aa r X i v : . [ m a t h . S G ] M a y CONTACT AND ISOCONTACT EMBEDDING OF π –MANIFOLDS KULDEEP SAHA
Abstract.
We prove some contact analogs of smooth embedding theorems for closed π –manifolds. We showthat a closed, k -connected, π –manifold of dimension (2 n + 1) that bounds a π –manifold, contact embedsin the (4 n − k + 3)-dimensional Euclidean space with the standard contact structure. We also prove someisocontact embedding results for π –manifolds and parallelizable manifolds. introduction Embedding of manifolds in the Euclidean spaces has long been a problem of great importance in geometrictopology. Over the years many remarkable results concerning embedding of manifolds were obtained. Thefirst major breakthrough in this direction was the Whitney embedding theorem [Wh], which says that every n -manifold can be smoothly embedded in R n . Later, Haefliger and Hirsch ([HH]) generalized Whitney’stheorem to show that a closed, orientable, k -connected n -manifold can be embedded in R n − k − .Recall that a manifold M is called a π -manifold, provided the direct sum of its tangent bundle withthe trivial real line bundle is trivial. In [Sa], Sapio improved the Haefliger-Hirsch embedding theorem for k -connected π -manifolds to produce embeddings with trivial normal bundle in R n − k − . In this note westudy the contact analogs of some of these embedding results.A contact manifold is an odd dimensional smooth manifold M n +1 , together with a maximally non-integrable hyperplane distribution ξ ⊂ T M . A contact form α representing ξ is a local 1-form on M suchthat ξ = Ker { α } . The contact condition is equivalent to saying that α ∧ ( dα ) n is a volume form. The 2-form dα then induces a conformal symplectic structure on ξ . If the line bundle T M/ξ over M is trivial, then thecontact structure is said to be co-orientable. For a co-orientable contact structure ξ , one can define a contactform α representing ξ on all of M . In this article, we will only consider co-orientable contact structures onclosed, orientable manifolds. We will denote a manifold M together with a contact structure ξ by ( M, ξ ).We will use ξ std to denote the standard contact structure on an odd dimensional Euclidean space R N +1 given by Ker { dz + Σ Ni =1 x i dy i } . The trivial real vector bundle of rank r over a space Z will be denoted by ε rZ and ε rZ ( C ) will denote the trivial complex vector bundle of complex rank r over Z . An embedding is alwaysassumed to be smooth. All the embedding results will be stated for closed manifolds. Definition 1.1 (Isocontact embedding) . ( M n +1 , ξ ) admits an isocontact embedding in ( V N +1 , η ), if thereis an embedding ι : M ֒ → V such that for all p in M, Dι ( T p M ) is transverse to η ι ( p ) and Dι ( T p M ) ∩ η ι ( p ) = Dι ( ξ p ). A manifold M n +1 contact embeds in ( V N +1 , η ) if there exists a contact structure ξ on M n +1 such that ( M, ξ ) has an isocontact embedding in ( V N +1 , η ).It follows from the definition that if α is a contact form representing ξ and β is a contact form representing η . Then ι ∗ ( β ) = h · α for some positive function h on M . Dι ( ξ ) is a conformal symplectic sub-bundle of( η | ι ( M ) , dβ ).Similarly, we can define the notion of an isocontact immersion. Definition 1.2 (Isocontact immersion) . An isocontact immersion of (
M, ξ ) in ( R N +1 , η ) is an immersion ι : ( M, ξ ) ( R N +1 , η ) such that Dι ( T M ) is transverse to η and Dι ( T M ) ∩ η = Dι ( ξ ).Gromov [Gr] reduced the existence of an isocontact embedding of a contact manifold ( M n +1 , ξ ) in acontact manifold ( V N +1 , η ), for N ≥ n + 2, to a problem in obstruction theory. Gromov [Gr] proved Mathematics Subject Classification.
Primary: 53D10. Secondary: 53D15, 57R17.
Key words and phrases. contact structures, embedding, h-principle. that any contact manifold ( M n +1 , ξ ) has an isocontact embedding in ( R n +3 , ξ std ). This result, which isessentially the contact analog of Whitney’s embedding theorem, was reproved later by A. Mori [Mo] for n = 1 and by D. M. Torres [Tor] for all n using different techniques. For isocontact embeddings of a contactmanifold ( M, ξ ) of co-dimension ≤ dim ( M ) −
1, there is a condition on the Chern classes of ξ . This conditioncomes from the normal bundle of the embedding. See the Remark 2.14 for a precise statement. So, onehas to restrict the isocontact embedding question to contact structures which satisfy that condition. Giventhis, a theorem of N. Kasuya ([Ka], Theorem 1 .
5) says that for 2-connected (2 n + 1)-contact manifolds,the Haefliger-Hirsch theorem has a contact analog giving isocontact embedding in ( R n +1 , ξ std ). Here, weinvestigate the contact analog of Sapio’s theorem for π -manifolds and also provide some contact embeddingresults for parallelizable manifolds. Before stating our results, we introduce some terminologies.In [Sa], Sapio introduced the notion of an almost embedding. A manifold M n almost embeds in a manifold W N , if there exists a homotopy sphere Σ n so that M n n smoothly embeds in W N . We want to defineanalogous notions for contact and isocontact embeddings. Recall that if ( M, ξ M ) and ( N n +1 , ξ N ) are twocontact manifolds, then by ( M n +1 N n +1 , ξ M ξ N ) we denote the contact connected sum of them. Fordetails on a contact connected sum we refer to chapter 6 of [Ge]. Definition 1.3 (Homotopy isocontact embedding) . ( M n +1 , ξ ) admits a homotopy isocontact embedding in( R N +1 , ξ std ), if there exists a contact homotopy sphere (Σ n +1 , η ) such that ( M n +1 n +1 , ξ η ) has anisocontact embedding in ( R N +1 , ξ std ). We say, M n +1 homotopy contact embeds in ( R N +1 , ξ std ), if there isa contact structure ξ on M n +1 such that ( M n +1 , ξ ) has an isocontact embedding in ( R N +1 , ξ std ).Before stating our results, we describe the contact structures we will be considering. For an isocontactembedding of ( M n +1 , ξ ) of co-dimension 2( N − n ) with trivial symplectic normal bundle, we need thatthe Chern classes c i ( ξ ) vanish for 1 ≤ i ≤ n . For details see the Remark 2.14. Note that by a theorem ofPeterson (Theorem 2 .
1, [Ke]), if ( M n +1 , ξ ) is torsion free, then this condition is true if and only if ξ is trivialas a complex vector bundle over the 2 n -skeleton of M . Consider the fibration map SO (2 n + 2) → Γ n +1 withfiber U ( n + 1), where Γ n +1 denotes the space of almost complex structures on R n +2 . Since T M ⊕ ε M istrivial for a π –manifold, one can postcompose a trivialization map to SO (2 n + 2) with the above fibrationmap to get an almost contact structure on M . For notions of almost complex and almost contact structuressee section 2 . Definition 1.4.
A contact structure on a π –manifold M , representing an almost contact structure thatfactors through a map from M to SO (2 n + 2) as mentioned above, will be called an SO -contact structure .We now state an analog of Sapio’s Theorem for contact π –manifolds. Theorem 1.5.
Let M n +1 be a k -connected, π –manifold. Assume that n ≥ k ≤ n −
1. Then(1) M n +1 homotopy contact embeds in ( R n − k +3 , ξ std ).(2) If n mod
4) and for all i ∈ { k + 1 , · · · , n − k } such that i ≡ , , , mod , H n − i +1 ( M ) =0, then for any contact structure ξ on M n +1 , ( M, ξ ) has a homotopy isocontact embedding in( R n − k +3 , ξ std ).(3) If n mod
4) and for all i ∈ { k + 1 , · · · , n − k } such that i ≡ , mod , H n − i +1 ( M ) = 0,then for any SO -contact structure ξ on M n +1 , ( M, ξ ) has a homotopy isocontact embedding in( R n − k +3 , ξ std ).(4) If M n +1 bounds a π –manifold, then we can omit “homotopy” in the above statements.We remark that in all the statements above, we get contact or isocontact embeddings with a trivialconformal symplectic normal bundle.Note that Theorem 1.5 provides criteria to find examples of isocontact embeddings of π -manifolds inthe standard contact euclidean space. For example, a straightforward application of statement 2 and 4 inTheorem 1.5 shows that every contact structure on S × S has an isocontact embedding in ( R , ξ std ).Here, we have k = 2 and n = 4. Similarly, one can check that all contact structures on S × S ( k = 3) and S × S ( k = 10) admit isocontact embeddings in ( R , ξ std ) and ( R , ξ std ) respectively. ONTACT AND ISOCONTACT EMBEDDING OF π –MANIFOLDS 3 The proof of Theorem 1 . Corollary 1.6.
Let M n +1 be an ( n − π –manifold that bounds a π –manifold. Then(1) M n +1 contact embeds in ( R n +5 , ξ std ).(2) If n ≡ , mod ξ , ( M, ξ ) has an isocontact embedding in( R n +5 , ξ std ).In particular, any contact homotopy sphere Σ n +1 that bounds a parallelizable manifold has an isocontactembedding in ( R n +5 , ξ std ), for n ≡ , , mod R , ξ std ). Remark 1.7. ( On optimal dimension of embedding ) In section 4 of [Sa], Sapio constructs a family of( r − ρ ( r ) − r − ρ ( r ) − M ( r ). Here, r = (2 a + 1)2 b +4 c , ≤ b ≤ , a, b, c ∈ Z ≥ and ρ ( r ) = 2 b + 8 c . Note that [2(2 r − ρ ( r ) − − r − ρ ( r ) − −
1] = 2 r −
1. Sapioshows that M ( r ) bounds a π -manifold. So, by Theorem 2.16, M ( r ) embeds in R r − . But M ( r ) does notembed in R r − . It follows from the discussion in section 3.1 that if we assume n − k ≥ n − k + 1 = 2(2 n + 1) − k −
1. Therefore, the familyof examples given by the manifolds M ( r ), for ρ ( r ) ≥
4, actually show that for n − k ≥
2, (4 n − k + 1) isthe optimal dimension of contact embedding.Using similar techniques as in Theorem 1.5 and Gromov’s h-principles for contact immersion and isocontactembedding (see 2.12 and 2.13) we prove the following result for parallelizable manifolds. Theorem 1.8.
Let M n +1 be a parallelizable manifold.(1) For any contact structure ξ on M n +1 , ( M n +1 , ξ ) contact immerses in ( R n +3 , ξ std ).(2) If M n +1 is 5-connected, then for n ≡ , mod
4) and n ≥
7, any contact structure ξ , ( M n +1 , ξ )has an isocontact embedding in ( R n − , ξ std ). Corollary 1.9.
Let M n +1 = N n − × ( S × S ). Where N n − is a π –manifold that embeds in R N +1 with trivial normal bundle. Then M n +1 contact embeds in ( R N +5 , ξ std ).In [BEM], S. Borman, Y. Eliashberg and E. Murphy defined the notion of an overtwisted contact ballin all dimensions. Any contact structure that admits a contact embedding of such an overtwisted ball iscalled an overtwisted contact structure. These contact structures were shown to satisfy the h-principle forhomotopy of contact structures. For details see Theorem 2 .
8. Using this, we prove a uniqueness result forembedding of certain π -manifolds in an overtwisted contact structure η ot on R N +1 , analogous to Theorem1 .
25 in [EF].
Theorem 1.10.
Let ( M k +3 , ξ ) be a contact π –manifold such that H i ( M ; Z ) = 0, for i ≡ , , , mod ι , ι : ( M k +3 , ξ ) → ( R N +1 , η ot ) be two isocontact embeddings with trivial conformal symplectic normalbundle such that both the complements of ι ( M ) and ι ( M ) in ( R N +1 , η ot ) are overtwisted. If ι and ι aresmoothly isotopic, then there is a contactomorphism χ : ( R N +1 , η ot ) → ( R N +1 , η ot ) such that χ · ι = ι .For example, any two isocontact embeddings of ( S k × S k +3 , ξ ) in ( R k +8 k +5 , η ot ) which satisfy thehypothesis of Theorem 1 .
10, are equivalent.We would like to mention that the problem of isocontact embedding of 3-manifolds in R has seen muchdevelopment in the past few years. The first result in this direction was given by Kasuya. In [Ka2], heproved that given a contact 3-manifold ( M, ξ ), the first Chern class c ( ξ ) is the only obstruction to isocontactembedding of ( M, ξ ) in some contact structure on R . The approach in [Ka2] was a motivation for the presentarticle. Afterwards, Etnyre and Furukawa [EF] showed that a large class of contact 3-manifolds embed in KULDEEP SAHA ( R , ξ std ). Recently, the existence and uniqueness question for co-dimension 2 isocontact embedding hasbeen completely answered by the works of Pancholi and Pandit [PP], Casals, Pancholi and Presas [CPP],Casals and Etnyre [CE] and Honda and Huang [HoH]. On the other hand explicit examples of co-dimension2 isocontact embeddings were produced in the works of Casals and Murphy [CM], Etnyre and Lekili [EL]and in [S].1.1. Acknowledgment.
The author is grateful to Dishant M. Pancholi for his help and support duringthis work. He would like to thank Suhas Pandit for reading the first draft of this note and for his helpfulcomments. He also thanks John Etnyre for clarifying some doubts regarding the proof of Theorem 1 .
25 in[EF]. Finally, he thanks the referee for various comments and suggestions which helped improve the article.The author is supported by the National Board of Higher Mathematics, DAE, Govt. of India.2. preliminaries
In this section we review some basic notions and results that will be relevant to us.2.1. h-principle for immersion.
Let f, g : M n +1 V N +1 be two immersions. We say that f is regularlyhomotopic to g , if there is a family h t : M V of immersions joining f and g . Being an immersion, f induces Df : T M → T V such that for all p ∈ M , Df restricts to a monomorphism Df p from T p M to T f ( p ) V . Definition 2.1 (Formal immersion) . A formal immersion of M in V is a bundle map F : T M → T V thatrestricts to a monomorphism F p on each tangent space T p M , for p ∈ M . It can be represented by thefollowing diagram, where f is any smooth map making it commutative. T M F −−−−→ T V y π y π M f −−−−→ V We say that F is a formal immersion covering f . The existence of a formal immersion from T M in T V is a necessary condition for the existence of an immersion of M in V . Definition 2.2 (Homotopy between formal immersions) . Two formal immersions, F and G are calledformally homotopic (or just homotopic) if there is a homotopy H t : T M → T V of formal immersions suchthat H = F and H = G .Two immersions f and g are called formally homotopic if Df and Dg are homotopic as formal immersions.To be precise there exists a formal homotopy H t covering a smooth homotopy f t joining f and g such that H = Df and H = Dg . Assume that dim ( V ) ≥ dim ( M ) + 1. Let Imm ( M, V ) denote the set of allimmersions of M in V and let M ono ( T M, T V ) denote the set of all formal immersions. Let I : Imm ( M, V ) → M ono ( T M, T V ) be the inclusion map given by the tangent bundle monomorphism induced by an immersion.
Theorem 2.3 (The Smale-Hirsch h-principle for immersion) . ([Hi]) The map I is a homotopy equivalence.So, I induces set bijection from π ( Imm ( M, V )) to π ( M on ( T M, T V )). This implies that the existenceof a formal immersion is also sufficient for the existence of an immersion. Moreover, the isomorphism that I induces from π ( Imm ( M, V )) to π ( M on ( T M, T V )) implies that if f and f are two immersions whichare formally homotopic, then they are regularly homotopic.We now discuss the obstruction theoretic problem for the existence of a formal immersion and the classifi-cation of formal immersions. For more details on immersion theory we refer to [Hi]. For related terminologiesand notions from fiber bundle and obstruction theory and we refer to [St]. Obstructions to formal immersion and homotopy:
Given a manifold N , T k N will denote thebundle of k -frames associated to T N . A formal immersion of M n into R N defines an SO ( n )–equivariantmap from T n M to V N,n . Here, V N,n denotes the real Stiefel manifold consisting of all oriented n -frames ONTACT AND ISOCONTACT EMBEDDING OF π –MANIFOLDS 5 in R N . According to [Hi], the homotopy classes of the formal immersions are in one-one correspondencewith the homotopy classes of such SO ( n )–equivariant maps to V N,n , i.e., with homotopy classes of crosssections of the associated bundle of T n M with fiber V N,n . Thus, the problem is reduced to looking at theobstructions to the existence of a section s of this associated bundle. Such obstructions lie in the groups H i ( M n ; π i − ( V N,n )), for 1 ≤ i ≤ n .Moreover, two formal immersions F and G are homotopic if and only if their corresponding sections s F and s G to the associated V N,n –bundle are homotopic. Thus, the homotopy obstructions between two formalimmersions F and G lie in H i ( M n ; π i ( V N,n )) for 1 ≤ i ≤ n .2.2. Almost contact structure.
We recall the notion of almost contact structure.
Definition 2.4.
Consider a real vector bundle p : E → B of rank 2 m . An almost complex structure on E is a smooth assignment of automorphisms J p : F p → F p for all point p in B such that J p = − Id .The set of all complex structures on R m is homeomorphic to Γ m = SO (2 m ) /U ( m ) ([Ge], Lemma 8 . . E is equivalent to the existence of a section of theassociated Γ m -bundle. Definition 2.5.
An almost contact structure on an odd dimensional manifold N n +1 is an almost complexstructure on its stable tangent bundle T N ⊕ ε N .Thus, an almost contact structure on N is an almost complex structure on N × R . So every almost contactstructure on N is given by a section of the associated Γ n +1 –bundle of T ( N × R ). Definition 2.6.
Two almost contact structures are said to be in the same homotopy class, if their corre-sponding sections to the associated Γ n +1 –bundle are homotopic.The existence of an almost contact structure on N is a necessary condition for the existence of a contactstructure. For open manifolds, Gromov ([Gr]) proved the following h-principle showing that this conditionis also sufficient. Theorem 2.7. (Gromov, [Gr]) Let K be a sub-complex of an open manifold V . Let ¯ ξ be an almost contactstructure on V which restricts to a contact structure in a neighborhood Op ( K ) of K . Then one can homotope¯ ξ , relative to Op ( K ), to a contact structure ξ on V .For closed manifolds, the corresponding h-principle follows from the work of Borman, Eliashberg andMurphy ([BEM]). In particular, they showed that in every homotopy class of an almost contact structurethere is at least one contact structure called overtwisted (see [BEM]). Two such overtwisted contact struc-tures are isotopic if and only if they are homotopic as almost contact structures. [BEM] gives a parametricversion of Theorem 2.7 that holds for both open and closed contact manifolds. Theorem 2.8 (Borman, Eliashberg and Murphy, [BEM]) . Let K ⊂ M n +1 be a closed subset. Let ξ and ξ be two overtwisted contact structures on M that agree on some Op ( K ). If ξ and ξ are homotopic asalmost contact structures over M \ K , then ξ and and ξ are homotopic as contact structures relative to Op ( K ).So, by Gray’s stability, ξ and ξ are isotopic contact structures.The obstructions to the existence of an almost contact structure on N n +1 lie in the groups H i ( N ; π i − (Γ n +1 )),for 1 ≤ i ≤ n + 1. The homotopy obstructions between two almost contact structures on N lie in the groups H i ( N ; π i (Γ n +1 )), for 1 ≤ i ≤ n + 1. The stable homotopy groups of Γ n were computed by Bott ([B]). Theorem 2.9 (Bott, [B]) . For q ≤ n − π q (Γ n ) = π q +1 ( SO ) = q ≡ , , , Z for q ≡ , Z for q ≡ , KULDEEP SAHA
Next, we define the notion of a null-homotopic almost contact structure on π –manifold. Definition 2.10.
When M n +1 is a π –manifold, T M ⊕ ε M ∼ = ε n +2 M always admits a trivial almost complexstructure that assigns over each point of M the standard complex structure J ( n + 1) on C n +1 . Thecorresponding homotopy class of almost contact structures on M is called null-homotopic.2.3. Obstructions to contact immersion.
Similar to the notion of formal immersion one can define a formal contact immersion or a contact monomorphism . Definition 2.11.
A formal immersion F : T M → T V of (
M, ξ ) in (
V, η ) is called a contact monomorphismif F ( ξ ) = F ( T M ) ∩ η .Gromov ([Gr]) proved the following h-principle for contact immersions. Theorem 2.12 (Gromov) . Let (
M, ξ ) and (
V, η ) be contact manifolds of dimensions 2 n + 1 and 2 N + 1respectively. Assume that n ≤ N −
1. A contact monomorphism F : T M → T V , covering an immersion f : M → V , is formally homotopic to F = df for some contact immersion f : M → V .For contact embedding, Gromov ([Gr]) proved the following theorem. The statement here is taken from[EM]. Theorem 2.13 (Gromov) . Let (
M, ξ ) and (
V, η ) be contact manifolds of dimension 2 n + 1 and 2 N + 1respectively. Suppose that the differential F = Df of an embedding f : ( M, ξ ) → ( V, η ) is homotopic viaa homotopy of monomorphisms F t : T M → T V covering f to a contact monomorphism F : T M → T V .(1) Open case: If n ≤ N − M is open, then there exists an isotopy f t : M → V suchthat the embedding f : M → W is contact and the differential Df is homotopic to F throughcontact monomorphisms.(2) Closed case: If n ≤ N −
2, then the above isotopy f t exists even if M is closed.Thus, if an embedding is regularly homotopic to a contact immersion, then it is isotopic to a contactembedding. We now discuss the analogous obstruction problem for the existence of a contact monomorphism. Obstructions to contact monomorphism :
Consider a symplectic vector space (
X, ω ). Let J be an ω -compatible almost complex structure (i.e., ω ( Ju, Jv ) = ω ( u, v ) and ω ( u, Ju ) > u, v ∈ X \ { } ).If Y is a symplectic subspace of ( X, ω ), then Y has to be a J -subspace of ( X, J ) and vice-versa. An al-most complex structure J ξ on the contact hyperplane bundle ξ = Ker { α } is called ξ –compatible, if it iscompatible with the conformal symplectic structure on ξ induced by dα . A contact monomorphism takes ξ to a symplectic sub-bundle of η . If J η is an η –compatible almost complex structure, then the contactmonomorphism takes ξ to a J η -sub-bundle of ( η, J η ). So, finding a contact monomorphism from ( T M, ξ ) to( T R N +1 , η ) is equivalent to finding a U ( n )–equivariant map from the complex n -frame bundle associatedto ξ to V C N,n . In other words, finding a contact monomorphism is equivalent to the existence of a section ofthe associated V C N,n –bundle of
T M (see 2 . V C N,n denotes the complex Stiefel manifold. Thus,(
M, ξ ) has a contact monomorphism in (
V, η ) if and only if all the obstructions classes in H i ( M ; π i − ( V C N,n ))vanish for 1 ≤ i ≤ n + 1. From contact immersion to contact embedding :
In the previous sections, we saw that any for-mal immersion of M n +1 in R N +1 is given by a section s F of the associated V N +1 , n +1 -bundle of T M .For a contact monomorphism F C , let s F C denote the corresponding section to the associated V C n ( η )–bundle. s F C also induces a section map s C to the associated V N +1 , n +1 -bundle via the inclusion V C N,n ⊂ V N, n ⊂ V N +1 , n +1 . Let ι : M ֒ → V be an embedding and let s ι denote the corresponding section to the associated V N +1 , n +1 -bundle. The homotopy obstructions between s ι and s C lie in the groups H i ( M ; π i ( V N +1 , n +1 )),for 1 ≤ i ≤ n + 1. If all of these obstructions vanish, then by Theorem 2.13, ι can be isotoped to a contactembedding. ONTACT AND ISOCONTACT EMBEDDING OF π –MANIFOLDS 7 For example, let (
V, η ) = ( R n +3 , ξ std ). Note that V n +3 , n +1 is (2 n +1)-connected and V C n +1 ,n is (2 n +2)-connected. So, all of the groups H i ( M ; π i − ( V C n +1 ,n ) and H i ( M ; π i ( V N +1 , n +1 ) vanish, for 1 ≤ i ≤ n + 1.By the Whitney embedding theorem, any smooth (2 n + 1)-manifold embeds into R n +3 . Thus, we getthe result of Gromov ([Gr]) saying that every contact manifold ( M n +1 , ξ ) has an isocontact embedding in( R n +3 , ξ std ). Remark 2.14.
When the embedding co-dimension is ≤ dim ( M ) −
1, there is a natural topological obstruc-tion to contact embedding. It has the following description. If ι : ( M n +1 , ξ ) ֒ → ( R N +1 , η ) is a contactembedding, then the normal bundle ν ( ι ) = ι ∗ ( η ) /ξ has an induced complex structure on it. So we have thefollowing relation of total Chern classes. c ( ξ ⊕ ν ( ι )) = ι ∗ ( η ) = 1. Let ¯ c j ( ξ ) denote the j th order cohomology class in (1 + c ( ξ ) + c ( ξ ) + ... + c n ( ξ )) − . Since the Euler classof the normal bundle of an embedding in R N +1 is zero, c N − n ( ν ( ι )) = 0 ⇔ ¯ c N − n ( ξ ) = 0This gives a condition on the Chern classes of ξ . Thus, for isocontact embedding of co-dimension ≤ dim ( M ) −
1, one has to restrict the problem on the contact structures whose Chern classes satisfy this condition. Forisocontact embedding with trivial symplectic normal bundle, the following holds. ξ ⊕ ν ( ι ) ∼ = ξ ⊕ ε N − nM ( C ) = η | ι ( M ) ∼ = ε NM ( C )Thus, c i ( ξ ⊕ ε N − nM ( C )) = 0 ⇔ c i ( ξ ) = 0, for 1 ≤ i ≤ n . Therefore, ¯ c N − n ( ξ ) = 0.2.4. Smooth embeddings in Euclidean space.
The following generalization of Whitney embeddingtheorem is due to Haefliger and Hirsch ([HH]).
Theorem 2.15 (Haefliger-Hirsch) . If M n is a closed orientable k -connected n -manifold (0 ≤ k ≤ ( n − M n embeds in R n − k − .If one further assumes that M n is a π –manifold, then we have the following result due to Sapio ([Sa]). Theorem 2.16 (Sapio) . Let M n be a k -connected, n -dimensional π –manifold ( n ≥
5, and k ≤ [ n/ n mod M n almost embeds in R n − k − with a trivial normal bundle.(2) If M n bounds a π –manifold, then M n embeds in R n − k − with a trivial normal bundle.3. contact embedding via h-principle Note that
T N n L ǫ k +1 N ∼ = ǫ n + k +1 N ⇔ T N n L ǫ N = ǫ n +1 N (see corollary 1 .
4, p-70 of [Kos]). Therefore, M n +1 is a π -manifold if and only if M n +1 embeds in the Euclidean space R d with a trivial normal bundle,for some d ≥ n + 2. The following lemma is the main ingredient to prove Theorem 1.5 . Lemma 3.1.
If an almost contact manifold M n +1 embeds in R N +1 with a trivial normal bundle, thenthere exists a contact structure ξ such that ( M, ξ ) isocontact embeds into ( R N +3 , ξ std ) ( N − n ≥ Proof.
By assumption, there is an embedding ι : M ֒ → R N +1 with normal bundle of embedding ν ( ι ) trivial.Since M is a π –manifold, T M ⊕ ǫ M ∼ = ǫ n +2 M . So, any section to the associated Γ n +1 -bundle of T M ⊕ ǫ M is given by a homotopy class of map s : M → Γ n +1 . By [BEM], in every homotopy class of an almostcontact structure there is a genuine contact structure. Fix a homotopy class of an almost contact structureon M n +1 . Let ξ be a contact structure representing it. Let E ( ν ) denote the total space of ν ( ι ). Since ξ isco-orientable, T E ( ν ) ⊕ ε ∼ = T M ⊕ ν ( ι ) ⊕ ε ∼ = ξ ⊕ ε ⊕ ε N − n ) . We now define a contact structure on the tubular neighborhood E ( ν ) of M , such that its restriction to M iscontact. Let α be a contact form representing ξ . Let ( r , θ , r , θ , ..., r N − n , θ N − n ) be a cylindrical co-ordinatesystem on D N − n ) . The 1-form ˜ α = α + Σ N − ni =1 ( r i dθ i ) defines a contact structure on E ( ν ) ∼ = M × D N − n KULDEEP SAHA that restricts to the contact structure ξ on M . Let J ξ be an almost complex structure on the stable tangentbundle of T M that induces the contact structure ξ on M . Put the standard complex structure J ( N − n ) onthe normal bundle ν ( ι ) and define an almost complex structure on T E ( ν ( ι )) ⊕ ǫ E given by J ξ ⊕ J ( N − n ).Note that over each fiber of E ( ν ), d ˜ α restricts to the standard symplectic structure on D N − n compatiblewith J ( N − n ). So, the almost contact structure associated to ˜ α is the same as the almost contact structureinduced by J ξ ⊕ J ( N − n ). If we can extend this almost contact structure on E ( ν ) to all of R N +1 , then byTheorem 2.7, we will get a contact embedding of ( M, ξ ) into ( R N +1 , η ) for some contact structure η .Now we show how to extend the section s ξ : M → Γ N +1 , given by J ξ ⊕ J ( N − n ), to all of R N +1 . Theobstructions to such an extension lie in H i +1 ( R N +1 , M ; π i (Γ N +1 )) ∼ = H i ( M ; π i (Γ N +1 )), for 1 ≤ i ≤ n + 1.Consider the section s η : M → Γ N +1 induced by η | ι ( M ) , for some contact structure η on R N +1 . If s ξ ishomotopic to s η , then the obstructions vanish and s ξ extends to all of R N +1 .Let In : Γ n +1 ֒ → Γ N +1 be the inclusion map given by J ( n + 1) J ( n + 1) ⊕ J ( N − n ). Consider thefibration Γ m j m −−→ Γ m +1 → S m [Ha]. Here, j m denotes the inclusion map that sends J ( m ) to J ( m ) ⊕ J (2).From this we get the following long exact sequence. ... −→ π i +1 ( S m ) −→ π i (Γ m ) −→ π i (Γ m +1 ) −→ π i ( S m ) −→ ... It follows that j m induces isomorphism on π i for i ≤ m − i = 2 m − n +1 j N ◦ j N − ◦ ... ◦ j n +1 −−−−−−−−−−−→ Γ N +1 is the same as the one defined by In . Here, j n +1 induces isomorphism on π i for i ≤ n and onto homomorphism for i = 2 n + 1. For l ≥ n + 2, j l ’sinduce isomorphisms on π i ’s, for all i ≤ n + 1. Therefore, In induces isomorphism on the i th -homotopygroup, for 1 ≤ i ≤ n and onto homomorphism for i = 2 n + 1. So, we can choose the homotopy class of ξ so that the homotopy class of the image of the corresponding section s ξ : M → Γ n +1 under In is thesame as the homotopy class of the map s η : M → Γ N +1 . Thus, there is a contact structure ξ on M suchthat the corresponding section s ξ : M → Γ N +1 extends to all of R N +1 . Therefore, ( M, ξ ) contact embedsin ( R N +1 , η ) for some contact structure η on R N +1 . By Theorem 2.13, ( R N +1 , η ) contact embeds in( R N +3 , ξ std ). Hence, ( M, ξ ) isocontact embeds in ( R N +3 , ξ std ). (cid:3) Remark 3.2.
For n ≥
4, the groups π n +1 (Γ n +1 ) have the following values [Ha]:(1) π n +1 (Γ n +1 ) = Z ⊕ Z for n ≡ Z ( n − for n ≡ Z for n ≡ Z ( n − for n ≡ π n +1 (Γ N +1 ) is either 0 or Z . So, for n ≥
4, the onto map induced by In on π n +1 has anon-trivial kernel. Thus, we can actually choose a homotopy class of almost contact structures on M whichis not null-homotopic and which isocontact embeds in ( R N +3 , ξ std ). Proof of Theorem 1.5 . Since every homotopy sphere Σ n +1 admits an almost contact structure, M n +1 ♯ Σ n +1 also admits an almost contact structure. By statement 2 of the Theorem 2.16, if M n +1 bounds a π –manifoldthen it satisfies the hypothesis of Lemma 3 .
1, for N = 2 n − k . By statement 1 of the Theorem 2.16, when-ever M n +1 does not bound a π –manifold, we can take connected sum with a suitable homotopy sphere andembed the resulting π –manifold in R n − k +1 with a trivial normal bundle. This hypothesis on the normalbundle of embedding is the only thing that we need to prove the present theorem. Therefore, it is enoughto prove statements (1) to (3), for manifolds that bound π –manifold. (1) The result follows from Lemma 3 . (2) As discussed in the proof of Lemma 3 .
1, given any contact structure ξ on M n +1 and an embedding ι : M → R n − k +1 with a trivial normal bundle ν , the obstructions to extend the almost contact structure on E ( ν ) to all of R n − k +1 , lie in the groups H i ( M ; π i (Γ n − k +1 )), for 1 ≤ i ≤ n + 1. Since M is k -connected,there are no obstructions in dimensions 1 to k . Being k -connected also implies that M r { pt. } deformationretracts onto the (2 n − k )-skeleton of M . So, the obstructions in dimensions (2 n − k + 1) to 2 n also vanish. ONTACT AND ISOCONTACT EMBEDDING OF π –MANIFOLDS 9 Thus, we now only consider values of i in { k + 1 , k + 2 , ..., n − k + 1 } . Since k ≤ n −
1, by the theorem ofBott (2.9), we get the following for 1 ≤ i ≤ n + 1. π i (Γ n − k +1 ) = f or i ≡ , , , mod Z f or i ≡ , mod Z f or i ≡ , mod i ≡ , , , mod i ≡ , , , mod H i ( M n +1 , ˜ G ) ∼ = H n − i +1 ( M n +1 , ˜ G ) = 0 by hypothesis. Here, ˜ G is either Z or Z . Moreover, for n mod π n +1 (Γ n − k +1 ) = 0. Hence, there is no obstructions in the top dimension. Thus, we can extend thealmost contact structure for any ξ and the result follows. (3) Note that every assumption in statement (2) holds for statement (3), except that now we have H i ( M n +1 , ˜ G ) ∼ = H n − i +1 ( M n +1 , ˜ G ) = 0 for i ≡ , mod i ≡ , mod .
1, both s ξ and s η factorsthrough the map ˜ j in the fibration U ( N + 1) → SO (2 N + 2) ˜ j N +1 −−−→ Γ N +1 . Since η was a contact structure on R N +1 , the assertion is clear for s η . The reason for s ξ is the following.Since ξ is an SO -contact structure, the almost contact structure associated to ξ , M → Γ n +1 , factors throughthe map SO (2 n + 2) ˜ j n +1 −−−→ Γ n +1 . Let ˆ i : SO (2 n + 2) → SO (2 N + 2) denote the inclusion map given by A A ⊕ I N − n ) . Recall that ˜ j takes a matrix A ∈ SO (2 m + 2) to A − J ( m + 1) A ∈ Γ m +1 . The assertionthen follows from the commutative diagram below. SO (2 n + 2) ˆ i −−−−→ SO (2 N + 2) y ˜ j n +1 y ˜ j N +1 Γ n +1 In −−−−→ Γ N +1 Thus, the homotopy obstructions come from the groups H i ( M ; π i ( SO (2 N + 2))). Since π i ( SO ) = 0 for i ≡ , mod .
1, we get an isocontactembedding of (
M, ξ ) into ( R n − k +3 , ξ std ). (cid:3) Proof of Corollary 1.6 . (1) Follows from Theorem 1.5 by putting k = n − (2) Following the proof of Theorem 1.5, we can see that the only obstructions to extending the almost con-tact structure on the normal bundle to all of R n +3 lie in the groups H n ( M ; π n ( SO )) and H n +1 ( M ; π n +1 ( SO )).Since both π n ( SO ) and π n +1 ( SO ) vanish for n ≡ , mod (cid:3) Proof of Theorem 1.8 . (1) The existence of a contact monomorphism from (
T M n +1 , ξ ) to ( T R N +1 , η st )is equivalent to the existence of a section s : M → V C N,n of the associated bundle of
T M . Since
T M is trivial,such a section always exists. Since any parallelizable manifold M n +1 immerses in R n +3 and has a contactmonomorphism in ( T R n +3 , ξ std ), by Theorem 2.12, ( M, ξ ) isocontact immerses in ( R n +3 , ξ std ). (2) Any section corresponding to a contact monomorphism also induces a section s to the associated V N +1 , n +1 -bundle of T M . Assume that M is (2 k − f : M n +1 → R n − k +3 . Let s f be the corresponding section to V n − k +3 , n +1 . The homotopyobstructions between s and s f lie in the groups H i ( M n +1 , π i ( V n − k +3 , n +1 )), for 1 ≤ i ≤ n + 1. Since M n +1 r D n +1 deformation retracts onto the (2 n − k + 1)-skeleton of M and V n − k +3 , n +1 is (2 n − k + 1)-connected, there are no obstructions till dimension 2 n . Therefore, the only homotopy obstruction lies in H n +1 ( M n +1 , π n +1 ( V n − k +3 , n +1 )). By [HM], for k = 3, and n ≡ , mod π n +1 ( V n − , n +1 ) = 0 .Thus, by Theorem 2.13, ( M n +1 , ξ ) has an isocontact embedding in ( R n − , ξ std ). (cid:3) Note that the proof of statement (2) in Theorem 1.8 does not necessarily require a parallelizable manifold.In general, the following can be said.
Proposition 3.3.
A 5-connected contact manifold ( M n +1 , ξ ) admits an isocontact embedding in ( R n − , ξ std )for n ≥ n ≡ , mod R n − , ξ std ). Proof of Corollary 1.9 . Consider R N as R N − × R . It is well known that R N \ ( R N − × { } ) can bedecomposed as R N − × S . This is the so called standard open book decomposition of R N (see [Ge] or [E]).Say, M is embedded in R N − . Then using this open book description we can see that M × S naturallyembeds in R N − × S ⊂ R N . Starting with an embedding of N n − in R N +1 with trivial normal bundle,we can then apply this procedure twice to get an embedding of N n − × ( S × S ) in R N +3 with trivialnormal bundle. The result then follows from Lemma 3 . (cid:3) Contact embedding of co-dimension ≥ . For embeddings of co-dimension ≥
4, we can actuallyget isocontact embedding in the standard contact structure. Let ι be an isocontact embedding of ( M, ξ )in ( R N +1 , η ). Any two contact structures on R N +1 are homotopic as almost contact structures. Let H t : T R N +1 → T R N +1 be a formal homotopy covering the identity map of R N +1 such that H = Id , H ( η ) = ξ std and H t ( η ) is an almost contact structure on R N +1 for all t ∈ (0 , H t · Dι gives aformal homotopy covering ι and H · Dι is a contact monomorphism of ( T M, ξ ) into ( T R N +1 , ξ std ). If weassume that N − n ≥
2, then by Theorem 2.13, ( M n +1 , ξ ) isocontact embeds in ( R N +1 , ξ std ). Thus, forembedding of co-dimension ≥
4, we also get contact embedding in the standard contact structure. Moreover,this shows that for n − k ≥
2, we can improve the dimension of embedding in Theorem 1.5 from 4 n − k + 3to 4 n − k + 1. Using this fact with Lemma 3 .
1, one can find interesting examples of contact embeddingof non-simply connected manifolds. For example, by [MR], the 7-dimensional real projective space RP embeds in R with trivial normal bundle. Thus, RP contact embeds in ( R , ξ std ). Let us look at anothersimple class of such examples. It is a well known theorem of Hirsch that every oriented, closed 3-manifold M embeds in R . By [CS], every closed, oriented 4-manifold V , whose second Stiefel-Whitney class andsignature vanish, embeds in R . Since, the Euler class of the normal bundle of an embedding vanishes, eachof these embeddings has trivial normal bundle. Thus, W = M × V embeds in R with trivial normalbundle. Hence, W contact embeds in ( R , ξ std ).We now prove Theorem 1.10. The idea of the proof is essentially contained in [EF] (proof of Theorem1 . Proof of Theorem 1.10 . .Since ι and ι are contact embeddings with trivial normal bundle, there exists a contact form α repre-senting ξ and suitable neighborhoods N j of ι j ( M ) in ( R N +1 , η ot ), for j = 1 ,
2, which are contactomorphicto ( M × D N − n ) , α + Σ N − ni =1 r i dθ i ). Now, ι and ι are isotopic and have isomorphic symplectic normalbundle. Thus, one can use the contact tubular neighborhood theorem for contact submanifold (Theorem2 . .
15, [Ge]) to get an ambient isotopy Φ t : R N +1 −→ R N +1 such that Φ restricts to a contactomor-phism from ( N , η ot | N ) to ( N , η ot | N ) and Φ ( ι ( M )) = ι ( M ). Moreover, since we can assume that theisotopy between ι and ι is supported in the complement of an overtwisted contact ball, Φ t can be cho-sen so that it restricts to the identity map on that overtwisted contact ball in the complement of N in( R N +1 , η ot ). Thus, the distribution (Φ ) ∗ η ot induces a contact structure on R N +1 that is overtwisted inthe complement of N . Now, we look at the homotopy obstructions between (Φ ) ∗ η ot and η ot relative to N . All such obstructions lie in H i ( R N +1 , N ; π i (Γ N +1 )), for 1 ≤ i ≤ n + 1. By Theorem 2.9, the group π i (Γ N +1 ) vanish for i ≡ , , , mod H i ( R N +1 , N ; π i (Γ N +1 )) ∼ = H i − ( M ; π i (Γ N +1 )) ∼ = H n +2 − i ( M ; π i (Γ N +1 )). The assumptions that 2 n + 1 is of the form 8 k + 3 and that the homology groups of M vanish for i ≡ , , , mod
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