# nLab diffeomorphism

Contents

### 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

tangent cohesion

differential cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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)