# nLab Reeb sphere theorem

Contents

### Context

#### Spheres

n-sphere

low dimensional n-spheres

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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)