# nLab diffeomorphism group

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

• (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

The diffeomorphism group $Diff(X)$ of a smooth manifold $X$ is the group of its diffeomorphisms: the automorphism group of $X$ as an object of the category SmoothMfd.

Beware that when $X$ is assumed orientable then sometimes, but not always, $Diff(X)$ is implicitly taken to be the group of orientation-preserving diffeomorphisms.

## Properties

### 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

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

### Homotopy type and mapping class group

The homotopy type $\Pi(Diff(\Sigma))$ of the diffeomorphism group $Diff(\Sigma)$ is of interest (e.g. Hatcher 12).

For instance this is the automorphism ∞-group of a manifold, regarded as a k-morphism in an (∞,n)-category of cobordisms.

Specifically, the group of connected components is the mapping class group

$\pi_0(\Pi(Diff(\Sigma))) = MCG(\Sigma) \,.$

#### For 1-manifolds

$\Pi(Diff(S^1))\simeq \Pi(O(2))$
$\Pi(Diff(D^1))\simeq \Pi(O(1))$

#### For 2-manifolds (surfaces)

###### Proposition

For $\Sigma$ a closed orientable surface, then the bare homotopy type of its diffeomorphism group is

1. if $\Sigma$ is the sphere then

\begin{aligned} \Pi(Diff(S^2)) & \simeq \Pi(O(3)) \\ & \simeq MCG(S^2)\times \Pi(SO(3)) \\ & \simeq \mathbb{Z}_2 \times \Pi(SO(3)) \end{aligned}
2. if $\Sigma$ is the torus then

\begin{aligned} \Pi(Diff(S^1 \times S^1)) & \simeq MCG(S^1 \times S^1)\times \Pi(S^1 \times S^1 ) \\ & \simeq GL_2(\mathbb{Z}) \times B(\mathbb{Z} \times\mathbb{Z}) \end{aligned}
3. in all other cases all higher homotopy groups vanish:

$\Pi(Diff(\Sigma)) \simeq MCG(\Sigma)$

The first statement is due to (Smale 58), see also at sphere eversion. The second and third are due to (Earle-Eells 67, Gramain 73).

#### For 3-manifolds

###### Proposition
$\Pi(Diff(S^1 \times S^2)) \simeq \Pi(O(2) \times O(3)) \times \Omega \Pi(SO(3)) \,.$

###### Theorem

(Smale conjecture)
The bare homotopy type of the diffeomorphism group of the 3-sphere is that of the orthogonal group $O(4)$

$\esh\big( Diff(S^3) \big) \;\simeq\; \esh \, O(4)) \,,$

the equivalence being exhibited by the canonical inclusion

$O(4) \hookrightarrow Diff(S^3) \,.$

Also

$\esh \, Diff(D^3) \;\simeq\; \esh \, O(3) \,.$

After being conjectured by Smale, this was proven in (Hatcher 1983).

Generally:

###### Theorem

For every smooth 3-manifold the canonical map

$\Pi(Diff(X)) \to \Pi(Homeo(X))$

sending diffeomorphisms to their underlying homeomorphisms of topological spaces is a weak homotopy equivalence.

That this follows from the Smale cojecture, theorem , was shown in (Cerf). For discussion see (Hatcher, 1978).

###### Proposition

If a 3-manifold $X$ is not a Seifert 3-manifold via an $S^1$-action then

$\Pi(Diff(X)) \simeq MCG(X) \,.$

If $X$ is Seifert via an $S^1$-action, then the component of $Diff(X)$ are typically $\Pi(S^1)$-s.

### Smooth structure

The observation that infinite-dimensional smooth groups such as diffeomorphism groups (and quantomorphism groups etc.) are naturally regarded as internal groups in diffeological spacesdiffeological groups – is due to

• Jean-Marie Souriau, Groupes différentiels, in Differential Geometrical Methods in Mathematical Physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math. 836, Springer, Berlin, (1980), pp. 91–128. (MathScinet)

### For 2-manifolds (surfaces)

• C.J. Earle, J. Eells, The diffeomorphism group of a compact Riemann surface, Bulletin of the American Mathematical Society 73(4) 557–559, 1967

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

• J. S. Dowker, Note on the structure constants for the diffeomorphisms of the two-sphere [arXiv:2301.09487]

### For 3-manifolds

• J. Cerf, Sur les difféomorphismes de la sphère de dimension trois ($\Gamma_4 = 0$), Lecture Notes in Math., vol. 53, Springer-Verlag, Berlin and New York, 1968_

• Allen Hatcher, Linearization in 3-dimensional topology, Proceedings of the international congress of Mathematicians, Helsinki (1978)

• Allen Hatcher, On the diffeomorphism group of $S^1\times S^2$, Proceedings of the AMS 83 (1981), 427-43 (pdf)

• Allen Hatcher, A proof of the Smale conjecture, $Diff(S^3) \simeq O(4)$, Annals of Mathematics 117 (1983) (jstor)

• 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)

### For 4-manifolds

For 4-manifolds the analogue of the Smale conjecture fails:

### General

• Alan Hatcher, A 50-Year View of Diffeomorphism Groups, talk at the 50th Cornell Topology Festival in May 2012 (pdf, pdf)