I have been browsing "Topological Degree Theory and Applications" by Cho, Chen and O'Regan as well as "Mapping Degree Theory" by Outerelo and Ruiz, but I have not been able to quite answer myself the following question:



> 
> Let $\gamma:\mathbb{S}^1\to\mathbb{R}^n$, $n\geq 3$, be a closed (rectifiable, piece-wise smooth, smooth, if necessary) curve and let $p\in\mathbb{R}^n\setminus\gamma(\mathbb{S}^1)$ be a point not lying on $\gamma$. Do we have a good notion of a winding number $w(\gamma,a)$ that generalizes the case $n=2$ ?
> 
> 
> 


*Remarks:*


1. If $\gamma$ is smooth with $D\gamma$ of rank $1$ everywhere, then $\gamma(\mathbb{S}^1)$ is a submanifold ($\gamma$ is injective by definition and $\mathbb{S}^1$ is compact). In particular, it is a connected manifold, and we have a well-defined topological degree $\deg(\gamma)$ since it does not depend on a choice of a regular value of $\gamma$ on the image. This is one straight-forward generalization of the case $n=2$ for curves in higher dimensions. There are probably other approaches as well.
2. Outerelo and Ruiz introduce $w(f,a)$ for continuous $f$ defined only on *hypersurfaces* $X:=\partial\bar{U}\subseteq\mathbb{R}^n$ (for some $U$ open and bounded). In particular, the construction does not seem to work for *curves* for $n\geq 3$. More generally, the topological degree appears to be defined only for maps whose domain and codomain are of equal dimension. (In this interpretation and modulo technical details, the winding number in the plane is the degree of a map defined on the unit disk which restricts on the boundary to our curve.)


I am hoping that (1.) could somehow be "relativized" to include an external vantage point $p$.



Is there a sensible notion of a winding number of a closed curve in $\mathbb{R}^n$, $n\geq 3$, with respect to a point not lying on it?
**No**, you cannot define a winding number if $n\geq 3$ since, as pointed out in a comment by Anthony Carapetis, any two curves in $\mathbb{R}^n\setminus\{ p\}$ are homotopic, and a winding number should be invariant under homotopies. You can however, define a linking number between two continuous disjoint images of spheres in $\mathbb{R}^n$ of dimensions $k$ and $n-k-1$. If $k=1$, the other sphere needs to have dimension $n-2$. In the case of a planar curve, $n=2$ and $k=1$, and the other sphere has dimension $0$ i.e., consists of two points. If one of the points is in the unbounded component of the image of the curve, the corresponding linking number will be equal to the winding number.


