Non-Complex Symplectic 4-Manifolds with Formula

Simply connected closed symplectic 4-manifolds with and K² = 0 are investigated. As aresult, it is confirmed that most of homotopy elliptic surfaces{E(1)_k|K is a fibred knot in S³} constructed by R. Fintusheland R. Stern in Invent. Math. 134 (1998) 363–400 are simplyconnected closed minimal symplectic 4-manifolds that do notadmit a complex structure. 2000 Mathematics Subject Classification57R17, 57R57 (primary), 14J26 (secondary).     

Small 3-Manifolds of Large Genus

In this paper we prove the Cheeger inequality for infinite weighted graphs endowed with 'corresponding' measure. This measure has already been developed in the study of tree lattices. Our graphs have finite volumes. A similar theory has already been developed for manifolds of finite volumes.     

Exotic smooth structures on Formula

Motivated by Stipsicz and Szabó's exotic 4-manifoldswith b₂⁺ = 3 and b₂^– = 8, we construct a family of simplyconnected smooth 4-manifolds with b₂⁺ = 3 and b₂^– = 8.As a corollary, we conclude that the topological 4-manifold     

Homotopy Self-Equivalences of 4-Manifolds

We establish a braid of interlocking exact sequences containing the group of homotopy self-equivalences of a smooth or topological 4-manifold. The braid is computed for manifolds whose fundamental group is finite of odd order. Research partially supported by NSERC Research Grant A4000. The authors thank the Max Planck Institut für Mathematik in Bonn, Germany, for hospitality and support. in final form: 19 November 2003 An erratum to this article is available at.     

Link Homology in 4-Manifolds

We define link homology in 4-manifolds, and show that it hasa close connection to linking numbers and intersection matricesof 4-manifolds. We also define null-homologous links in 4-manifolds.We give a necessary and sufficient condition for links to benull-homologous in 4-manifolds. This condition implies thatfor any 4-manifold with second Betti number n, there are (n+ 2)-component links which are not nullhomologous in the 4-manifold.     

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。     

The Geography Problem of 4-Manifolds with Various Structures

We review results about the geography problem of complex, symplectic and Einstein 4-manifolds. Finally we discuss the same problem for Lefschetz fibrations. We show that in the geography of symplectic 4-manifolds and Lefschetz fibrations the slopes = c₁ ²/c₂ do not admit gaps — following similar results of Sommese in the complex case.     

Einstein Metrics with Prescribed Conformal Infinity on 4-Manifolds

This paper considers the existence of conformally compact Einstein metrics on 4-manifolds. A reasonably complete understanding is obtained for the existence of such metrics with prescribed conformal infinity, when the conformal infinity is of positive scalar curvature. We find in particular that general solvability depends on the topology of the filling manifold. The obstruction to extending these results to arbitrary boundary values is also identified. While most of the paper concerns dimension 4, some general results on the structure of the space of such metrics hold in all dimensions. Received: April 2006, Revision: November 2006, Accepted: February 2008     

4维完备Einstein 流形的一些注释

本文给出了在开的定向的4维流形上存在有界曲率的完备Einstein度量的障碍.     

Cyclic Group Actions on 4-Manifolds

Let X be a closed, oriented Riemannian 4-manifold. Suppose that a cyclic group Z( _p (p is prime) acts on X by an orientation preserving isometry with an embedded Riemann surface as fixed point set. We study the representation of Z _p on the Spin^c-bundles and the Z _p-invariant moduli space of the solutions of the Seiberg–Witten equations for a Spin^c-structure X. When the Z _p action on the determinant bundle det L acts non-trivially on the restriction L| over the fixed point set , we consider -twisted solutions of the Seiberg-Witten equations over a Spin^c-structure ' on the quotient manifold X/Z _p X', (0,1). We relate the Z _p -invariant moduli space for the Spin^c-structure on X and the -twisted moduli space for the Spin^c-structure on X'. From this we induce a one-to-one correspondence between these moduli spaces and calculate the dimension of the -twisted moduli space. When Z _p acts trivially on L|, we prove that there is a one-to-one correspondence between the Z _p -invariant moduli space M( ^Zp and the moduli space M (") where ' is a Spin^c-structure on X' associated to the quotient bundle L/Z _p X'. vskip0pt When p = 2, we apply the above constructions to a Kahler surface X with b ₂ ⁺ (X) > 3 and H ₂(X;Z) has no 2-torsion on which an anti-holomorphic involution acts with fixed point set , a Lagrangian surface with genus greater than 0 and []2H ₂(H ;Z). If _K ^X 2 > 0 or K _X ² = 0 and the genus g()> 1, we have a vanishing theorem for Seiberg–Witten invariant of the quotient manifold X'. When K _X ² = 0 and the genus g()= 1, if there is a Z ²-equivariant Spin^c-structure on X whose virtual dimension of the Seiberg–Witten moduli space is zero then there is a Spin^c-structure " on X' such that the Seiberg-Witten invariant is ±1.     

4维完备Einstein流形的一些注释

本文给出了在开的定向的4维流形上存在有界曲率的完备Einstein度量的障碍．     

Minimal Surfaces in Riemannian 4-Manifolds

((Without abstract)) Submitted: December 1996, final version: June 1997     

Ricci Flow on Open 4-Manifolds with Positive Isotropic Curvature

In this note we prove the following result: Let X be a complete, connected 4-manifold with uniformly positive isotropic curvature, with bounded geometry, and with no essential incompressible space form. Then X is diffeomorphic to $\mathbb{S}^{4}$ , or $\mathbb{RP}^{4}$ , or $\mathbb{S}^{3}\times\mathbb {S}^{1}$ , or $\mathbb{S}^{3}\widetilde{\times} \mathbb{S}^{1}$ , or a possibly infinite connected sum of them. This extends work of Hamilton and Chen–Zhu to the noncompact case. The proof uses Ricci flow with surgery on complete 4-manifolds, and is inspired by recent work of Bessières, Besson, and Maillot.     

Haken 3-Manifolds in Small Covers*

The authors prove that a 3-dimensional small cover M is a Haken manifold if and only if M is aspherical or equivalently the underlying simple polytope is a flag polytope. In addition, they find that M being Haken is also equivalent to the existence of a Riemannian metric with non-positive sectional curvature on M.     

Connected Sums of 4-Manifolds as Codimension-k Fibrators

The main result provides mild conditions under which a closed,orientable, PL 4-manifold N = N₁#N₂ with ₁(N_e) residually finite(e=1,2) is a codimension-5 PL fibrator. The paper also presentsa rich variety of conditions on a closed 4-manifold N⁴ underwhich every PL map between manifolds, where the domain is orientableand all point inverses are copies of N⁴, must be an approximatefibration.     

Compactness Theorems for Critical Riemannian 4-Manifolds with Integral Bounds on Curvature

In this article, we prove the compactness of the set of critical 4-manifolds with L ^p-bound on the negative part of the Ricci curvature tensor, p > 2. In earlier work we proved this under the assumption that the Ricci curvature is pointwise bounded from below.     

从怪异7维球面到怪异4维欧氏空间

本文简要地介绍了微分拓扑学在20世纪深入发展中一系列卓越的成果,包括由J.Milnor发现的第一个怪异7维球面,不存在光滑化的组合流形的构作,当n≠4时瓗n上只有唯一的微分结构的证明,Freedman关于拓扑4维流形的定理,它推导出4维广义Poincaré猜测的解决,Donaldson关于光滑4维流形的定理,进而得到瓗4上存在怪异微分结构等重要内容.     

Gluing Affine 2-Manifolds with Polygons

Affine structures on surfaces are constructed by gluing polygons. The geometry of the affine surface depends on the shape of the polygon(s) and the particular gluing transformations used. The affine version of the Poincaré fundamental polygon theorem expresses the fundamental group and holonomy of the surface in terms of the gluing data. The theorem may be used to construct all complete affine structures on the 2-torus. The space of inequivalent holonomy representations of such structures is homeomorphic to R2.     

3-Manifolds with nonnegative Ricci curvature

For a complete noncompact 3-manifold with nonnegative Ricci curvature, we prove that either it is diffeomorphic to ?³ or the universal cover splits. This confirms Milnor’s conjecture in dimension 3.