A remark on the contactomorphism group of overtwisted contact spheres
aa r X i v : . [ m a t h . S G ] O c t A remark on the contactomorphism groupof overtwisted contact spheres
Eduardo Fern´andez, Fabio Gironella
Abstract
We show the existence of elements of infinite order in some homotopy groupsof the contactomorphism group of overtwisted spheres. It follows in particularthat the contactomorphism group of some high dimensional overtwisted spheresis not homotopically equivalent to a finite dimensional Lie group.
Let (
M, ξ ) be a closed contact manifold. These short notes are concerned with the rela-tionship between the topology of the connected component Diff ( M ) of the identity inthe group of diffeomorphisms of M and its subgroup Diff ( M, ξ ) consisting of contacto-morphisms of (
M, ξ ). More precisely, throught the notes we will always assume contactstructures to be cooriented and contactomorphisms to be coorientation–preserving.The path components of the group of contactomorphisms of particular contact man-ifolds have been studied by several authors in the literature; see for instance Dymara[2001], Giroux [2001], Ding and Geiges [2010], Lanzat and Zapolsky [2018], Massot andNiederkr¨uger [2016], Giroux and Massot [2017], Vogel [2018], Gironella [2018, 2019].Higher–order homotopy groups have also been studied: for instance, Eliashberg [1992],Casals and Presas [2014], Casals and Sp´aˇcil [2016] contain results for the case of thestandard tight (2 n + 1)–contact sphere. In this notes, we deal with the case of over-twisted spheres (cf. Borman, Eliashberg, and Murphy [2015]).Let ( S n +1 , ξ ot ) be any overtwisted sphere, and consider the natural inclusion i : Diff (cid:0) S n +1 , ξ ot (cid:1) ֒ → Diff (cid:0) S n +1 (cid:1) .For any k ∈ N , denote K n +1 k the kernel of the homomorphism π k ( i ) : π k (cid:0) Diff (cid:0) S n +1 , ξ ot (cid:1)(cid:1) → π k (cid:0) Diff (cid:0) S n +1 (cid:1)(cid:1) . Theorem 1.
Let k ∈ N be such that ≤ k + 1 ≤ n − . The group K n +14 k +1 containsan infinite cyclic subgroup. Under some conditions on the dimension, Theorem 1 can be improved in the caseof the fundamental group and the fifth homotopy group as follows:
Theorem 2. (i) The group K contains a subgroup isomorphic to Z ⊕ Z . ii) Let n ≥ . The group K n +11 contains a subgroup isomorphic to Z ⊕ Z .(iii) Let n ≥ . The group K n +15 contains a subgroup isomorphic to Z ⊕ Z . From the methods developed in the paper we are also able to show the following
Theorem 3. (i) Let n ≥ . The group K n +33 contains an infinite cyclic subgroup.(ii) Let n ≥ . The group K n +74 contains an infinite cyclic subgroup. As the even–order higher homotopy groups of a finite dimensional Lie group arefinite (see for instance F´elix, Oprea, and Tanr´e [2008, Example 2.51]), Theorem 3immediately implies:
Corollary 4.
For n ≥ , Diff (cid:0) S n +7 , ξ ot (cid:1) is not homotopy equivalent to a finitedimensional Lie group. The proofs of Theorems 1, 2 and 3 use four main ingredients. The first is thenotion of overtwisted group introduced in Casals, del Pino, and Presas [2018], whichrelies on the flexibility results for overtwisted contact manifolds from Eliashberg [1989],Borman, Eliashberg, and Murphy [2015]. The second is the existence of a long exactsequence relating the homotopy groups of the space of contact structures on S n +1 tothose of Diff (cid:0) S n +1 , ξ ot (cid:1) and of Diff (cid:0) S n +1 (cid:1) ; see Section 2.1. The last ingredientsare the description of the rational homotopy groups of Diff (cid:0) S n +1 (cid:1) from Farrell andHsiang [1978] and the description of some homotopy groups of the homogeneous spaceΓ n = SO(2 n ) /U ( n ) from Bott [1959], Massey [1961], Harris [1963], Kachi [1978], Mukai[1990].We point out that these methods could also be applied to the case of any overtwistedcontact manifold ( M n +1 , ξ ) such that both the homotopy type of the space of almostcontact structures on M and the diffeomorphism group of M can be (at least partially)understood. Acknowledgments
The authors are extremely grateful to Fran Presas for explainingthem the construction of the overtwisted group and encouraging them to write downthis note, as well as to Javier Mart´ınez Aguinaga for very interesting discussions onthe problem. The first author is supported by the Spanish Research Projects SEV–2015–0554, MTM2016–79400–P, and MTM2015–72876–EXP as well as by a Beca dePersonal Investigador en Formaci´on UCM. The second author is supported by thegrant NKFIH KKP 126683.
Let (
M, ξ ) be a closed contact manifold. In this section, the spaces Diff ( M ) andDiff ( M, ξ ) are be considered as pointed spaces, with base point Id. Similarly, Cont (
M, ξ )is considered with base point ξ .As shown for instance in Giroux and Massot [2017] (and, more in detail, in Massot[2015]), the natural map 2iff ( M ) −→ Cont (
M, ξ ) ϕ ϕ ∗ ξ is a locally–trivial fibration with fiber Diff ( M, ξ ); see also Geiges and Gonzalo Perez[2004] for a proof of the fact that the map is a Serre fibration (which is enough forwhat follows). In particular, it induces a long exact sequence of homotopy groups . . . → π k +1 (Cont ( M, ξ )) → π k (Diff ( M, ξ )) → π k (Diff ( M )) → π k (Cont ( M, ξ )) → . . . (1) S n +1 Recall that, given an oriented smooth manifold M n +1 , an almost contact structure isa triple ( ξ, J, R ), where ξ ⊆ T M is a cooriented hyperplane distribution, J : ξ → ξ is a complex structure on ξ , R = h v i ⊆ T M is a trivial line sub–bundle defining thecoorientation of ξ and ξ ⊕ R ∼ = T M as oriented vector bundles. Fixing an auxiliaryRiemannian metric g on M which is adapted to J and such that w is of norm 1 andorthogonal to ξ , one can see that ( ξ, J, R ) is equivalent to a reduction of the structuregroup SO(2 n + 1) of the principal bundle Fr SO ( M ) of orthonormal oriented frames of T M to its subgroup U( n ) = U( n ) × ⊆ SO(2 n + 1). The space of such reductions isthe space of sections Γ( M ; X ) of a fiber bundle π : X = Fr SO ( M ) / U( n ) → M , withtypical fiber SO(2 n + 1) / U( n ). Such space Γ( M ; X ) is naturally identified with thespace of almost contact structures on M , which we denote AlmCont( M ).Recall also (see Geiges [2008, Lemma 8.2.1]) that there is an identificationΓ n +1 := SO (2 n + 2) . U ( n + 1) ≃ SO (2 n + 1) . U ( n ) . (2)In particular, the fiber bundle π can also be seen as a fibrationΓ n +1 XM π (3)Denote the trivial real line bundle over M by ε = h w i . Then, the Riemannianmetric g on M naturally extends to a metric on T M ⊕ ε , still denoted g , by declaringthe vector w to be orthogonal to T M and of norm 1. Let now Complex (
T M ⊕ ε ) bethe space of complex structures on the oriented bundle T M ⊕ ε , which are compatiblewith the metric g (i.e. g ( J., J. ) = g ( ., . )). Notice that this space can be identified withthe space of sections of a fiber bundle over M with fiber the space of complex structureson R n +2 compatible with the standard metric, i.e. Γ n +1 .Given any almost contact structure ( ξ, J, R ), one can naturally extend J to a com-plex structure ˜ J : T M ⊕ ε → T M ⊕ ε on T M ⊕ ε , by defining ˜ Jv = − w . This gives aninclusion j : AlmCont( M ) ֒ → Complex(
T M ⊕ ε ).In fact, Equation (2) says that i is a diffeomorphism. More precisely, denoting theprojection on the first factor by pr : T M ⊕ ε → T M , the mapΦ : Complex(
T M ⊕ ε ) −→ AlmCont( M ) J ( T M ∩ J ( T M ) , J | T M ∩ J ( T M ) , h pr ( Jw ) i )3s the inverse of i . As a consequence: Lemma 5.
If the vector bundle
T M is stably trivial of type over R , i.e. T M ⊕ ε istrivializable (as real vector bundle), the fiber bundle π : X → M is trivializable. For the rest of the section we focus on the case of almost contact structures on S n +1 .According to Lemma 5, the fiber bundle π : X → S n +1 is trivial. Once fixed anytrivialization, one can then identify AlmCont( S n +1 ) = Map( S n +1 , Γ n +1 ). Remark 6.
The homotopy groups π k (Γ n +1 ), in the stable range 1 ≤ k ≤ n , werecomputed in Bott [1959]: they are of period 8 and the first eight groups are, in order,0 , Z , , , , Z , Z , Z . Moreover, some of the first unstable groups π n +1+ k (Γ n +1 ) werecomputed in Massey [1961], Harris [1963], Kachi [1978], Mukai [1990]. More precisely,we will use the fact that the following unstable homotopy groups contain a cyclicsubgroup: π n +3 (Γ n +1 ), π n +7 (Γ n +1 ), π n +7 (Γ n +2 ) and π n +12 (Γ n +4 ). Lemma 7.
All the path connected components of the space
AlmCont( S n +1 ) are home-omorphic.Proof. Let J ∈ Γ n +1 be the standard (almost) complex structure on R n +2 , anddenote ξ : S n +1 → Γ n +1 z J the almost contact structure with constant value J . Consider then any other almostcontact structure ξ : S n +1 → Γ n +1 . Because Γ n +1 is path–connected, up to homotopy,we can moreover assume that ξ ( N ) = J , where N denotes the north pole of S n +1 .Denote by AlmCont ξ ( S n +1 ) and AlmCont ξ ( S n +1 ) the path connected compo-nents of ξ and ξ , respectively. Consider the U( n +1)–principal bundle p : SO(2 n +2) → Γ n +1 , A A · J · A − . By Bott periodicity, π n (U( n + 1)) = 0. In particular, thehomomorphism π n +1 ( p ) : π n +1 (SO(2 n + 2)) → π n +1 (Γ n +1 )is surjective, so that there exists a lift b ξ : S n +1 → SO(2 n +2) of ξ such that b ξ ( N ) = Id.The desired homeomorphism is then given byΦ ˆ ξ : AlmCont ξ ( S n +1 ) −→ AlmCont ξ ( S n +1 ) η b ξ · η where b ξ · η : S n +1 → Γ n +1 z ˆ ξ ( z ) · η ( z )is defined by using the left action of SO(2 n + 2) on Γ n +1 .4 roposition 8. For each k ∈ N there is an isomorphism π k (cid:0) AlmCont( S n +1 ) (cid:1) ∼ = π k (Γ n +1 ) ⊕ π n + k +1 (Γ n +1 ) Proof.
For k = 0 we argue as follows. Recall that [ S n , X ] = π n ( X, x ) /π ( X, x ), forany pointed topological space (
X, x ). Hence, π (cid:0) AlmCont( S n +1 ) (cid:1) = [ S n +1 , Γ n +1 ] = π n +1 (Γ n +1 ) /π (Γ n +1 ) = π n +1 (Γ n +1 ) ,and the statement follows from the fact that, according to Remark 6, Γ n +1 = SO(2 n +2) / U( n + 1) is simply connected.We now prove the statement for π k with k ≥
1. According to Lemma 7, we canconsider AlmCont (cid:0) S n +1 (cid:1) as space pointed at ξ ≡ J : S n +1 → Γ n +1 . Similarly,we consider Γ n +1 as space pointed at J . There is then a natural Serre fibration (ofpointed spaces) ev N : AlmCont( S n +1 ) → Γ n +1 ξ ξ ( N )The fiber over J is the space F := AlmCont ξ ( N )= J ( S n +1 ) of almost contact struc-tures which evaluate at J on the north pole, which is naturally considered as pointed at ξ . In particular, F = Map(( S n +1 , N ) , (Γ n +1 , J )), so that π k ( F ) = π n + k +1 (Γ n +1 ).Moreover, the map s : Γ n +1 → AlmCont ξ ( S n +1 ) J ξ J where ξ J ≡ J , defines a section of the fibration. In particular, the boundary map inthe long exact sequence of homotopy groups associated to the Serre fibration ev N istrivial, and every obtained short exact sequence of groups splits. In other words, π k (cid:0) AlmCont( S n +1 ) (cid:1) ∼ = π k (Γ n +1 ) ⊕ π k ( F ) = π k (Γ n +1 ) ⊕ π n + k +1 (Γ n +1 ) . Let M be a (2 n + 1)–dimensional manifold. We denote in this section the subspacesof contact and almost contact structures on M with a fixed overtwisted disk ∆ ⊂ M respectively by Cont OT ( M, ∆ ) ⊆ Cont ( M ) and AlmCont ( M, ∆ ) ⊆ AlmCont ( M ). Theorem 9 (Eliashberg [1989], Borman, Eliashberg, and Murphy [2015]) . The fol-lowing forgetful map induces a weak homotopy equivalence:
Cont OT ( M, ∆ ) → AlmCont ( M, ∆ ) , Notice that the overtwisted disk is not allowed to move in this results. However,an easy corollary is the fact that the forgetful mapCont OT ( M ) → AlmCont ( M ) (4)5nduces a bijection at π –level, where Cont OT ( M ) denotes the space of overtwistedcontact structures on M . This can be seen by introducing an overtwisted disk in aneighborhood of a (properly chosen) point of M , and using Theorem 9.To deal with the higher–order homotopy groups, one needs the existence of a con-tinuous choice of overtwisted disks in order to run the same argument. Definition 1 (Casals, del Pino, and Presas [2018]) . Let 0 ≤ k ≤ n . The overtwisted k –group of M , denoted OT k ( M ), is the subgroup of π k (Cont OT ( M )) made of thoseclasses that admit a representative ξ : S k → Cont OT ( M ) for which there is a certificateof overtwistedness , i.e. a continuous map∆ : S k → Emb PL (cid:0) D n , M (cid:1) := { ψ : D n ֒ → M piece–wise linear embedding } such that, for each p ∈ S k , ∆( p ) is overtwisted for ξ ( p ).Homotopy classes in OT k ( M ) are called overtwisted . A homotopy class which isnot overtwisted is called tight .In these terms, Equation (4) says that the map OT ( M ) → π (AlmCont ( M )) isa bijection. For higher–order homotopy groups one then has the following: Proposition 10 (Casals, del Pino, and Presas [2018, Proposition 33]) . Let ( M, ξ ot ) be any closed overtwisted contact manifold. For each ≤ k ≤ n , the inclusion Cont OT ( M ) ֒ → AlmCont ( M ) induces an isomorphism OT k ( M ) ∼ −→ π k (AlmCont ( M )) . Moreover, OT k ( M ) < π k (Cont OT ( M ) , ξ ot ) = π k (Cont ( M ) , ξ ot ) is a normal subgroupfor k > and, thus, the set of tight classes Tight k ( M ) = π k (Cont ( M ) , ξ ot ) / OT k ( M ) has group structure. In particular, for any ≤ k ≤ n there is an isomorphism π k (Cont ( M ) , ξ ot ) ∼ = OT k ( M ) ⊕ Tight k ( M ) . To the authors’ knowledge, the only known example of a non–trivial tight class iscontained in Vogel [2018], where the author exhibits an order 2 loop of overtwistedcontact structures on S , based at the only overtwisted structure on S having Hopfinvariant −
1, which does not admit a certificate of overtwistedness. It follows that thistight loop cannot come from a loop of diffeomorphisms in the long exact sequence inEquation (1). In particular, its image via the boundary map is a non–trivial element(of order 2) in the contact mapping class group.
We start by recalling some known facts in algebraic topology. Recall the followingstandard homotopy equivalence (see for instance Antonelli, Burghelea, and Kahn [1972,Lemma 1.1.5] for a proof):Diff (cid:0) S n +1 (cid:1) ∼ ←− Diff (cid:0) D n +1 , ∂ (cid:1) × SO(2 n + 2) . (5)6ere, the group Diff (cid:0) D n +1 , ∂ (cid:1) of diffeomorphisms of the disk relative to its bound-ary which are smoothly isotopic to the identity is understood as the subgroup ofDiff (cid:0) S n +1 (cid:1) of diffeomorphisms which fixes (a neighborhood of) the north hemi-sphere, and the arrow is the natural inclusion map. Moreover, some of the rationalhomotopy groups of the first factor of the right–hand side of Equation (5) are com-pletely characterized (see also Weiss and Williams [2001, Section 6]): Theorem 11 (Farrell and Hsiang [1978]) . Let ≤ k < min { n − , n − } . Then π k (cid:0) Diff (cid:0) D n +1 , ∂ (cid:1)(cid:1) ⊗ Q = ( if k , Q if k ≡ . Let’s now go back to contact topology and prove the statements announced in theintroduction.
Proof (Theorem 1).
Let ξ ot be any overtwisted structure on S n +1 , and k ∈ N suchthat 1 ≤ k + 1 ≤ n −
1. The relevant part of the long exact sequence in Equation (1)is the following: π k +2 (Diff (cid:0) S n +1 (cid:1) ) π k +2 (Cont (cid:0) S n +1 (cid:1) ) K n +14 k +1 According to Propositions 8 and 10, there is an isomorphism π k +2 (cid:0) Cont (cid:0) S n +1 , ξ ot (cid:1)(cid:1) ∼ = π k +2 (Γ n +1 ) ⊕ π n +4 k +3 (Γ n +1 ) ⊕ Tight k ( S n +1 ) . Moreover, under this isomorphism, the projection on the first factor π k +2 (cid:0) Cont (cid:0) S n +1 , ξ ot (cid:1)(cid:1) → π k +2 (Γ n +1 ) .is just the map induced by the evaluation at the north pole ev N . As Diff (cid:0) D n +1 , ∂ (cid:1) i ⊂ Diff (cid:0) S n +1 (cid:1) is the subgroup of diffeomorphisms fixing the north hemisphere, it followsthat the following composition is trivial: π k +2 (Diff (cid:0) D n +1 , ∂ (cid:1) ) π k +2 (Diff (cid:0) S n +1 (cid:1) ) π k +2 (Cont (cid:0) S n +1 (cid:1) ) π k +2 (Γ n +1 ) π k +2 ( i ) π k +2 ( ev N ) Moreover, according to Bott periodicity, π k +2 (SO(2 n + 2)) = 0. In particular, thefollowing composition is also trivial: π k +2 (Diff (cid:0) S n +1 (cid:1) ) π k +2 (Cont (cid:0) S n +1 (cid:1) ) π k +2 (Γ n +1 ) π k +2 ( ev N ) Now, according to Remark 6, π k +2 (Γ n +1 ), hence π k +2 (cid:0) Cont (cid:0) S n +1 (cid:1)(cid:1) , containsa subgroup Z . It then follows from the exact sequence that K n +14 k +1 must have at leastone element of infinite order, as desired. 7 roof (Theorem 2). According to Hatcher [1983], the Smale Conjecture holds for S ;in particular, π (cid:0) Diff (cid:0) S (cid:1)(cid:1) = 0. Moreover, since Γ = SO(4) / U(2) = S it followsfrom Propositions 8 and 10 that the groupOT ( S ) ∼ = π (cid:0) S (cid:1) ⊕ π (cid:0) S (cid:1) ∼ = Z ⊕ Z is a subgroup of π (cid:0) Cont (cid:0) S (cid:1) , ξ ot (cid:1) . Item (i) then follows from the exact sequence inEquation (1).Since π (SO(4 n + 1)) = π (SO(4 n + 1)) = 0, Theorem 11 implies that π (cid:0) Diff (cid:0) S n +1 (cid:1)(cid:1) ⊗ Q = 0 for n ≥
3, and π (cid:0) Diff (cid:0) S n +1 (cid:1)(cid:1) ⊗ Q = 0 for n ≥ π (Γ n +1 ) for n ≥ π (Γ n +1 ) for n ≥ π n +3 (Γ n +1 ) and π n +7 (Γ n +1 ). Items (ii) and (iii) then follow from the exact sequence in Equation (1)and from Propositions 8 and 10. Proof (Theorem 3).
Since π (SO(4 n + 4)) is trivial, it follows from the identificationin Equation (5) and from Theorem 11 that π (cid:0) Diff (cid:0) S n +3 (cid:1)(cid:1) ⊗ Q = 0 for n ≥ π n +7 (Γ n +2 ) contains a subgroup Z .Similarly, π (SO(8 n + 8)) = 0 thus Equation (5) and Theorem 11 imply that π (cid:0) Diff (cid:0) S n +7 (cid:1)(cid:1) ⊗ Q = 0 for n ≥
2. According to Remark 6, π n +12 (Γ n +4 ) ∼ = Z .The statement then follow from the exact sequence in Equation (1) and from Propo-sitions 8 and 10. References
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E. Fern´andez,
Universidad Complutense de Madrid, Departamento de ´Algebra, Geometr´ıay Topolog´ıa, Facultad de Matem´aticas, and Instituto de Ciencias Matem´aticas CSIC-UAM-UC3M-UCM, Madrid, Spain [email protected]
F. Gironella,
Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary [email protected], [email protected]@renyi.hu, [email protected]