On face numbers of manifolds with symmetry

Necessary conditions on the face numbers of Cohen-Macaulay simplicial complexes admitting a proper action of the cyclic group of a prime order are given. This result is extended further to necessary conditions on the face numbers and the Betti numbers of Buchsbaum simplicial complexes with a proper -action. Adin's upper bounds on the face numbers of Cohen-Macaulay complexes with symmetry are shown to hold for all (d−1)-dimensional Buchsbaum complexes with symmetry on n?3d−2 vertices. A generalization of Kühnel's conjecture on the Euler characteristic of 2k-dimensional manifolds and Sparla's analog of this conjecture for centrally symmetric 2k-manifolds are verified for all 2k-manifolds on n?6k+3 vertices. Connections to the Upper Bound Theorem are discussed and its new version for centrally symmetric manifolds is established.     

3-manifolds which are orbit spaces of diffeomorphisms

In a very general setting, we show that a 3-manifold obtained as the orbit space of the basin of a topological attractor is either S²×S¹ or irreducible.We then study in more detail the topology of a class of 3-manifolds which are also orbit spaces and arise as invariants of gradient-like diffeomorphisms (in dimension 3). Up to a finite number of exceptions, which we explicitly describe, all these manifolds are Haken and, by changing the diffeomorphism by a finite power, all the Seifert components of the Jaco-Shalen-Johannson decomposition of these manifolds are made into product circle bundles.     

Complexity of Geometric Three-manifolds

We compute for all orientable irreducible geometric 3-manifolds certain complexity functions that approximate from above Matveev's natural complexity, known to be equal to the minimal number of tetrahedra in a triangulation. We can show that the upper bounds on Matveev's complexity implied by our computations are sharp for thousands of manifolds, and we conjecture they are for infinitely many, including all Seifert manifolds. Our computations and estimates apply to all the Dehn fillings of M 6 ₁ ³ (the complement of the three-component chain-link, conjectured to be the smallest triply cusped hyperbolic manifold), whence to infinitely many among the smallest closed hyperbolic manifolds. Our computations are based on the machinery of the decomposition into ‘bricks’ of irreducible manifolds, developed in a previous paper. As an application of our results we completely describe the geometry of all 3-manifolds of complexity up to 9.     

Cartesian powers of 3-manifolds

Previous results of the authors completely determine when the n-fold self-products of two 3-dimensional lens spaces are diffeomorphic; in particular, if n is odd then the fundamental group determines the diffeomorphism type. We prove that for all other irreducible geometric 3-manifolds with trivial first Betti number, the n-fold products of such manifolds with themselves are homeomorphic for some n?2 if and only if the manifolds themselves are homeomorphic and obtain partial results for other cases. The proofs use an assortment of techniques from 3-dimensional topology and group theory.     

Pseudo-Riemannian 3-manifolds with prescribed distinct constant Ricci eigenvalues

We study three-dimensional pseudo-Riemannian manifolds having distinct constant principal Ricci curvatures. These spaces are described via a system of differential equations, and a simple characterization is proved to hold for the locally homogeneous ones. We then generalize the technique used in [O. Kowalski, F. Prüfer, On Riemannian 3-manifolds with distinct constant Ricci eigenvalues, Math. Ann. 300 (1994) 17-28] for Riemannian manifolds and construct explicitly homogeneous and non-homogeneous pseudo-Riemannian metrics in R³, having the prescribed principal Ricci curvatures.     

Families of simply connected 4-manifolds with the same Seiberg-Witten invariants

This article presents several new constructions of infinite families of smooth 4-manifolds with the property that any two manifolds in the same family are homeomorphic. While the construction gives strong evidence that any two of these manifolds of are not diffeomorphic, they cannot be distinguished by Seiberg-Witten invariants. Whether these manifolds are, or are not, diffeomorphic seems to be a very difficult question to answer. For one of these constructions, each member of the family is symplectic with the further property that each contains nullhomologous tori with the property that infinitely many log transformations on these tori yield nonsymplectic 4-manifolds. This is detected by calculations of Seiberg-Witten invariants. The surgery in question can be performed on any 4-manifold which contains as a codimension 0 submanifold a punctured surface bundle over a punctured surface and a nontrivial loop in the base which has trivial monodromy. A starting point for another class of examples in this paper is a family of examples which show that the Parshin-Arakelov theorem for holomorphic Lefschetz fibrations is false in the symplectic category. Such families are constructed by means of knot surgery on ellipitic surfaces. It is shown that for a fixed homeomorphism type X (of a simply connected elliptic surface) and a fixed integer g?3, there are infinitely many genus g Lefschetz fibrations on nondiffeomorphic 4-manifolds, all homeomorphic to X.     

Multiple closed geodesics on 3-spheres

This paper is devoted to a study on closed geodesics on Finsler and Riemannian spheres. We call a prime closed geodesic on a Finsler manifold rational, if the basic normal form decomposition (cf. [Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math. 187 (1999) 113-149]) of its linearized Poincaré map contains no 2×2 rotation matrix with rotation angle which is an irrational multiple of π, or irrational otherwise. We prove that if there exists only one prime closed geodesic on a d-dimensional irreversible Finsler sphere with d?2, it cannot be rational. Then we further prove that there exist always at least two distinct prime closed geodesics on every irreversible Finsler 3-dimensional sphere. Our method yields also at least two geometrically distinct closed geodesics on every reversible Finsler as well as Riemannian 3-dimensional sphere. We prove also such results hold for all compact simply connected 3-dimensional manifolds with irreversible or reversible Finsler as well as Riemannian metrics.     

On the Constant Type of Almost Hermitian Manifolds

We suggest a most natural generalization of the notion of constant type for nearly Kählerian manifolds introduced by A. Gray to arbitrary almost Hermitian manifolds. We prove that the class of almost Hermitian manifolds of zero constant type coincides with the class of Hermitian manifolds. We show that the class of G ₁-manifolds of zero constant type coincides with the class of 6-dimensional G ₁-manifolds with a non-integrable structure. Finally, we prove that the class of normal G ₂-manifolds of nonzero constant type coincides with the class of 4-dimensional G ₂-manifolds with a nonintegrable structure.     

Generalized Hopfian Property,a Minimal Haken Manifold,and Epimorphisms Between 3-Manifold Groups

We address the question that if π₁-surjective maps between closed aspherical 3-manifolds have the same rank on π₁ they must be of non-zero degree. The positive answer is proved for Seifert manifolds, which is used in constructing the first known example of minimal Haken manifold. Another motivation is to study epimorphisms of 3-manifold groups via maps of non-zero degree between 3-manifolds. Many examples are given. Received September 16, 1999, Revised August 31, 2000, Accepted March 29, 2001.     

Cyclic branched coverings of some pretzel links

We construct infinite families of closed connected orientable 3-manifolds obtained from certain triangulated 3-cells by pairwise identifications of their boundary faces. Our combinatorial constructions extend and complete a particular polyhedral scheme which Kim and Kostrikin used in [10] and [11] to define a series of spaces denoted M ₃(n). Then we determine geometric presentations of the fundamental groups, and prove that many of the constructed manifolds are n-fold (non-strongly) cyclic coverings of the ³-sphere branched over some specified pretzel links.     

Contact structures on product five-manifolds and fibre sums along circles

Two constructions of contact manifolds are presented: (i) products of S ¹ with manifolds admitting a suitable decomposition into two exact symplectic pieces and (ii) fibre connected sums along isotropic circles. Baykur has found a decomposition as required for (i) for all closed, oriented 4-manifolds. As a corollary, we can show that all closed, oriented 5-manifolds that are Cartesian products of lower-dimensional manifolds carry a contact structure. For symplectic 4-manifolds we exhibit an alternative construction of such a decomposition; this gives us control over the homotopy type of the corresponding contact structure. In particular, we prove that \mathbb CP²×S¹{{\mathbb {CP}}^2\times S^1} admits a contact structure in every homotopy class of almost contact structures. The existence of contact structures is also established for a large class of 5-manifolds with fundamental group \mathbbZ₂{{\mathbb{Z}}_2} .     

Construction of pseudo-isometries for treelike hyperbolic 3-manifolds of infinite volume

We introduce a family of rigid hyperbolic 3-manifolds of infinite volume with possibly infinitely many ends: the treelike manifolds. These manifolds generalize a family of constructive non compact surfaces – the equational surfaces – for which the homeomorphism problem is decidable. The proof of rigidity relies firstly on Thurston's theorem of compactness of the Teichmüller space of acylindrical compact 3-manifolds, and secondly, on Sullivan's rigidity theorem. To cite this article: O. Ly, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

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.     

Minimal distortion morphs generated by time-dependent vector fields

A morph between two Riemannian n-manifolds is an isotopy between them together with the set of all intermediate manifolds equipped with Riemannian metrics. We propose measures of the distortion produced by some classes of morphs and diffeomorphisms between two isotopic Riemannian n-manifolds and, with respect to these classes, prove the existence of minimal distortion morphs and diffeomorphisms. In particular, we consider the class of time-dependent vector fields (on an open subset Ω of Rn+1 in which the manifolds are embedded) that generate morphs between two manifolds M and N via an evolution equation, define the bending and the morphing distortion energies for these morphs, and prove the existence of minimizers of the corresponding functionals in the set of time-dependent vector fields that generate morphs between M and N and are L² functions from [0,1] to the Sobolev space .     

Cartesian product stabilization of 3-manifolds

The Cartesian product of a closed, orientable prime geometric 3-manifold and a closed orientable surface is unique except for the case of the Cartesian product of a special class of Seifert manifolds and a torus. The same type of uniqueness holds for stabilization of 3-manifolds by an n-dimensional torus. Cartesian squares of Seifert fibered 3-manifolds are completely classified.     

L 2-topological invariants of 3-manifolds

Summary We give results on theL ²-Betti numbers and Novikov-Shubin invariants of compact manifolds, especially 3-manifolds. We first study the Betti numbers and Novikov-Shubin invariants of a chain complex of Hilbert modules over a finite von Neumann algebra. We establish inequalities among the Novikov-Shubin invariants of the terms in a short exact sequence of chain complexes. Our algebraic results, along with some analytic results on geometric 3-manifolds, are used to compute theL ²-Betti numbers of compact 3-manifolds which satisfy a weak form of the geometrization conjecture, and to compute or estimate their Novikov-Shubin invariants.Oblatum 6-V-1993 & 20-VI-1994Partially supported by NSF-grant DMS 9101920Partially supported by NSF-grant DMS 9103327     

Combinatorial 3-Manifolds with Transitive Cyclic Symmetry

In this article we give combinatorial criteria to decide whether a transitive cyclic combinatorial d-manifold can be generalized to an infinite family of such complexes, together with an explicit construction in the case that such a family exists. In addition, we substantially extend the classification of combinatorial 3-manifolds with transitive cyclic symmetry up to 22 vertices. Finally, a combination of these results is used to describe new infinite families of transitive cyclic combinatorial manifolds and in particular a family of neighborly combinatorial lens spaces of infinitely many distinct topological types.     

Topology of compact space forms from Platonic solids. I.

The problem of classifying, up to isometry, the orientable 3-manifolds that arise by identifying the faces of a Platonic solid was completely solved in a nice paper of Everitt [B. Everitt, 3-manifolds from Platonic solids, Topology Appl. 138 (2004) 253-263]. His work completes the classification begun by Best [L.A. Best, On torsion-free discrete subgroups of PSL₂(C) with compact orbit space, Canad. J. Math. 23 (1971) 451-460], Lorimer [P.J. Lorimer, Four dodecahedral spaces, Pacific J. Math. 156 (2) (1992) 329-335], Prok [I. Prok, Classification of dodecahedral space forms, Beiträge Algebra Geom. 39 (2) (1998) 497-515], and Richardson and Rubinstein [J. Richardson, J.H. Rubinstein, Hyperbolic manifolds from a regular polyhedron, Preprint]. In this paper we investigate the topology of closed orientable 3-manifolds from Platonic solids. Here we completely recognize those manifolds in the spherical and Euclidean cases, and state topological properties for many of them in the hyperbolic case. The proofs of the latter will appear in a forthcoming paper.