On certain classes of hyperbolic 3-manifolds

We study the topological structure and the homeomorphism problem for closed 3-manifolds M(n,k) obtained by pairwise identifications of faces in the boundary of certain polyhedral 3-balls. We prove that they are (n/d)-fold cyclic coverings of the 3-sphere branched over certain hyperbolic links of d+1 components, where d= (n/k). Then we study the closed 3-manifolds obtained by Dehn surgeries on the components of these links. Received: 27 November 1998 / Accepted: 12 May 1999     

Lins-Mandel graphs representing 3-manifolds

In this paper we find necessary and sufficient conditions for a Lins-Mandel 4-coloured graph to represent a 3-manifold.     

Some examples of aspherical 4-manifolds that are homology 4-spheres

In this paper, Problem 4.17 on Kirby's problem list is solved by constructing infinitely many homotopy types of aspherical 4-manifolds that are homology 4-spheres.     

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.     

On 4-manifolds with zero-entropy metrics

We sharpen a recent result of Paternain and Petean by showingthat a closed 4-manifold which admits a Riemannian metric withzero topological entropy and infinite fundamental group eitheris aspherical or has a finite covering space homeomorphic toS³ x S¹ or S² x S¹ x S¹.     

On asymptotic cones and quasi-isometry classes of fundamental groups of 3-manifolds

We apply the concept of asymptotic cone to distinguish quasi-isometry classes of fundamental groups of 3-manifolds. We prove that the existence of a Seifert component in a Haken manifold is a quasi-isometry invariant of its fundamental group.This research was partially supported by the grant SFB 256 Nichtlineare partielle Differentialgleichungen and the NSF grant DMS-9306140 (Kapovich).     

On 4-manifolds with singular torus actions

Mathematische Annalen -     

On the fundamental homology classes of a real algebraic variety

It is proved that there is only one relation between the homology classes determined by the real points of a special real algebraic variety. This relation is equal to the sum of all the homology classes. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 216–219, August, 1999.     

On nonnegatively curved 4-manifolds with discrete symmetry

Let M be the closed, simply connected, 4-manifold with nonnegative sectional curvature, called a nonnegatively curved 4-manifold, with an effective and isometric Z _m -action for a positive integer m ≧ 61⁷. Assume that Z _m acts trivially on the homology of M. The goal of this short paper is to prove that if the fixed point set of any nontrivial element of Z _m has at most one two-dimensional component, then M is homeomorphic to S ⁴, # _i ^l =1S ² × S ², l = 1, 2, or # _j ^k = 1 ± CP ², k = 1, 2, 3, 4, 5. The main strategy of this paper is to give an upper bound of the Euler characteristic χ(M) under the homological assumption of a Z _m -action as above by using the Lefschetz fixed point formula.     

Chebotarev-type theorems in homology classes

We describe how closed geodesics lying in a prescribed homology class on a negatively curved manifold split when lifted to a finite cover. This generalizes a result of Zelditch in the case of compact hyperbolic surfaces.

     

Closed orbits in homology classes

Dedicated to Pr. Akihiko Morimoto for his 60th birthday     

On the algebra of handle operations in 4-manifolds

The possibility is considered that the Andrews-Curtis conjecture, if false, may provide obstructions to the cancellation of 1- and 3-handles in a simply connected 4-manifold. An algebra is developed for the simultaneous algebraic reading of attaching maps for 2-handles and their duals, and it is shown that obstructions to 1- and 3-handle cancellation cannot bedetected in this algebra.