
> 
> Is there a natural geometric generalization of the [winding number](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Fredholm_operator),
the [Pontryagin index](https://www.encyclopediaofmath.org/index.php/Pontryagin_number), 
and to [Bott periodicity](https://en.wikipedia.org/wiki/Bott_periodicity_theorem),
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.



Generalization of winding number to higher dimensions
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.)


