RIGIDITY OF COMPACT SURFACES IN HOMOGENEOUS 3-MANIFOLDS WITH CONSTANT MEAN CURVATURE

In this paper, we establish a rigidity theorem for compact constant mean curva- ture surfaces of the Berger sphere in terms of the surfaces＇ geometric invariants. This extends the previous similar result on compact minimal surfaces of the Berger sphere.

An Algorithm of Witten's Invariants of Some 3-manifolds

In this paper, I give out an algorithm of the witten' s invariants of the 3-manifolds obtained by surgery on the links whose corresponding graphs may be no longer trees.

Sufficient Conditions for Amalgamated 3-Manifolds to be ∂-Irreducible

In this paper,the authors give some sufficient conditions for an amalgamated 3-manifold along a compact connected surface F with boundary to be ∂-irreducible in terms of distances between some kinds of vertex subsets of the curve complex and the arc complex of F.

Minimal Euler characteristics of 4-manifolds with 3-manifold groups

Letπ=π_(1)(M)for a compact 3-manifold M,andχ_(4),p and q^(*)be the invariants of Hausmann and Weinberger(1985),Kotschick(1994)and Hillman(2002),respectively.For a certain class of compact 3-manifolds M,including all those not containing two-sided RP2’s,we determineχ_(4)(π).We address when p(π)equalsχ_(4)(π)and when q^(*)(π)equalsχ_(4)(π),and answer a question raised by Hillman(2002).

Finiteness of Mapping Degree Sets for 3-Manifolds

By constructing certain maps, this note completes the answer of the question： For which closed orientable 3-manifold N, is the set of mapping degrees D（M, N） finite for any closed orientable 3-manifold M？

3-Manifold Containing Separating Incompressible Surfaces of All Positive Genera

In this paper, we shall prove that for any orientable 3-manifold M, there is a link L - K U K1 U K2 U K3 with four components in M, such that the complement of L, say ML, contains separating essential closed surfaces of all positive genera.

Nielsen Theory on 3-manifolds Covered by S^2× R

Let f ： M→ M be a self-map of a closed manifold M of dimension dim M ≥ 3. The Nielsen number N（f） of f is equal to the minimal number of fixed points of f＇ among all self-maps f＇ in the homotopy class of f. In this paper, we determine N（f） for all self-maps f when M is a closed 3-manifold with S^2× R geometry. The calculation of N（f） relies on the induced homomorphisms of f on the fundamental group and on the second homotopy group of M.

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

We address the question that if π_1-surjective maps between closed aspherical 3-manifolds have the same rank on π_1 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.

Eigenvalues under the Backward Ricci Flow on Locally Homogeneous Closed 3-manifolds

In this paper, we study the evolving behaviors of the first eigenvalue of the Laplace- Beltrami operator under the normalized backward Ricci flow, construct various quantities which are monotonic under the backward Ricci flow and get upper and lower bounds. We prove that in cases where the backward Ricci flow converges to a sub-Riemannian geometry after a proper rescaling, the eigenvalue evolves toward zero.

On embedding infinite cyclic covers in compact 3-manifolds

We show that there are knots whose infinite cyclic covers do not embed in any compact 3-manifold,answering a question of Jiang et al.(2006).

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.

THE SURFACE AREA PRESERVING MEAN CURVATURE FLOW IN QUASI-FUCHSIAN MANIFOLDS

In this paper, we consider the surface area preserving mean curvature flow in quasi-Fuchsian 3-manifolds. We show that the flow exists for all times and converges exponentially to a smooth surface of constant mean curvature with the same surface area as the initial surface.

On the Cohomology of Some Complex Hyperbolic Arithmetic 3-Manifolds

Given an arithmetic lattice of the unitary group U(3,1)arising from a hermitian form over a CM-field,we show that all unitary representations of U(3,1)with nonzero cohomology contribute to the cohomology of the attached arithmetic complex 3-manifold,at least when we pass to a finite-index subgroup of the given arithmetic lattice.

Lattice of Finite Group Actions on Prism Manifolds

The set of finite group actions (up to equivalence) which operate on a prism manifold M, preserve a Heegaard Klein bottle and have a fixed orbifold quotient type, form a partially ordered set. We describe the partial ordering of these actions by relating them to certain sets of ordered pairs of integers. There are seven possible orbifold quotient types, and for any fixed quotient type we show that the partially ordered set is isomorphic to a union of distributive lattices of a certain type. We give necessary and sufficent conditions, for these partially ordered sets to be isomorphic and to be a union of Boolean algebras.

Topologically minimal surfaces versus self-amalgamated Heegaard surfaces

Let V ∪SW be a Heegaard splitting of M,such that αM = α-W = F1 ∪ F2 and g(S) = 2g(F1)= 2g(F2). Let V * ∪S*W * be the self-amalgamation of V ∪SW. We show if d(S) 3 then S* is not a topologically minimal surface.

H'-splittings of Seifert Manifolds with Boundary

It is known that each compact connected orient able 3-manifold M with boundary admits an H’-splitting H1∪FH2,where F is a compact connected orientable surface properly embedded in M and splits M into two handlbodies H_(1) and H_(2).In this paper,we show that a non-completely L-reducible and minimal H’-splitting surface for a compact connected irreducible orientable anannular Seifert 3-manifold with boundary is horizontal,and give a necessary and sufficient condition for an amalgamation of two compact connected orientable 3-manifolds along a compact connected surface to be a Seifert manifold with boundary,and describe a characteristic of some H’-splittings to denote a Seifert 3-manifold with boundary.For a compact connected orientable Seifert manifold M with a semi-bundle structure M_(1)∪_(F)M_(2),we give an upper bound of the genus of the base surface.

Relations among Chern-Simons-Witten-Jones invariants

Various relations and non-homotopy invariance of the invariants are given.

A necessary and sufficient condition for a surface sum of two handlebodies to be a handlebody

The main results of the paper are that we give a necessary and sufficient condition for a surface sum of two handlebodies along a connected surface to be a handlebody as follows:(1)The annulus sum H=H1∪AH2 of two handlebodies H1 and H2 is a handlebody if and only if the core curve of A is a longitude for either H1 or H2;(2)Let H=H1∪Sg,b H2 be a surface sum of two handlebodies H1 and H2 along a connected surface S=Sg,b,b 1,ni=g(Hi)2,i=1,2.Suppose that S is incompressible in both H1 and H2.Then H is a handlebody if and only if there exists a basis J={J1,...,Jm}with a partition(J1,J2)of J such that J1 is primitive in H1 and J2 is primitive in H2.

Kirby-Melvin's τ_r'and Ohtsuki's τ for lens spaces

Based on the previous results on quantum invariants, explicit formulae for T_r~’ of lens spaces are obtained. As a byproduct, the Ohtsuki invariants of even lens spaces are obtained.

On free Z_p-torus actions in dimensions two and three

We confirm the Halperin-Carlsson conjecture for free Z_p-torus actions(p is a prime) on 2-dimensional finite CW-complexes and free Z_2-torus actions on closed 3-manifolds.