What is the winding number?

Question

I tried to study the concept of winding number in a general way (the algebraic topology way) but i only find for example the definition from differential geometry and then i find this Winding number question. In this question is defined the winding number in the plane, but i think that this definition can be extended to an arbitrary topological space. And then the definition would be: If $X$ is a topological space and $\gamma:S^1\rightarrow X$ is a loop that not contain a point $p$, the winding number $W( \gamma, p)$ is an integer $n$ that $\gamma$ represents $n$ times the canonical generator in the fundamental group $\pi_1(X\setminus \{p\})$. In this definition what is the meaning of canonical generator? is this the algebraic topology standard definition? that $\gamma$ represents $n$ times the "canonical generator" of the fundamental group means that iterating $n$ times the operation group with the "canonical generator" we obtain $\gamma$ i.e. if the "canonical generator" is $z$ and the group operation is $*$ then $z*z*z*z*z...*z$ is $\gamma$ when we repeat $n$ $z$'s?

Answer 1

The thing that makes the notion of winding number work in the punctured plane is that $\pi_1(\mathbb{R}^2 - p) \cong \mathbb{Z}$. Fixing an orientation (i.e. which generator to call '1' and which to call '-1') allows us to convert loops to integers, and this is the winding number.

For more general spaces, $\pi_1(X)$ is the generalization of winding number: each loop in $X$ corresponds to some element of the homotopy group.

There are some cases where it bears closer resemblance to the winding number: e.g. in $\mathbb{R}^3 - Y$ where $Y$ is a line. Or when $Y$ is a loop, we get (I think) the linking number from knot theory.

One could also think of elements $H_1(X)^*$ -- the group of homomorphisms $H_1(X) \to \mathbb{Z}$ -- as each being a generalized winding number about a different 'hole' in $X$.

But maybe what you're interested in is the engulfing number: instead of mapping a loop to the punctured plane, you can map a sphere to punctured $3$-space, and the engulfing number is the degree to which the image of the sphere engulfs the hole. This corresponds to $\pi_2(\mathbb{R}^3 - p)$. And this easily generalizes to even higher dimensions.