I want to verify the following "geometric" proof that if we have a continuous $f:S^n \rightarrow S^n$ then $\textit{deg}f=degSf$ , $Sf$ denoting the suspended map from $S^{n+1}$ to $S^{n+1}$.
It is safe to assume that $f$ is onto, since if it isn't, neither is $Sf$ so both maps have degree zero.
Conjecture: For every surjective $f:S^n \rightarrow S^n$ there exists $y$ such that $\#f^{-1}(y)=\textit{finite}$ .
If the conjecture is true then we can compute the degree of $f$ and $Sf$ from the preimages of above $y$ either considered in $S^n$ or in the "equatorial" $S^n \hookrightarrow S(S^n)=S^{n+1}$ and since $Sf|_{eq(S^n)}=f$ the local degrees are the same and therefore the total degrees.
(Attempted) Proof of conjecture: $f$ can be homotoped to $\tilde{f}$ that is both onto and smooth (not sure about how to do both). By Sard's theorem we can find a a regular value, since the critical values have measure $0$ and the image of our $\tilde{f}$ is the whole $S^n$. But this means that $\tilde{f}^{-1}(y)$ is a smooth compact manifold of $n-n=0$ dimension therefore is just finite points.
I would appreciate comments and/or corrections to the following proof. I am aware that $degf=degSf$ can easily be proved algebraicaly, I am just trying to produce a different proof.
Best Answer
Your conjecture is false because space filling curves exist. Here is an "explicit example" Take a surjective map $f:S^1\rightarrow S^2$, suspend it to $Sf:S^2\rightarrow S^3$. Then postcomposing with the Hopf fibration $\nu:S^3\rightarrow S^2$ will give you a map with the desired properties: $\nu^{-1}(y)$ will be a circle for every point and as the map $Sf$ is surjective $Sf^{-1}\nu^{-1}(y)$ will have an infinite number of points for all $y$.
I think it is a good idea to homotope your map to a smooth one: You can homotope to a smooth curve and then suspend. The map might not be smooth at the north and southpole but everywhere else it will be smooth. By a small perturbation (leaving the equator untouched) you can make the map smooth. The degree can be calculated for a regular value on the equator which you can relate to the degree of the unsuspended map.