nLab
Reeb sphere theorem

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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}{}& ʃ &\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 XX admits a differentiable function XX \to \mathbb{R} with exactly two critical points which are non-degenerate, then XX 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 minx_{min} be the critical point at which ff attains its minimum and f maxf_{max} the critical point at which it attains its maximum.

By the fact that XX is a differential manifold we find an closed neighbourhood BXB \subset X of x minx_{min} which is diffeomorphic to an closed ball.

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

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

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

This implies that for every yy \in \mathbb{R} with f(x min)<y<f(x max)f(x_{min}) \lt y \lt f(x_{max}) there exists tt \in \mathbb{R} such that the flow ϕ f\phi_{\nabla f} along f\nabla f satisfies ϕ f(x,t)y\phi_{\nabla f}(x,t) \geq y for all xBx \in \partial B. In particular this is the case for yy the maximum of f| Cf|_{C}. This implies that CC is contained in the image of BB under some flow of f\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 XX 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 nn-spheres.

References

  • 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)

Last revised on June 26, 2017 at 13:56:52. See the history of this page for a list of all contributions to it.