, and so represents the abstract solution to our problem: all possible topological data of a curve $\gamma$ is encoded by its image $[\gamma]$ in this space. But how do we work with abstract data like $[\gamma]$? This is what algebraic topology is all about. To access this information we again turn to algebraic topology, and specifically to homotopy groups to provide a tool to convert this topological data to something concrete.

Homotopy Groups

The first natural set of algebraic invariants to consider are the homotopy groups: given a (pointed,connected) topological space $X$ we can for each $n$ consider the set of (pointed) homotopy classes of maps $[\mathbb{S}^n,X]$. For each $n\geq 1$ this set naturally carries the structure of a group ($[\mathbb{S}^0,X]$ is just a set), and for each $n \geq 2$ this group is abelian (thus the choice of basepoint has no effect). With their group structure, we denote these sets of homotopy classes as $\pi_n(X):=[\mathbb{S}^n,X]$ We denote the collection of all homotopy groups as $\pi_\bullet(X)$.

The important property of these algebraic gadgets is how they let us turn homotopy classes of functions into algebraic data. Given two spaces $X,Y$ and a map $f\colon X\to Y$, we may use this map to take any element $[\eta]$ of $\pi_n(X)$ and produce an element of $\pi_n(Y)$ via composition: define $f_\star([\eta])\in\pi_n(Y)$ as $f_\star([\eta])=[f\circ\eta]$

That is, elements of $f\in C(X,Y)$ induce group homomorphisms $f_\star\in \mathrm{Hom}(\pi_n(X),\pi_n(Y))$. It is directly visible from this definition that if $f\sim g$ are homotopic then $f_\star=g_\star$: thus, $\star$: thus this assignment factors through the quotient by homotopy classes of maps and we get a map $\star\colon [X,Y]\to \mathrm{Hom}(\pi_n(X),\pi_n(Y))$ As we can do this for each $n$, this actually results in an infinite sequence of group homomorphisms $\star\colon [X,Y]\to \mathrm{Hom}(\pi_\bullet(X),\pi_\bullet(Y))$

$$[f]\mapsto f_\star:\begin{pmatrix} \pi_0(X)\to\pi_0(Y)\\\ \pi_1(X)\to\pi_1(Y)\\\ \vdots\\\ \pi_n(X)\to\pi_n(Y)\\\ \vdots\\\ \end{pmatrix}$$

The natural question here of course is, is this any good? Sure we can use $[f]$ to produce this collection of maps, but do the resulting maps remember $[f]$?
Unfortunately this question is very hard (its false in general, but open in many interesting classes of spaces).
Nonetheless, in the situations we will be interested in (winding number and some generalizations) things can be worked out explicitly, and we will find $f_\star$ to be a complete invariant: that is, if $f_\star=g_\star$ then $[f]=[g]$.

The Homotopy Theory of $[\mathbb{S}^1,\mathbb{R}^2\smallsetminus 0]$

The homotopy groups of the circle are very simple: $\pi_1(\mathbb{S}^1)=\mathbb{Z}$ and for $n\geq 2$, $\pi_n(\mathbb{S}^1)=0$ (to prove this, recall that for $n\geq2$ homotopy groups are invariant under taking covers: and since the universal cover of $\mathbb{S}^1$ is contractible, all its higher homotopy vanishes). Since $\mathbb{R}^2\smallsetminus 0$ is homotopy equivalent to the circle, it has the same homotopy groups: $\mathbb{Z}$ in dimension $1$ and $0$ above that. Thus, the algebraic data carried by a homotopy class of closed curve $[\gamma]$ is the sequence of homomorphisms

$$[\gamma]\mapsto\gamma_\star\colon\begin{pmatrix} \pi_1(\mathbb{S}^1)\to \pi_1(\mathbb{R}^2\smallsetminus 0)\\\ 0\to 0\\\ \vdots\\\ 0\to 0\\\ \vdots \end{pmatrix}$$

This collection of maps is a complete homotopy invariant, so the entirety of the topological information carried by $[\gamma]$ is recorded by the maps $\gamma_\star$. But - all but one of these maps is the zero map between trivial groups! Thus, the entirety of the topological data of $[\gamma]$ is carried by $\gamma_\star\colon\pi_1(\mathbb{S}^1)\to \pi_1(\mathbb{R}^2\smallsetminus 0)$ Since each of these groups is abstractly isomorphic to $\mathbb{Z}$, and $\gamma_\star$ is a group homomorphism, after choosing identifications $\gamma_\star$ becomes an endomorphism of $\mathbb{Z}$: such maps are easily classified, every $\phi\in\mathrm{End}(\mathbb{Z})$ is of the form $\phi(x)=nx$ for some $n$. That is, the topological data of $[\gamma]$ is fully characterized by a single integer $n$.

Getting Access to this Integer

The only remaining problem is a means of actually computing this integer stored by $\pi_1$. While each of $\pi_1(\mathbb{S}^1)$, $\pi_1(\mathbb{R}^2\smallsetminus 0)$ are abstractly isomorphic to the integers their actual elements are defined as homotopy classes: which are exactly the kind of thing we are trying to avoid working with! So, we need to do some more algebraic topology to recover this data in a more user-friendly means. Since we are working only with $\pi_1$ here, the tool for the job is covering spaces: these let us replace slippery concepts like homotopy classes with concrete deck transformations: given a covering space $p\colon\widetilde{X}\to X$ the deck group of $p$ is the set of self-homeomorphisms of $\widetilde{X}$ which project to the identity under $p$: $\mathrm{deck}(\widetilde{X}\to X)=\{\phi\colon\widetilde{X}\to\widetilde{X}\mid p\phi=\mathrm{id}_X\}$

When $\widetilde{X}$ is the universal covering space of $X$, then we have a natural identification $\mathrm{deck}(\widetilde{X}\to X)\cong\pi_1(X)$ where a deck transformation $\phi$ with $\phi(x)=y$ is sent to the loop $p\gamma$ for $\gamma$ any path connecting $x$ to $y$ in $\widetilde{X}$.

To begin our computation, we first choose a universal cover of $\mathbb{R}^2\smallsetminus 0$ so that we can replace its homotopy group with the deck group. The simplest choice is better known as polar coordinates: $p\colon \mathbb{R}^2\to \mathbb{R}^2\smallsetminus 0$ $(r,\theta)\mapsto (r\cos\theta,r\sin\theta)$

But before we can put this to use, we have to overcome a slight difficulty: our loop $\gamma\colon\mathbb{S} ^1\to\mathbb{R}^2\smallsetminus 0$ does not necessarily lift to the universal cover, as $\gamma$ only satisfies the lifting criterion if it sends $\mathbb{S}^1$ to the trivial element of $\pi_1(\mathbb{R}^2\smallsetminus 0)$. (And, this only occurs if $\gamma$ does not wind around the origin!)

To fix this, we must also choose a universal cover of $\mathbb{S}^1$. Again, the simplest such choice is already well-known to us from calculus as a parameterization of the unit circle: $c\colon \mathbb{R}\to\mathbb{S}^1$ $t\mapsto (\cos t,\sin t)$ Now, we may lift the map $\gamma$ to a map $\widetilde{\gamma}$ between the universal covers (Such a lift is usually called ‘parameterizing the curve in polar coordinates’). $\widetilde{\gamma}\colon\mathbb{R}\to\mathbb{R}^2$ $t\mapsto (r(t),\theta(t))$

This map, together with the covers and the original $\gamma$ form a commuting square: draw this

Because $c(0)=c(2\pi)$ in the covering of the circle, $\gamma(c(0))=\gamma(c(2\pi))$, and thus by commutativity, $p(\widetilde{\gamma}(0))=p(\widetilde{\gamma}(2\pi))$. This means that $\widetilde{\gamma}(0)$ and $\widetilde{\gamma}(2\pi)$ both lie above the same point of $\mathbb{R}^2\smallsetminus 0$, and thus there is a some deck transformation $\phi\in\mathrm{deck}(\mathbb{R} ^2\to\mathbb{R}^2\smallsetminus 0)$ relating them.

As the deck transformations of $(r,\theta)\mapsto (r\cos\theta,r\sin\theta)$ are all maps of the form $(r,\theta)\mapsto (r,\theta +2\pi n)=(r,\theta)+2\pi(0,n)$ we see that there must be some $n$ such that $\widetilde{\gamma}(2\pi)=\phi(\widetilde{\gamma}(0))=\widetilde{\gamma}(0)+2\pi(0,n)$

This integer $n$ represents the deck transformation encoded by $\widetilde{\gamma}$, which in turn represents the induced homomorphism $\gamma_\star\colon\pi_1(\mathbb{S}^1)\to\pi_1(\mathbb{R}^2\smallsetminus 0)$, which itself represents the entirety of the topological data of the homotopy class $[\gamma]$ in $[\mathbb{S}^1,\mathbb{R} ^2\smallsetminus 0]$. “Solving for $n$”, we may recover this topological data directly from the lift:

$\frac{\widetilde{\gamma}(2\pi)-\widetilde{\gamma}(0)}{2\pi}=(0,n)$ Which, recalling that $\widetilde{\gamma}(t)=(r(t),\theta(t))$ implies $n=w(\gamma)=\frac{\theta(2\pi)-\theta(0)}{2\pi}$