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Is there a natural geometric generalization of the winding number to higher dimensions?

I know it primarily as an important and useful index for closed, plane curves (e.g., the Jordan Curve Theorem), and for its role in Cauchy's theorem integrating holomorphic functions. I would be interested to learn of generalizations that essentially replace the role of the circle $\mathbb{S}^1$ with $\mathbb{S}^n$.

I've encountered references to the Fredholm index, the Pontryagin index, and to Bott periodicity, but none seem to be straightforward geometric generalizations of winding number.

This is an entirely naive question, and references and high-level descriptions would be appreciated, and more than suffice.

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    $\begingroup$ For maps $S^n\rightarrow S^n$, or more generally between closed oriented manifolds of the same dimension, there is the notion of the degree. $\endgroup$ – Thomas Rot Jan 8 '17 at 1:44
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    $\begingroup$ The top homology group of a closed oriented manifold is canonically isomorphic to $\mathbb Z$. The induced map on the top homology groups can then be identified with the multiplication with a number, which is the degree. For non-oriented manifolds there is a notion of degree modulo two. Any algebraic topology book ought to explain this, it is for sure in Hatcher's book $\endgroup$ – Thomas Rot Jan 8 '17 at 1:49
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    $\begingroup$ @ThomasRot You don't even need oriented, orientable is enough (since the endomorphism ring of a free abelian group of rank 1 is still canonically $\mathbb{Z}$). $\endgroup$ – Denis Nardin Jan 8 '17 at 11:20
  • $\begingroup$ @denis Nardin: I don't think I understand that. $\endgroup$ – Thomas Rot Jan 8 '17 at 12:34
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    $\begingroup$ @ThomasRot I probably was just channeling my inner pedant. If $M$ is an orientable $n$-manifold an orientation is the same thing as an isomorphism $H_n(M)\cong \mathbb{Z}$. But even without choosing an orientation there is a canonical isomorphism $End(H_n(M))\cong \mathbb{Z}$, so the degree of $f:M\to M$ does not depend on the orientation. $\endgroup$ – Denis Nardin Jan 8 '17 at 12:39
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This is a very naive answer which I am sure you already considered, but isn't the most obvious generalization just given by the topological degree (https://en.wikipedia.org/wiki/Degree_of_a_continuous_mapping)?

The winding number of $f:S^1\rightarrow \mathbb{R}^2$ around $p$ is just the degree of the composition of $f$ with the radial projection from $p$, considered as a map from $S^1$ to $S^1$. It is obvious how to do the same thing for general $n$.

(This should surely just be a comment.)

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A smooth function $f$ with image on the unit circle in $\mathbb{C}$ has winding number: $$\text{wind} f=\frac1{2\pi i}\int f'\bar{f}=\frac1{2\pi i}\sum\hat{f'}(n)\hat{f}(n)=\sum n\vert\hat{f}(n)\vert^2.$$ This formulation allows generalizing the winding number to higher dimensions (eg. $f:\mathbb{S}^n\rightarrow\mathbb{S}^n$) for functions in Sobolev spaces $H^{1/2}$. Notice the natural fit of this space via the characterization of $f\in H^{1/2}$ through its Fourier coefficients: $$\sum \vert n\vert\,\vert\hat{f}(n)\vert^2<\infty.$$ The first result in this direction was due to L. Boutet de Monvel and O. Gabber. Since then the concept was extended as "degree of a map" (obviously the Topologist were aware of topological degree) much further to VMO spaces, etc. A good survey of results and developements on the subject can be found here:

Ha?m Brezis, New questions related to the topological degree, The unity of mathematics, 137–154, Progr. Math. 244, Birkh?user Boston, Boston, MA, 2006

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The linking number is one of a natural generalizations of the winding number, see my answer to a related question: http://www.2874565.com/a/297440/121665

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Perhaps it is worth mentioning that in their book "Mapping Degree Theory" Outerelo and Ruiz give a general definition of the winding number based on the topological (Brouwer-Kronecker) degree, which was already mentioned in a previous answer. The point being that, while it is a special case of the mapping degree, it is a generalization of the usual winding number that is interesting on its own.

Theorem-Definition: Let $U\subseteq\mathbb{R}^n$ be bounded and open, let $X:=\partial U$ be the boundary hypersurface, let $f\in C^0(X,\mathbb{R}^n)$, let $\bar{f}:\bar{U}\to\mathbb{R}^n$ be a continuous extension of $f$ (the set of such extensions is non-empty by Tietze's extension theorem), and let $p\in\mathbb{R}^n\setminus f(X)$. Then $\deg(\bar{f},U,a)$ does not depend on the choice of continuous extension $\bar{f}$, and one defines $$ w(f,a):=\deg(\bar{f},U,a) $$

The winding number inherits many of the nice properties of the topological degree : being locally constant on the codomain, homotopy invariance, being $0$ for points outside of $\bar{f}(\bar{U})$ and others. More importantly, the smooth version of the topological degree can be recovered from this notion of a winding number.

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In your question you mentioned the word "Fredholm index".

So I would like to say that in the circle case there are two different interpretations of Fredholm index of certain linear operators in terms of winding number. So it would be interesting to consider a possible generalization of these $1$ dimensional facts to higher dimensional spheres.

1)If I remember the following theorem correctly, there is a Theorem by Veku in "Topology and analysis, the Atiyah-Singer index formula and gauge-theoretic physics, by B. Booss and D. D. Bleecker" which says:

Theorem: If $X$ is a non vanishing vector field along $S^1\subset \mathbb{R}^2$, not necessarily tangent to $S^1$, then the fredholm index of the pair operator $(\Delta, \partial/\partial X)$ is equal to the "winding number of $X$. In the above pair operator $\Delta$ is the standard Laplace operator on the interior of circle and the derivational operator $\partial/\partial X$ is restricted to the boundary.

So it would be a good idea to consider an appropriate generalization of this fact. For every smooth self map on sphere one can consider an arbitrary extension to whole $\mathbb{R}^{n+1}$ and try to find an appropriate generalization. However an immediate plain generalization is not true but one should consider a modified generalization. I was thinking to this question about 7 years ago and I observed that a plain generalization is not true since the corresponding operator on $S^3$ is not a Fredholm operator. I had intension to discuss these materials in my following talk I presented in Timisoara but I did not had enough time to present all of the materials of this abstract, since my time was 20 minutes:

http://at.yorku.ca/c/b/d/z/37.htm

(Sorry, the talk abstract I wrote is very snafu)

2) Many years ago I learned from a speaker in Non commutative geometry
that the multiplication by a non vanishing complex map $f$ on $S^1$ is a fredholm operator on $L^1(S^1)$ whose index is $- W(f)$, the winding number of $f$. So it would be interesting to consider an $S^3$ analogy or even more a compact Lie group analogy.

The following paper contains an explanation of a similar situation

http://www.ams.org/journals/proc/2005-133-05/S0002-9939-04-07642-7/S0002-9939-04-07642-7.pdf

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