A'campo, N.

Kotschick, D.

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CONTACT STRUCTURES, FOLIATIONS, AND THEFUNDAMENTAL GROUPN. A'CAMPO AND D. KOTSCHICK1. IntroductionIt was proved in [12] that every finitely presentable group is the fundamentalgroup of an orientable closed 5-manifold which admits an almost contact structure.In Section 2 of this paper we use a recent result of A. Weinstein [18] to prove thefollowing stronger result.THEOREM 1. Every finitely presentable group is the fundamental group of anorientable closed ^-dimensional contact manifold.The corresponding result is trivial in dimensions 2«— 1 ^ 7 , because for n ~£ 4every finitely presentable group can be realized as the fundamental group of anorientable closed w-manifold M. The unit cotangent bundle of an arbitraryRiemannian metric on M has the same fundamental group and a tautological contactstructure.Thinking of a contact structure as a completely non-integrable hyperplanedistribution, there is a question dual to the one answered above: Does the existenceof an integrable ^-dimensional distribution, that is, a foliation, on an orientableclosed w-manifold M restrict the fundamental group of M, or not? The answer is asfollows.THEOREM 2. Ifn^5 and n^k^0orn = 4 and ke{0,2,4}, then every finitelypresentable group is the fundamental group of an orientable closed n-dimensionalmanifold with a k-dimensional foliation of class C00 with trivial Godbillon-Veyinvariant.The proof of Theorem 2, given in Section 3, is elementary and explicit in all cases.Even in the codimension 1 case we do not use Thurston's deep existence theorem [17].As with Theorem 1, the result is trivial in dimensions n ^ 7, and the interest lies insorting out the low-dimensional cases.Theorem 2 is sharp in several respects, as follows.REMARK 1.1. If one requires the foliations to be analytic rather than smooth,then a result of A. Haefliger [8] shows that the fundamental group is restricted fork = n— 1, in that no finite groups are possible.Received 20 July 1992.1991 Mathematics Subject Classification 57M05.This paper was presented at the Joint Meeting of the American and London Mathematical Societiesin Cambridge, June/July 1992.Bull. London Math. Soc. 26 (1994) 102-106CONTACT STRUCTURES, FOLIATIONS, AND 7^ 103REMARK 1.2. Considering smooth foliations defined globally by a closeddifferential form restricts the fundamental group if k = n — 1, because in that case thefirst Betti number is positive. (We owe this remark to A. Haefliger.)REMARK 1.3. A closed oriented n-manifold carries a /c-dimensional foliation fork = 1 or n— 1 only if its Euler characteristic vanishes. For « = 4 this imposes non-trivial restrictions on the fundamental group. Again, the first Betti number has to bepositive, but this is not sufficient; compare [10].The results of this paper (and of [16, 18]) can be thought of as an instance ofGromov's Homotopy Principle [7] applied to contact structures and to foliations.For orientable closed manifolds of dimension 3, not all finitely presentable groupsarise as fundamental groups. However, those which do arise do not obstruct theexistence of contact structures [14] or of foliations [13]. Our proof of Theorem 1 is ageneralization of J. Martinet's argument [14] for the 3-dimensional case.2. Proof of Theorem 1Recall that a submanifold Y of a contact manifold X is said to be isotropic if itstangent bundle is contained in the contact distribution. The contact distribution hasa natural conformal symplectic structure, and we denote by JL the symplecticorthogonal with respect to this structure. If Y is isotropic, then TY a (TY)1, so thatthe conformal symplectic normal bundle CSN(X, Y) = (TYY/TY is defined.Weinstein [18] proved that the manifold obtained from X by surgery on an isotropic/c-sphere with trivialized conformal symplectic normal bundle is again a contactmanifold.Now let F be a finitely presented group. For each generator we take a copy ofS1 x S4 with its standard contact structure, and form the connected sum X as acontact manifold using the case k = 0 of Weinstein's theorem, or [15]. Each relationof the presentation corresponds to a unique homotopy class of a map S1 -* X, and wecan represent these classes by disjoint embedded circles. Let Y be the union of thesecircles. According to the lemma below, we may assume that Y is an isotropicsubmanifold. The conformal symplectic normal bundle CSN(Ar, Y) is of rank 2, andit follows from the connectedness of Sp (2) that it is trivial. Performing surgery andusing the case k = 1 of Weinstein's theorem, we obtain an orientable contact 5-manifold with fundamental group P.This completes the proof of Theorem 1 modulo the following lemma.LEMMA 2.1. Let X be an orientable contact manifold, and <f>\Sx -*• X an embedding.Then (/> is isotopic to an embedding with isotropic image.Proof. The case dimZ= 3 is a special case of Lemma 2.2.1 in [2].The general case can be reduced to this as follows. Let a be a contact formon X. Choose a vector field v along (j> such that a^t)(v^t)) = 1, with vm and <j>(t)linearly independent for all teS1. Choose a second vector field w along 0 such thatV)K<o) = ° and104 N. A'CAMPO AND D. KOTSCHICKNow let M be a 3-dimensional submanifold of a tubular neighbourhood of <j>{Sl) inX, with ^ ( 5 ' ) c M and T^t)M = Spanfu^, w^t),<p(t)}. After shrinking M, ifnecessary, a|M is a contact form on M. An isotopy of 0 in M to an embedding withisotropic image exists by [2] and produces the desired isotopy.REMARK 2.1. Instead of using Weinstein's theorem [18], we could phrase theargument in the framework of Eliashberg [3]. This is completely equivalent, andshows that the contact structures so constructed are holomorphically fillable.REMARK 2.2. After we had found the above proof of Theorem 1, R. E. Gompf[6], in reply to a question from the authors, proved the analogous theorem forsymplectic 4-manifolds. That case is much more difficult. Using the argument of [12],it yields another proof of Theorem 1.3. Proof of Theorem 2All the manifolds considered in this section are orientable, closed and smooth.Let F be a finitely presentable group. We shall say that T is («, ^ -realizableif it is (isomorphic to) the fundamental group of an H-manifold with a smoothA>dimensional foliation. Each of the cases in Theorem 2, except the trivial ones(k e {0,«}), is covered by at least one of the following five existence lemmas. Lemma 3.6proves that the foliations constructed have trivial generalized Godbillon-Veyinvariants, completing the proof of Theorem 2.LEMMA 3.1. Ifn^-5, then V is (n, \)-realizable.Proof One can easily construct an w-manifold X with fundamental groupn^X) = F and Euler characteristic x(X) — 0- Then, by the Poincare-Hopf theorem,X has a vector field without zeros. Its integral curves yield the desired foliation.LEMMA 3.2. Ifn^l and 2^k^ n-2 or n = 6 and ke{2,4}, then T is (n,k)-realizable by a product foliation.Proof The assumptions imply m = max {k, n — k} ^ 4 and mm {k,n—k) ^ 2.Thus we can find an w-manifold X with nx(X) = V, such that X x Sn~m shows that Tis (n, ^-realizable by a product foliation.LEMMA 3.3. Ifn^S, then T is (n,n — ^-realizable.Proof. Let X be a 4-manifold with n^X) s F. By replacing X with its connectedsum with a number of copies of CP2 if necessary, we may assume x(X) >The manifold X has a vector field v with zeros Plt P2,...,PX, all of index +1. LetD\,D\, ...,D* be a collection of small disjoint disks centred at the points Pv Weidentify these with the standard unit disk in C2 and denote by Sf c D\ the sphere ofradius \. In the outside collar of Sf we can deform the vector field v to make its flowlines asymptotic to the fibres of the Hopf fibration of Sf. Doing this near each Sfproduces a non-singular 1-dimensional foliation J5" on the complement of the disksof radius | around the zeros off, tangent and restricting to the Hopf fibration on eachboundary component.CONTACT STRUCTURES, FOLIATIONS, AND nx 105The complex coordinates around Pt can be used to 'blow up' these points in astandard way. This replaces a disk of radius \ around Pt by a Z)2-bundle Ei over S2.This does not change the fundamental group, because topologically it amounts toforming the connected sum with one copy of CP2 at each Pt. We now replace X withthis connected sum.Let n: Y -*• X be an ^-bundle which has degree 1 on one 2-sphere free in n2(X)and is trivial otherwise. The exact sequence of homotopy groups induced by nshows n1(Y)^n1(X)(^r). The pullback n*{fF) is a 2-dimensional foliation ony\(|J (Et x S1)), tangent and restricting to the product of the Hopf fibration with S1on each boundary component. One can fill in standard Reeb foliations on each fibreD2 x S1 of each Ei x S1 to obtain a smooth 2-dimensional foliation on Y. This showsthat F is (5,2)-realizable.We have rank7r2(F) = rankn2(X) — 1 ^ n — 5. If n > 5, we can take another circlebundle Z -> Ywith nx(Z) = nx{Y) and rank7r2(Z) = rankTT2(Y) — 1, and pull back thefoliation. Iteration shows that T is («,« —3)-realizable.LEMMA 3.4. If« ^ 4, then T is (n,n — 2)-realizable.Proof. Let X be an oriented 4-manifold with nx(X) = T, and X the samemanifold with the opposite orientation. It was proved in [11] that there exists an asuch that the connected sum Y = X#aCP2 has an almost complex structure. Apply-ing the same argument to P, we find that there is a b such that Z = X% aCP2 # bCP2and Z both have almost complex structures. (The construction of the second structuredoes not destroy the first one because forming the connected sum with CP2, unlikewith CP2, is a complex operation; compare the proof of the previous lemma.)This pair of almost complex structures for Z and Z corresponds to a 2-planebundle in TZ; compare [9]. According to Thurston [16], any 2-plane bundle can behomotoped to a completely integrable one. Thus T is (4,2)-realizable, and (n,n — 2)-realizability follows by pulling back this foliation to the total space of iterated S1-bundles, as in the proof of the previous lemma.LEMMA 3.5. Ifn^S, then T is (n,n-\)-realizable. Ifn^l, then the realizationcan be done by pulling back the standard 2-dimensional foliation of S3.Proof. If n ^ 7, then there is an (« — 3)-dimensional manifold Zwith nx{X) = T.On Xx Sz we pull back the standard foliation from the second factor.For the general case, let X be a 4-manifold with n^X) s T and n2(X) large. Wefoliate X x Sl by the product foliation X x {/}, / e S1. From this we obtain a 5-manifoldY with n^Y) = T and a codimension 1 foliation J5" by surgery on {pt} x S1: foliate(A \^£)4) x S1 by Reeb spiraling applied to the above product foliation so that theboundary Sz x S1 becomes a leaf, and fill in the foliation of Sz x D2 obtained in [1,Corollaire 1]. This shows that T is (5,4)-realizable.The general case now follows by pulling back 8F to the total space of an iteratedS^-bundle as in the proof of Lemma 3.3.The Godbillon-Vey invariant referred to in Theorem 2 is the well-knowngeneralization of the definition in [5]. If J5" is a fc-dimensional foliation on an n-106 N. A'CAMPO AND D. KOTSCHICKmanifold X, then GV (J27) is a cohomology class in H2{n-h)+1(X, U), natural underpullback, which vanishes for elementary foliations like products, bundles and thestandard 2-dimensional foliation of S3.LEMMA 3.6. The foliations constructed in Lemmas 3.1-3.5 have trivial God-billon- Vey invariants.Proof. The foliations in Lemma 3.2 are products. The foliations in Lemma 3.1have trivial GV (J5") for dimension reasons, as do the ones in the basic cases n e {4,5}and k = 2 in Lemmas 3.3 and 3.4. In Lemma 3.5, the basic case for n ^ 7 is S3, so thatby the naturality of GV (#") there is nothing left to prove. The second constructiongiven in the proof of Lemma 3.5 leads to foliations almost without holonomy, so thatGV($F) = 0 by a result of G. Duminy; see [4].References1. N. A'CAMPO, ' Feuilletages de codimension 1 sur les varietes simplement connexes de dimension 5',Comment. Math. Helv. 47 (1972) 514-525.2. Y. ELIASHBERG, 'Classification of overtwisted contact structures on 3-manifolds', Invent. Math. 98(1989) 623-637.3. Y. ELIASHBERG, 'Topological characterization of Stein manifolds of dimension > 2', Internal. J.Math. 1 (1990) 29-46.4. E. GHYS, 'I/invariant de Godbillon-Vey', Seminaire Bourbaki, Aste'risque 177-178 (1989) 155-181.5. C. GODBILLON and J. VEY, ' Un invariant des feuilletages de codimension 1', C. R. Acad. Sci. Paris 273(1971) 92-95.6. R. E. GOMPF, 'Some new symplectic 4-manifolds', Doga Mat., to appear.7. M. GROMOV, Partial differential relations, Ergeb. Math. Grenzgeb. (3) (Springer, Berlin, 1986).8. A. HAEFLIGER, 'Structures feuilletees et cohomologie a valeurs dans un faisceau des groupoides',Comment. Math. Helv. 32 (1958) 248-329.9. F. HIRZEBRUCH and H. HOPF, ' Felder von Flachenelementen in 4-dimensionalen Mannigfaltigkeiten',Math. Ann. 136(1958) 156-172.10. F. E. A. JOHNSON, 'The fundamental group and homological restrictions in dimension four', Preprint.11. D. KOTSCHICK, 'All fundamental groups are almost complex', Bull. London Math. Soc. 24 (1992)377-378.12. D. KOTSCHICK, 'All fundamental groups are almost contact', Bull. London Math. Soc. 24 (1992)493-494.13. W. B. R. LICKORISH, 'A foliation for 3-manifolds', Ann. of Math. 82 (1965) 414-420.14. J. MARTINET, 'Formes de contact sur les varietes de dimension 3', Proceedings of LiverpoolSingularities Symposium II, Lecture Notes in Math. 209 (Springer, New York, 1971).15. C. MECKERT, 'Forme de contact sur la somme connexe de deux varietes de contact de dimensionimpaire', Ann. Inst. Fourier 32 (1982) 251-260.16. W. P. THURSTON, 'The theory of foliations of codimension greater than one', Comment. Math. Helv.49(1974)214-231.17. W. P. THURSTON, 'Existence of codimension-one foliations', Ann. of Math. 104 (1976) 249-268.18. A. WEINSTEIN, 'Contact surgery and symplectic handlebodies', Hokkaido Math. J. 20 (1991) 241-251.Mathematisches InstitutUniversitat BaselRheinsprung 214051 BaselSwitzerland

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