nLab Reeb sphere theorem

Idea

The Reeb sphere theorem in differential topology says that:

If a compact differentiable manifold $X$ admits a differentiable function $X \to \mathbb{R}$ with exactly two critical points which are non-degenerate, then $X$ is homeomorphic to an n-sphere (with its Euclidean metric topology).

In fact this holds true even if the two critical points happen to be degenerate (Milnor 64, theorem 1’ on p. 166)

Proof

Let $x_{min}$ be the critical point at which $f$ attains its minimum and $f_{max}$ the critical point at which it attains its maximum.

By the fact that $X$ is a differential manifold we find an closed neighbourhood $B \subset X$ of $x_{min}$ which is diffeomorphic to an closed ball.

By the assumption that there are no other critical points, $f$ is monotonically increasing along the flow lines $\gamma$ of $\nabla f$ on $X \setminus \{x_{min}, x_{max}\}$, in that the function $\nabla_f(d \gamma_t) = \nabla f(\nabla f) = {\vert \nabla f\vert^2 }$ is strictly positive:

$$\vert\nabla f\vert^2 \colon X \setminus \{x_{min}, x_{max}\} \to (0,\infty) \subset \mathbb{R} \,.$$

Hence for $C \subset X$ a compact, the extreme value theorem implies that this function attains its minimum.

This implies that for every $y \in \mathbb{R}$ with $f(x_{min}) \lt y \lt f(x_{max})$ there exists $t \in \mathbb{R}$ such that the flow $\phi_{\nabla f}$ along $\nabla f$ satisfies $\phi_{\nabla f}(x,t) \geq y$ for all $x \in \partial B$. In particular this is the case for $y$ the maximum of $f|_{C}$. This implies that $C$ is contained in the image of $B$ under some flow of $\nabla f$. But this image is, by the nature of the flow, diffeomorphic to a Euclidean ball.

In conclusion this shows that every compact subspace of $X$ is contained in an open subspace diffeomorphic to a Euclidean ball. With this the claim follows by the Brown-Stallings lemma.

Applications

Notice that the Reeb sphere theorem does not speak of diffeomorphism to an n-sphere, just about homeomorphism. Indeed, the theorem may be used as an ingredient in the construction of exotic smooth structures on some $n$-spheres.

References

Due to:

• Georges Reeb, Sur les points singuliers d’une forme de Pfaff completement integrable ou d’une fonction numerique, Comptes Rendus Acad. Sciences Paris 222 (1946) 847-849 $[$crid:1571417125676878592$]$

Review:

• John Milnor, theorem 4.1 in Morse theory (pdf)

• John Milnor, Differential topology, chapter 6 in T. L. Saaty (ed.) Lectures On Modern Mathematic II 1964 (web, pdf)

See also

• Wikipedia, Reeb sphere theorem