nLab
diffeomorphism group

Context

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& Rh & \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)

Manifolds and cobordisms

Contents

Definition

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

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

Properties

Relation to homotopy equivalences

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

α:Π(Σ)Π(Σ) \alpha \colon \Pi(\Sigma) \stackrel{\simeq}{\longrightarrow} \Pi(\Sigma)

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

a:ΣΣ a \colon \Sigma \stackrel{\simeq}{\longrightarrow} \Sigma

i.e. that given α\alpha there is aa with

αΠ(a). \alpha \simeq \Pi(a) \,.

Homotopy type and mapping class group

The homotopy type Π(Diff(Σ))\Pi(Diff(\Sigma)) of the diffeomorphism group Diff(Σ)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

π 0(Π(Diff(Σ)))=MCG(Σ). \pi_0(\Pi(Diff(\Sigma))) = MCG(\Sigma) \,.

For 1-manifolds

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

For 2-manifolds (surfaces)

Proposition

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

  1. if Σ\Sigma is the sphere then

    Π(Diff(S 2)) Π(O(3)) MCG(S 2)×Π(SO(3)) 2×Π(SO(3)) \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 Σ\Sigma is the torus then

    Π(Diff(S 1×S 1)) MCG(S 1×S 1)×Π(S 1×S 1) GL 2()×B(×) \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:

    Π(Diff(Σ))MCG(Σ) \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
Π(Diff(S 1×S 2))Π(O(2)×O(3))×ΩΠ(SO(3)). \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)O(4)

Π(Diff(S 3))Π(O(4)), \Pi(Diff(S^3)) \simeq \Pi(O(4)) \,,

the equivalence being exhibited by the canonical inclusion

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

Also

Π(Diff(D 3))Π(O(3)). \Pi(Diff(D^3)) \simeq \Pi(O(3)) \,.

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

Generally:

Theorem

For every smooth 3-manifold the canonical map

Π(Diff(X))Π(Homeo(X)) \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 1, was shown in (Cerf). For discussion see (Hatcher, 1978).

Proposition

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

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

If XX is Seifert via an S 1S^1-action, then the component of Diff(X)Diff(X) are typically Π(S 1)\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 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

For 3-manifolds

  • J. Cerf, Sur les difféomorphismes de la sphère de dimension trois (Γ 4=0\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×S 2S^1\times S^2, Proceedings of the AMS 83 (1981), 427-43 (pdf)

  • Allen Hatcher, A proof of the Smale conjecture, Diff(S 3)O(4)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)

General

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

Revised on March 6, 2015 07:19:00 by Urs Schreiber (195.113.30.252)