A proof of the Fundamental Theorem of Algebra based on Differential Topology

The following proof of Fundamental Theorem of Algebra, which uses techniques of Differential Topology, can be found in [1, pp. 8-9].

Theorem. Every non-constant polynomial $P \in \mathbb{C}[z]$ must have a zero.

Proof. Using the stereographic projection from the point $\infty=(0, \, 0, \, 1)$, we can extend the polynomial function $P \colon \mathbb{C} \to \mathbb{C}$ to a function $f \colon \mathbb{S}^2 \to \mathbb{S}^2$ such that $f(\infty)=\infty$. A straightforward use of the change of coordinates $z=1/w$ shows that $f$ is smooth on the whole of $\mathbb{S}^2$, including the point $\infty$.

Let us remark that $f \colon \mathbb{S}^2 \to \mathbb{S}^2$ has a finite number of critical points (and so a finite number of critical values), corresponding to the zeros of the derivative $P’$ and possibly to the point $\infty.$

If $0\in \mathbb{C}$ corresponds to a critical value for $f \colon \mathbb{S}^2 \to \mathbb{S}^2$ then, by definition, it has a preimage via $P \colon \mathbb{C}^2 \to \mathbb{C}^2$, and there is nothing to prove.

We may therefore assume that $0\in \mathbb{C}$ corresponds to a regular value for $f \colon \mathbb{S}^2 \to \mathbb{S}^2$. The locus of these regular values is the complement of finitely many points on the sphere, so it is connected.

From the Implicit Function Theorem for differentiable manifolds, it follows that $f \colon \mathbb{S}^2 \to \mathbb{S}^2$ is a local diffeomorphism when restricted to the locus of regular values. So the number of points which are preimages of a fixed one is locally constant on such locus, hence constant (being the locus connected).

Since $\deg(P) \geq 1$, there are surely (infinitely many) points $y \in \mathbb{S}^2$ such that $f^{-1}(y) $ is non-empty. Thus, the same must be true for the point corresponding to $0 \in \mathbb{C}.$

In other words, $0$ is in the image of $P$. $\square$

References.

[1] J. W. Milnor: Topology from the Differentiable Viewpoint, Princeton University Press 1965.

Whitney approximation theorem: $\mathbb{S}^n$ is simply connected for $n \geq 2$

Let me give a proof of the fact that the $n$-sphere $\mathbb{S}^n \subset \mathbb{R}^{n+1}$ is simply connected for $n \geq 2$, which is based on differential topology methods (instead than on the Seifert-Van Kampen theorem). We will use the following two results:

(1) Whitney Approximation Theorem: Every continuous map $f \colon M \to N$ between smooth manifolds is homotopic to a smooth map.

(2) Sard Theorem: Let $f \colon  M \to N$ be a smooth map between smooth manifolds, and let $\operatorname{Crit}(f)$ be the locus of points of $M$ where the differential df has rank $< \dim(N)$. Then the image of $\operatorname{Crit}(f)$ has zero Lebesgue measure in $N$. In particular, if $\dim(M) < \dim(N)$ then $f$ cannot be surjective.

Given a continuous loop $c \colon [0, \, 1] \to \mathbb{S}^n$, we want to show that $c$ is homotopically trivial. By (1), it is not restrictive to assume that $c$ is piecewise smooth. Then, by (2), $c$ cannot be surjective (recall that we are assuming $n \geq 2$).

Now, take a point $p$ not in the image of $c$. Then we can see $c$ as a loop in $\mathbb{S}^n-\{p\}$, which is a contractible space (it is homeomorphic to $\mathbb{R}^n$ via the stereographic projection). So $c$ is homotopically trivial.

Remark. The regularization provided by (1) is essential here. In fact, there exist continuous, non-smooth loops "of Peano type", whose image is the whole sphere.