# Intuitive explanation of degree

I do understand the for $$f : \mathbb{S}^{1} \longmapsto \mathbb{S}^{1}$$ the degree of a map (thinking $$f$$ as a closed curve $$\gamma$$ defined on $$[0,1]$$) can be seen as "how many times a closed curve wraps itself on $$\mathbb{S}^{1}"$$ in the codomain, but this seems to me a very dependant explanation due to the fact that every $$n$$-degree map from $$\mathbb{S}^{1}$$ to itself is homotopic to $$z^{n}$$.

Defining the degree for a $$f: M \longmapsto N$$ of class $$C^{\infty}$$ where $$M$$ is compact and orientable, $$N$$ connected and orientable, both without boundary with dim$$M$$= dim$$N$$ as deg $$f =\sum\limits_{x \in f^{-1}(y)}\text{sgn}df_x$$, where $$y \in \text{RegVal}(f)$$

Are there any chance to visualize why this should at least generalize for manifolds in $$\mathbb{R}^{3}$$ the degree of maps from $$\mathbb{S}^{1}$$ to itself ? I was interested in finding a $$k$$-degree map from $$\mathbb{S}^{n} \longmapsto \mathbb{S}^{n}$$ and I found this question, which was related to this other question. The answers given by Alex Becker in the first and Jared in the second I think it would be awesome, If I just understood what they really meant.

Especially in the second one, I can't see how such map should be smooth or intuitevely understand why the degree of that map should be one with the definition given above; while in the first I don't understand how would have come to my mind to think to such a map, since I'm unable to visualize what a map of degree $$k$$ should look like with the definition of degree as sum.

Any help would be appreciated

• Here's an example to start off with. Suspend the map $z\mapsto z^k$ ($k$ an integer) on the circle to $S^2$. So you're wrapping each latitude circle around itself $k$ times and fixing the north and south poles. What's the degree? Check the computation. – Ted Shifrin Jan 13 at 20:35
• @TedShifrin I think the map from the suspension has also degree $k$. You think the better way to costruct a map of any degree is to start from $\mathbb{S}^{1}$ and iterate on the suspension ? – jacopoburelli Jan 13 at 20:37
• If so, this can be done only using with degree theory and topology? – jacopoburelli Jan 13 at 20:38
• You can certainly do that. There are lots of other things to do, too. Do something interesting on a ball so that everything outside it maps to a fixed point of $S^n$. – Ted Shifrin Jan 13 at 20:39
• I don't know what you mean by your "done only using ..." question. – Ted Shifrin Jan 13 at 20:40