nLab diffeomorphism

Context

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

Definition

Given two $k$-times differentiable manifolds (or smooth manifolds), then a diffeomorphism

$f \;\colon\; X \longrightarrow Y$

is a differentiable function such that there exists an inverse differentiabe function $f^{-1}$ (a function which is an inverse function on the underlying sets and is itself differentiable to the given degree).

Diffeomorphisms are the isomorphisms in the corrresponding category Diff of differentiable manifolds/smooth manifolds.

Properties

Relation to homeomorphisms

It is clear that

Observation

Every diffeomorphism is in particular a homeomorphism between the underlying topological spaces.

The converse in general fails. There exist differentiable maps with only continuous inverse. There are also differentiable bijections whose inverse is not even continuous.

Example

The function $f : \mathbb{R} \to \mathbb{R}$ given by $x \mapsto x^3$ is a homeomorphism but not a diffeomorphism. The diffeomorphism property fails at the origin, where the differential $d f : T_0 \mathbb{R} \to T_0 \mathbb{R}$ is not onto.

But there is a rich collection of theorems about cases when the converse is true after all.

Definition

For $n \in \mathbb{N}$, the open n-ball $\mathbb{B}^n$ is the open subset

$\mathbb{B}^n = \{ \vec x \in \mathbb{R}^n | \sum_{i = 1}^n (x^i)^2 \lt 1 \} \subset \mathbb{R}^n$

of the Cartesian space $\mathbb{R}^n$ of all points of distance lower than 1 from the origin. This inherits the structure of a smooth manifold from the embedding into $\mathbb{R}^n$.

Theorem

In dimension $d \in \mathbb{N}$ for $d \neq 4$ we have:

every open subset of $\mathbb{R}^d$ which is homeomorphic to $\mathbb{B}^d$ is also diffeomorphic to it.

See the first page of (Ozols) for a list of references.

What’s a good/canonical textbook reference for this?

Remark

In dimension 4 the analog statement fails due to the existence of exotic smooth structures on $\mathbb{R}^4$.

Theorem

For $X$ and $Y$ smooth manifolds of dimension $d = 1$, $d = 2$ or $d = 3$ we have:

if there is a homeomorphism from $X$ to $Y$, then there is also a diffeomorphism.

See the corollary on p. 2 of (Munkres).

Relation to homotopy equivalences

For the following kinds of manifolds $\Sigma$ it is true that every homotopy equivalence

$\alpha \colon \Pi(\Sigma) \stackrel{\simeq}{\longrightarrow} \Pi(\Sigma)$

(hence every equivalence of their fundamental infinity-groupoids) is homotopic to a diffeomorphism

$a \colon \Sigma \stackrel{\simeq}{\longrightarrow} \Sigma$

i.e. that given $\alpha$ there is $a$ with

$\alpha \simeq \Pi(a) \,.$

A review of results and relevant literature is also on the first page of (Hass-Scott 92)-

• V. Ozols, Largest normal neighbourhoods Proceedings of the American Mathematical Society Vol. 61, No. 1 (Nov., 1976), pp. 99-101 (jstor) (AMS: pdf)

• James Munkres, Obstructions to the smoothing of piecewise-differentiable homeomorphisms Bull. Amer. Math. Soc. Volume 65, Number 5 (1959), 332-334. (Euclid)(AMS: pdf)

• Zieschang, Vogt and Coldeway, Surfaces and planar discontinuous groups

• Friedhelm Waldhausen, On Irreducible 3-Manifolds Which are Sufficiently Large, Annals of Mathematics

Second Series, Vol. 87, No. 1 (Jan., 1968), pp. 56-88 (JSTOR)

• Joel Hass, Peter Scott, Homotopy equivalence and homeomoprhism of 3-manifolds, Topology, Vol. 31, No. 3 (1992) pp. 493-517 (pdf)