> 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.