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} .     

A cut-and-paste method for computing the Seifert volumes

We use methods from gauge theory to compute the Seifert volumes of 3-manifolds. As applications, we are able to find the Seifert volumes of several hyperbolic manifolds obtained by surgery on 2-bridge knots.     

On fibering and splitting of 5-manifolds over the circle

Our main result is a generalization of Cappell's 5-dimensional splitting theorem. As an application, we analyze, up to internal s-cobordism, the smoothable splitting and fibering problems for certain 5-manifolds mapping to the circle. For example, these maps may have homotopy fibers which are in the class of finite connected sums of certain geometric 4-manifolds. Most of these homotopy fibers have non-vanishing second mod 2 homology and have fundamental groups of exponential growth, which are not known to be tractable by Freedman–Quinn topological surgery. Indeed, our key technique is topological cobordism, which may not be the trace of surgeries.     

On fibering and splitting of 5-manifolds over the circle

Our main result is a generalization of Cappell's 5-dimensional splitting theorem. As an application, we analyze, up to internal s-cobordism, the smoothable splitting and fibering problems for certain 5-manifolds mapping to the circle. For example, these maps may have homotopy fibers which are in the class of finite connected sums of certain geometric 4-manifolds. Most of these homotopy fibers have non-vanishing second mod 2 homology and have fundamental groups of exponential growth, which are not known to be tractable by Freedman-Quinn topological surgery. Indeed, our key technique is topological cobordism, which may not be the trace of surgeries.     

The Fibered Isomorphism Conjecture for Complex Manifolds

In this paper we show that the Fibered Isomorphism Conjecture of Farrell and Jones, corresponding to the stable topological pseudoisotopy functor, is true for the fundamental groups of a class of complex manifolds. A consequence of this result is that the Whitehead group, reduced projective class groups and the negative K-groups of the fundamental groups of these manifolds vanish whenever the fundamental group is torsion free. We also prove the same results for a class of real manifolds including a large class of 3-manifolds which has a finite sheeted cover fibering over the circle.     

Quasi-isometries preserve the geometric decomposition of Haken manifolds

We prove quasi-isometry invariance of the canonical decomposition for fundamental groups of Haken 3-manifolds with zero Euler characteristic. We show that groups quasi-isometric to Haken manifold groups with nontrivial canonical decomposition are finite extensions of Haken orbifold groups. As a by-product we describe all 2-dimensional quasi-flats in the universal covers of non-geometric Haken manifolds. Oblatum 27-III-1996 & 5-IX-1996     

Kac-Moody群及其旗流形的庞加莱级数和有理同伦型

研究了不定型的Kac-Moody群及其旗流形的有理上同调.通过从庞加莱级数提取关于同调的信息,能够决定Kac-Moody群及其旗流形的有理上同调环.因为这些空间都是有理formal的空间,也决定了它们的有理同伦群及有理同伦型.     

Quasi-K?hler groups, 3-manifold groups, and formality

In this note, we address the following question: Which 1-formal groups occur as fundamental groups of both quasi-K?hler manifolds and closed, connected, orientable 3-manifolds. We classify all such groups, at the level of Malcev completions, and compute their coranks. Dropping the assumption on realizability by 3-manifolds, we show that the corank equals the isotropy index of the cup-product map in degree one. Finally, we examine the formality properties of smooth affine surfaces and quasi-homogeneous isolated surface singularities. In the latter case, we describe explicitly the positive-dimensional components of the first characteristic variety for the associated singularity link.     

Contact self-dual geometry of quasi-Sasakian 5-manifolds

We construct a self-dual geometry of quasi-Sasakian 5-manifolds. Namely, we intrinsically define the notion of contact conformally semiflat (i.e., contact self-dual or contact antiself-dual) almost contact metric manifolds and also obtain a number of results concerning contact conformally semiflat quasi-Sasakian 5-manifolds. Themost important results concerning Sasakian and cosymplectic manifolds reveal interesting relationships between the characteristics of these manifolds such as contact self-duality and constancy of the Φ-holomorphic sectional curvature, contact anti-self-duality and Ricci flatness, etc.     

Bounding volume by systoles of 3-manifolds

We prove a new systolic volume lower bound for non-orientablen-manifolds, involving the stable 1-systole as well as the codimension-1systole with coefficients in ₂. As an application, we provethat Lusternik–Schnirelmann category and systolic categoryagree for non-orientable closed manifolds of dimension 3, extendingour earlier result in the orientable case. Finally, we provethe homotopy invariance of systolic category.     

Almost contact metric 5-manifolds and connections with torsion

We study 5-dimensional Riemannian manifolds that admit an almost contact metric structure. We classify these structures by their intrinsic torsion and review the literature in terms of this scheme. Moreover, we determine necessary and sufficient conditions for the existence of metric connections with vectorial, totally skew-symmetric or traceless cyclic torsion that are compatible with the almost contact metric structure. Finally, we examine explicit examples of almost contact metric 5-manifolds from this perspective.     

On a class of hyperbolic 3-manifolds and groups with one defining relation

We construct compact hyperbolic 3-manifolds with totally geodesic boundary, arbitrarily many of the same volume. The fundamental groups of these 3-manifolds are groups with one defining relation. Our main result is a classification of these manifolds up to homeomorphism, resp. isometry.     

Covering degrees are determined by graph manifolds involved

W.Thurston raised the following question in 1976: Suppose that a compact 3-manifold M is not covered by (surface) ×S¹ \times S^1 or a torus bundle over S¹ S^1 . If M₁ M_1 and M₂ M_2 are two homeomorphic finite covering spaces of M, do they have the same covering degree?¶For so called geometric 3-manifolds (a famous conjecture is that all compact orientable 3-manifolds are geometric), it is known that the answer is affirmative if M is not a non-trivial graph manifold.¶In this paper, we prove that the answer for non-trivial graph manifolds is also affirmative. Hence the answer for the Thurston's question is complete for geometric 3-manifolds. Some properties of 3-manifold groups are also derived.     

Exotic torus manifolds and equivariant smooth structures on quasitoric manifolds

In 2006 Masuda and Suh asked if two compact non-singular toric varieties having isomorphic cohomology rings are homeomorphic. In the first part of this paper we discuss this question for topological generalizations of toric varieties, so-called torus manifolds. For example we show that there are homotopy equivalent torus manifolds which are not homeomorphic. Moreover, we characterize those groups which appear as the fundamental groups of locally standard torus manifolds. In the second part we give a classification of quasitoric manifolds and certain six-dimensional torus manifolds up to equivariant diffeomorphism. In the third part we enumerate the number of conjugacy classes of tori in the diffeomorphism group of torus manifolds. For torus manifolds of dimension greater than six there are always infinitely many conjugacy classes. We give examples which show that this does not hold for six-dimensional torus manifolds.     

Non-separable surfaces in cubed manifolds

We show that there are 3-manifolds with cubings of non-positive curvature such that their fundamental groups are not subgroup separable (LERF). We also give explicit examples of non-separable surfaces in certain cubed manifolds.

     

Incompressible surfaces in punctured-torus bundles

We derive in this paper the classification up to isotopy of the incompressible surfaces in hyperbolic 3-manifolds which fiber over the circle with fiber a once-punctured torus. From this classification it follows that most of the 3-manifolds obtained by compactifying these bundles via a circle at infinity are closed hyperbolic 3-manifolds which contain 1.0 incompressible surfaces, i.e., are not Haken manifolds.     

Conjugate Points on a Type of Khler Manifolds

We study conjugate points on a type of Khler manifolds, which are submanifolds of Grassmannian manifolds. And then we give the applications to the study of the index of geodesics and homotopy groups.     

On medial links and hyperbolic 3-manifolds with large isometry groups

Starting from the regular Platonic solids we construct links, generalizing the Borromean rings, with few components but large finite symmetry groups. We consider the 3-manifolds obtained by equivariant surgeries on these links, most of them hyperbolic, and the quotient orbifolds obtained from these group actions, among them various of the smallest known hyperbolic 3-orbifolds. Also, various of the manifolds obtained by equivariant surgery on these links are maximally symmetric hyperbolic 3-manifolds.     

The Canonical Decompositions of Some Family of Compact Orientable Hyperbolic 3-Manifolds with Totally Geodesic Boundary

L. Paoluzzi and B. Zimmermann constructed a family of compact orientable hyperbolic 3-manifolds with totally geodesic boundary, and classified them up to homeomorphism. Our main purpose is to determine the canonical decompositions of these manifolds. Using the result, we can obtain an alternative proof of the classification theorem of these manifolds and determine their isometry groups. We also determine their unknotting tunnels. Some of these manifolds are related to certain spatial graphs, so-called Suzukis Brunnian graphs. The properties of these manifolds enable us to obtain those of the graphs. Moreover, we give an affirmative answer to Kinoshitas problem concerning these graphs. In the Appendix, we calculate the volume of these manifolds.