On foliated circle bundles over closed orientable 3-manifolds

We show that there exists a family of smooth orientable circle bundles over closed orientable 3-manifolds each of which has a codimension-one foliation transverse to the fibres of class C ⁰ but has none of class C ³ . There arises a necessary condition induced from the Milnor-Wood inequality for the existence of a foliation transverse to the fibres of an orientable circle bundle over a closed orientable 3-manifold. We show that with some exceptions this necessary condition is also sufficient for the existence of a smooth transverse foliation if the base space is a closed Seifert fibred manifold. Received: May 13, 1996     

On 4-manifolds homotopy equivalent to circle bundles over 3-manifolds

We give criteria for a closed 4-manifold to be homotopy equivalent to the total space of an S¹-bundle over a closed 3-manifold. In the aspherical case the conditions are that the Euler characteristic be 0 and that the fundamental group have an infinite cyclic normal subgroup such that the quotient group has one end and finite cohomological dimension. Under further assumptions on this quotient group we characterize the total spaces of such bundles over -or H² × E¹-manifolds and over E³-, Nil³- or Sol³-manifolds up to s-cobordism and homeomorphism respectively.     

CR-structures on three dimensional circle bundles

Oblatum 5-VIII-1991 & 9-III-1992     

Minimal triangulations of sphere bundles over the circle

For integers d?2 and ε=0 or 1, let S1,d−1(ε) denote the sphere product S¹×Sd−1 if ε=0 and the twisted sphere product if ε=1. The main results of this paper are: (a) if then S1,d−1(ε) has a unique minimal triangulation using 2d+3 vertices, and (b) if then S1,d−1(ε) has minimal triangulations (not unique) using 2d+4 vertices. In this context, a minimal triangulation of a manifold is a triangulation using the least possible number of vertices. The second result confirms a recent conjecture of Lutz. The first result provides the first known infinite family of closed manifolds (other than spheres) for which the minimal triangulation is unique. Actually, we show that while S1,d−1(ε) has at most one (2d+3)-vertex triangulation (one if , zero otherwise), in sharp contrast, the number of non-isomorphic (2d+4)-vertex triangulations of these d-manifolds grows exponentially with d for either choice of ε. The result in (a), as well as the minimality part in (b), is a consequence of the following result: (c) for d?3, there is a unique (2d+3)-vertex simplicial complex which triangulates a non-simply connected closed manifold of dimension d. This amazing simplicial complex was first constructed by Kühnel in 1986. Generalizing a 1987 result of Brehm and Kühnel, we prove that (d) any triangulation of a non-simply connected closed d-manifold requires at least 2d+3 vertices. The result (c) completely describes the case of equality in (d). The proofs rest on the Lower Bound Theorem for normal pseudomanifolds and on a combinatorial version of Alexander duality.     

Nonnegatively and positively curved invariant metrics on circle bundles

We derive and study necessary and sufficient conditions for an -bundle to admit an invariant metric of positive or nonnegative sectional curvature. In case the total space has an invariant metric of nonnegative curvature and the base space is odd dimensional, we prove that the total space contains a flat totally geodesic immersed cylinder. We provide several examples, including a connection metric of nonnegative curvature on the trivial bundle that is not a product metric.

     

Elliptic open books on torus bundles over the circle

As an application of the construction of open books on plumbed 3-manifolds, we construct elliptic open books on torus bundles over the circle. In certain cases these open books are compatible with Stein fillable contact structures and have minimal genus.      

On subharmonic functions dominated by certain functions

Given two kinds of functionsf(X) andh(y) defined on them-dimensional Euclidean spaceR ^m (m≧1) and the set of positive real numbers respectively, we give an estimation of growth of subharmonic functionsu(P) defined onR ^m+n (n≧1) such that $$u(P) \leqq f\left( X \right)h\left( {\left\| Y \right\|} \right)$$ for anyP=(X, Y),X ∈R ^m, Y ∈R ⁿ, where ‖Y ‖ denotes the usual norm ofY. Using an obtained result, we give a sharpened form of an ordinary Phragmén-Lindelöf theorem with respect to the generalized cylinderD ×R ⁿ, with a bounded domainD inR ^m.     

On codimension one foliations with Morse singularities on three-manifolds

A closed, connected oriented three-manifold supporting a codimension one oriented smooth foliation with Morse singularities having more centers than saddles and without saddle connections is diffeomorphic to the three-sphere. The use of the Reeb Stability theorem in place of the Poincaré-Bendixson theorem paves the way to a three-dimensional version, for foliations with singularities of Morse type, of a classical result of Haefliger. Finally, we give an example of a codimension one C^∞ foliation in the closed ball , with only one singularity which is of saddle type 2-2 and transverse to the boundary S³=∂B⁴.     

On compactly dominated spaces

Acta Mathematica Hungarica -     

Curvature homogeneous Lorentzian three-manifolds

We study three-dimensional curvature homogeneous Lorentzian manifolds. We prove that for all Segre types of the Ricci operator, there exist examples of nonhomogeneous curvature homogeneous Lorentzian metrics in .      

Tangent circle bundles admit positive open book decompositions along arbitrary links

In the present paper we generalize the divide lying in the unit disk, introduced by A'Campo, to compact, oriented, smooth surfaces, and prove a fibration theorem for generalized divides. As a consequence, we will show that, for any link L in the tangent circle bundle Y to the compact surface, there exists an additional knot K such that the link L∪K is the binding of a “positive” open book decomposition of Y.     

On JB*-triples defined by fibre bundles

We associate JB^*-triples to certain fibre bundles and describe their automorphisms, conjugations and real forms. In particular, we construct a JB^*-triple that does not admit a single conjugation.     

Isoperimetric inequalities for the fundamental groups of torus bundles over the circle

We give upper bounds for isoperimetric functions of semi-direct products in terms of the asymptotic behaviour of ||A ^k || ask . In the caseA Sp(n, ) we show that these bounds are sharp. This enables us to describe infinite families of nilpotent groups whose Dehn functions are bounded above and below by polynomials of degree the nilpotency class plus 1. We also recover the isoperimetric inequalities of cocompact lattices inNil andSol.Partially supported by NSF grant DMS-9203500.