Bounding edge degrees in triangulated -manifolds

In this note we prove that every closed orientable -manifold has a triangulation in which each edge has degree , or .

     

Determining Knotsw and Links by Cyclic Branched Coverings

It is well known that different knots or links in the 3-sphere can have homeomorphic n-fold cyclic branched coverings. We consider the following problem: for which values of nis a knot of link determined by itsn-fold cyclic branched covering? We consider the class of hyperbolic resp.2π/n-hyperbolic links. The isometry or symmetry groups of such links are finite, and their n-fold branched coverings are hyperbolic 3-manifolds. Our main result states that if ndoes not divide the order of the finite symmetry group of such a link, then the link is determined by its n-fold branched covering. In a sense, the result is best possible; the key argument of its proof is algebraic using some basic result about finite p-groups. The main result applies, for example, to the cyclic branched coverings of the 2-bridge links; in particular, it gives a classification of the maximally symmetricD₆-manifolds which are exactly the 3-fold branched coverings of the 2-bridge links.     

Dihedral Branches Coverings of Hyperbolic Orbifolds

We consider the class of hyperbolic 3-orbifolds whose underlying topological space is the 3-sphere S ³ and whose singular set is a trivalent graph with singular index 2 along each edge (an important special case occurs when the trivalent graph is the 1-skeleton of a hyperbolic polyhedron). Our main result is a classification of the D-branched coverings of these orbifolds (where D₂ is the dihedral group of order 4) under some general conditions on their isometry groups or the lengths of their geodesics.     

Hyperbolic 3-manifolds as cyclic branched coverings

There is an extensive literature on the characterization of knots in the 3-sphere which have the same 3-manifold as a common n-fold cyclic branched covering, for some integer . In the present paper, we study the following more general situation. Given two integers m and n, how are knots K ₁ and K ₂ related such that the m-fold cyclic branched covering of K ₁ coincides with the n-fold cyclic branched covering of K ₂. Or, seen from the point of view of 3-manifolds: in how many different ways can a given 3-manifold occur as a cyclic branched covering of knots in S ³. Under certain hypotheses, we solve this problem for the basic class of hyperbolic 3-manifolds and hyperbolic knots (the other basic class is that of Seifert fiber spaces resp. of torus and Montesinos knots for which the situation is well understood; the general case can then be analyzed using the equivariant sphere and torus decomposition into Seifert fiber spaces and hyperbolic manifolds). Received: December 7, 1999; revised version: May 22, 2000     

Branched Coverings and Embedded Surfaces in Four-manifolds

In this paper, we get a lower bound on the genera of surfaces re, presenting certain divisible classes in some 4-manifold X with Hi（X; Z） finite.     

不可定向四维流形的一些例子

利用Taubes的结果，给出了一些紧致带边不可定向四维拓扑流形，它们的Kirby-Sibenmann不变量平凡，但这些流形不可光滑化。     

Certain 4-manifolds with non-negative sectional curvature

In this paper, we study certain compact 4-manifolds with non-negative sectional curvature K. If s is the scalar curvature and W. is the self-dual part of Weyl tensor, then it will be shown that there is no metric g on S ^• × S ^• with both (i) K > 0 and (ii) ÷ s − W _• ⩾ 0. We also investigate other aspects of 4-manifolds with non-negative sectional curvature. One of our results implies a theorem of Hamilton: “If a simply-connected, closed 4-manifold M^• admits a metric g of non-negative curvature operator, then M^• is one of S^•, ℂP^• and S^•×S^•”. Our method is different from Hamilton’s and is much simpler. A new version of the second variational formula for minimal surfaces in 4-manifolds is proved.      

On Optimal 4-Dimensional Metrics

We completely determine, up to homeomorphism, which simply connected compact oriented 4-manifolds admit scalar-flat, anti-self-dual Riemannian metrics. The key new ingredient is a proof that the connected sum of five reverse-oriented complex projective planes admits such metrics. Supported in part by NSF grant DMS-0604735.     

A universal planar graph under the minor relation

We construct an infinite planar graph that contains every planar graph as a minor. © 1999 John Wiley & Sons, Inc. J. Graph Theory 32: 191–206, 1999     

Cyclic Group Action of Composite Order on Indefinite 4-manifolds

In this paper, we study the possibilities for several kinds of topological, locally linear cyclic group actions of non-prime order on some closed, simply connected 4-manifolds with indefinite intersection form. Especially, we discuss the existence of locally linear pseudofree C9 action on this kind of 4-manifolds.     

A Banach space block finitely universal for monotone bases

A reflexive Banach space with a basis is constructed having the property that every monotone basis is block finitely representable in each block basis of .

     

On 4-manifolds homotopy equivalent to surface bundles over surfaces

We show that a closed 4-manifold is homotopy equivalent to the total space of a surface bundle over a surface if the obviously necessary conditions on the fundamental group and Euler characteristic hold. When the base is the 2-sphere we need also conditions on the characteristic classes of the manifold. (Our results are incomplete when the base is the projective plane.) In most cases we can show the manifold is s-cobordant to the total space of the bundle.     

The stable topology of 4-manifolds

The stable theory (which allows connected sums with S²×S²) is unified and extended using current 4-manifold techniques. Principal new results are a stable 5-dimensional s-cobordism theorem, and the fact that 1-connected smooth 4-manifold pairs stably have handle decompositions with no 1-handles.     

Bordism of spin 4-manifolds with local action of tori

We prove that bordism group of spin -manifolds with singular -structure, the notion introduced by Cheeger and Gromov, is an infinite cyclic group and is detected by singnature. In particular we obtain that the theorem of Atiyah and Hirzebruch of vanishing of Â-genus in case of action on spin -manifolds is not valid in case of -structures on spin -manifolds.

     

Compactifications and universal spaces in extension theory

We show that for each countable simplicial complex the following conditions are equivalent:

• iff for any space .
• There exists a -invertible map of a metrizable compactum with onto the Hilbert cube.

     

Q-curvature flow on 4-manifolds

We formulate an appropriate gradient flow in order to study the evolution of the Q-curvature to a prescribed function on a 4-manifold. For a class of prescribed functions, we show convergence and describe the asymptotic behaviour at infinity.     

2-Compressible non-Heegaard surfaces in small 3- manifolds

In the present paper, we investigate the non-Heegaard 2-compressible surfaces in small 3-manifolds, and show that such surfaces have some good properties.     

An isoperimetric inequality in the universal cover of the punctured plane

We find the largest ? (approximately 1.71579) for which any simple closed path α in the universal cover of R²?Z², equipped with the natural lifted metric from the Euclidean two-dimensional plane, satisfies L(α)≥?A(α), where L(α) is the length of α and A(α) is the area enclosed by α. This generalizes a result of Schnell and Segura Gomis, and provides an alternative proof for the same isoperimetric inequality in R²?Z².