nLab diffeomorphism group

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

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

• for $\Sigma$ any surface (Zieschang-Vogt-Coldeway)

• for $\Sigma$ a Haken 3-manifold (Waldhausen)

• for $\Sigma$ any hyperbolic manifold of finite volume and of dimension $\geq 3$ (by Mostow rigidity theorem) (check)

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

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

(Hatcher 81)

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.

References

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 spaces – diffeological 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:

• Tadayuki Watanabe, Some exotic nontrivial elements of the rational homotopy groups of $Diff(S^4)$ (arXiv:1812.02448)

• Tadayuki Watanabe, Addendum to: Some exotic nontrivial elements of the rational homotopy groups of $Diff(S^4)$ (homological interpretation) (arXiv:2109.01609)

  (via graph complexes)

General

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