Symplectic tori in rational elliptic surfaces

Let E(1)_p denote the rational elliptic surface with a single multiple fiber f_p of multiplicity p. We construct an infinite family of homologous non-isotopic symplectic tori representing the primitive 2-dimensional homology class [f_p] in E(1)_p when p>1. As a consequence, we get infinitely many non-isotopic symplectic tori in the fiber class of the rational elliptic surface . We also show how these tori can be non-isotopically embedded as homologous symplectic submanifolds in other symplectic 4-manifolds.     

On a class of hyperbolic 3-orbifolds of small volume and small heegaard genus associated to 2-bridge links

We consider a class of hyperbolic 3-orbifoldsO(α/β); the underlying topological space of such an orbifold is the 3-sphere and the singular set is obtained by adding the two standard (upper and lower) unknotting tunnels to a 2-bridge linkL(α/β) (and associating branching order two to both unknotting tunnels). These 3-orbifolds are extremal with respect to the notion of Heegaard genus or Heegaard number of 3-orbifolds; it is to be expected that they are also extremal with respect to the volume, that is the smallest volume hyperbolic 3-orbifolds should belong to this or some closely related class. We show that an orbifoldO(α/β) has a uniqueD ₂-covering by an orbifoldℒ _n(α/β) wose space is the 3-sphere and whose singular set is the same 2-bridge linkL(α/β) used for the construction ofO(α/β); moreoverO(α/β) is hyperbolic if and only ifℒ _n(α/β) is hyperbolic. As the volumes of the orbifoldsℒ _n(α/β) are known resp. can be computed, this allows to compute the volumes of the orbifoldsO(α/β). The problem of computation of volumes remains open for some closely related classes of 3-orbifolds which are also extremal with respect to the Heegaard genus (for example associating a branching order bigger than two to one or both unknotting tunnels).     

A Hadamard product involving N-matrices

We show that for any pair M,N of n by n M-matrices, the Hadamard (entry-wise) product M°N _-1 is again an M-matrix. For a single M-matrix M, the matrix M°M ^-1 is also considered.     

Smooth free involution of $$H{\mathbb{C}}P^{3}$$ and Smith conjecture for imbeddings of S3 in S6

This paper establishes an equivalence between existence of free involutions on and existence of involutions on S ⁶ with fixed point set an imbedded S ³, then a family of counterexamples of the Smith conjecture for imbeddings of S ³ in S ⁶ are given by known result on . In addition, this paper also shows that every smooth homotopy complex projective three-space admits no orientation preserving smooth free involution, which answers an open problem (Petrie T in Lecture Notes in Mathematics, Springer, New York, Vol. 298, pp. 234–259, 1972). Moreover, the study of existence problem for smooth orientation preserving involutions on is completed. B.-h. Li is partially supported by the 973 Program of China. Z. Lü is partially supported by the scholar fund of the Ministry of Education in China and grants from NSFC (No. 10371020 and No. 10671034).     

Branched spines and Heegaard genus of 3-manifolds

We prove that an invariant of closed 3-manifolds, called the block number, which is defined via flow-spines, equals the Heegaard genus, except for S ³ and S ² × S ¹. We also show that the underlying 3-manifold is uniquely determined by a neighborhood of the singularity of a flow-spine. This allows us to encode a closed 3-manifold by a sequence of signed labeled symbols. The behavior of the encoding under the connected sum and a criterion for reducibility are studied.     

Local torus actions modeled on the standard representation

We introduce the notion of a local torus action modeled on the standard representation (for simplicity, we call it a local torus action). It is a generalization of a locally standard torus action and also an underlying structure of a locally toric Lagrangian fibration. For a local torus action, we define two invariants called a characteristic pair and an Euler class of the orbit map, and prove that local torus actions are classified topologically by them. As a corollary, we obtain a topological classification of locally standard torus actions, which includes the topological classifications of quasi-toric manifolds by Davis and Januszkiewicz and of effective T²-actions on four-dimensional manifolds without nontrivial finite stabilizers by Orlik and Raymond. We discuss locally toric Lagrangian fibrations from the viewpoint of local torus actions. We also investigate the topology of a manifold equipped with a local torus action when the Euler class of the orbit map vanishes.     

Symmetric group actions on homotopy S 2 × S 2

Let X be a closed smooth 4-manifold which is homotopy equivalent to S ² × S ². In this paper we use Seiberg-Witten theory to prove that if X admits a spin symmetric group S ₄ action of even type with b ₂ ⁺ (X/S ₄) = b ₂ ⁺ (X), then as an element of R (S ₄), Ind _S4 D _X = k ₁ (1 − θ) + k ₂(ψ₁ − ψ₂) for some integers k ₁ and k ₂, where 1, θ, ψ₁, ψ₂ are irreducible characters of S ₄ of degree 1, 1, 3, and 3 respectively. Authors’ address: Ximin Liu and Hongxia Li, Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, P.R. China     

Fixity and free group actions on products of spheres

A representation G U(n) of degree n has fixity equal to the smallest integer f such that the induced action of G on U(n) /U(n-f-1) is free. Using bundle theory we show that if G admits a representation of fixity one, then it acts freely and smoothly on We use this to prove that a finite p-group (for p > 3) acts freely and smoothly on a product of two spheres if and only if it does not contain ( /p)³ as a subgroup. We use propagation methods from surgery theory to show that a representation of fixity f < n - 1 gives rise to a free action of G on a product of f + 1 spheres provided the order of G is relatively prime to (n - 1)!. We give an infinite collection of new examples of finite p-groups of rank r which act freely on a product of r spheres, hence verifying a strong form of a well-known conjecture for these groups. In addition we show that groups of fixity two act freely on a finite complex with the homotopy type of a product of three spheres. A number of examples are explicitly described.     

Noncyclic Covers of Knot Complements

Hempel has shown that the fundamental groups of knot complements are residually finite. This implies that every nontrivial knot must have a finite-sheeted, noncyclic cover. We give an explicit bound, Φ (c), such that if K is a nontrivial knot in the three-sphere with a diagram with c crossings then the complement of K has a finite-sheeted, noncyclic cover with at most Φ (c) sheets.The author is supported by an NSF Postdoctoral Fellowship at Cornell University.     

On extensions of a symplectic class

Let F be a fibration on a simply-connected base with symplectic fiber (M,ω). Assume that the fiber is nilpotent and T2k-separable for some integer k or a nilmanifold. Then our main theorem, Theorem 1.8, gives a necessary and sufficient condition for the cohomology class [ω] to extend to a cohomology class of the total space of F. This allows us to describe Thurston?s criterion for a symplectic fibration to admit a compatible symplectic form in terms of the classifying map for the underlying fibration. The obstruction due to Lalond and McDuff for a symplectic bundle to be Hamiltonian is also rephrased in the same vein. Furthermore, with the aid of the main theorem, we discuss a global nature of the set of the homotopy equivalence classes of fibrations with symplectic fiber in which the class [ω] is extendable.     

Surgery on the Novikov complex

Let M be a closed connected smooth manifold with dim M=n6, and : ₁(M) Z be an epimorphism. Denote by the group ring of ₁(M) and let ^– be its Novikov completion. Let D _* be a free-based finitely generated chain complex over ^– . Assume that D _ii=0 for i1 and in–1 and that D _* has the same simple homotopy type as the Novikov-completed simplicial chain complex of the universal covering M. Let N be an integer. We prove that D _* can be realized, up to the terms of ^– of degree N as the Novikov complex of a Morse map : M S ¹, belonging to . Applications to Arnold's conjectures and to the theory of fibering of M over S ¹ are given.     

Separation by a codimension-1 map with a normal crossing point

Letf:M ^n–1N ⁿ be an immersion with normal crossings of a closed orientable (n–1)-manifold into an orientablen-manifold. We show, under a certain homological condition, that iff has a multiple point of multiplicitym, then the number of connected components ofN–f(M) is greater than or equal tom+1, generalizing results of Biasi and Romero Fuster (Illinois J. Math. 36 (1992), 500–504) and Biasi, Motta and Saeki (Topology Appl. 52 (1993), 81–87). In fact, this result holds more generally for every codimension-1 continuous map with a normal crossing point of multiplicitym. We also give various geometrical applications of this theorem, among which is an application to the topology of generic space curves.     

An effective algorithm to get linking numbers of 3-manifolds

Summary A colored triangulation of a 3-manifoldM ³ is a decomposition into tetrahedra so that each vertex of them receive one of the colors 0, 1, 2, 3 in such a way that each tetrahedron has four differently colored vertices. From the combinatorics of the dual of a colored triangulation forM ³ we provide an easy algorithm to get a special kind of intersection matrix; from this matrix and from the torsion coefficients of the firstZ-homology group ofM ³ we provide a formula which yields its linking numbers.     

Compactifying coverings of 3-manifolds

If a finitely presented groupG is negatively curved, automatic or asynchronously automatic thenG has an asynchronously bounded “almost prefix closed” combing. Results in [Br₁] and [E] imply that the fundamental group of any closed 3-manifold satisfying Thurston's geometrization conjecture has an asynchronously bounded, almost prefix closed combing. MAIN THEOREM. IfM is a compactP ²-irreducible 3-manifold,π ₁ (M) has an asynchronously bounded, almost prefix closed combing, andH, a subgroup ofπ ₁ (M), is quasiconvex with respect to this combing, then the cover ofM corresponding toH is a missing boundary manifold.     

The weak isovariant Borsuk-Ulam theorem for compact Lie groups

In this paper we shall deal with a weak version of the Borsuk-Ulam theorem for G-isovariant maps, which we call the weak isovariant Borsuk-Ulam theorem. One of the results is that the weak isovariant Borsuk-Ulam theorem in linear G-spheres holds for an arbitrary compact Lie group G. On the contrary the weak isovariant Borsuk-Ulam theorem in semilinear G-(homology) spheres holds if and only if G is solvable. Received: 2 April 2002     

On certain classes of hyperbolic 3-manifolds

We study the topological structure and the homeomorphism problem for closed 3-manifolds M(n,k) obtained by pairwise identifications of faces in the boundary of certain polyhedral 3-balls. We prove that they are (n/d)-fold cyclic coverings of the 3-sphere branched over certain hyperbolic links of d+1 components, where d= (n/k). Then we study the closed 3-manifolds obtained by Dehn surgeries on the components of these links. Received: 27 November 1998 / Accepted: 12 May 1999     

A primary obstruction to topological embeddings¶and its applications

For a proper continuous map f:M→N between topological manifolds M and N with m≡ dimM < dimN≡m+k, a primary obstruction to topological embeddings θ(f) ∈H ^c _{m − k} (M; Z ₂) has been defined and studied by the authors in {9, 8, 2, 3], where H ^c _* denotes the singular homology with closed support. In this paper, we study the obstruction from the viewpoint of differential topology and give various applications. We first give some characterizations of embeddings among generic differentiable maps, which are refinements of the results in [9, 10]. Then we give a result concerning the number of connected components of the complement of the image of a codimension-1 continuous map with a normal crossing point, which generalizes the results in [6, 4, 5, 9]. Finally we give a simple proof of a theorem of Li and Peterson [20] about immersions of m-manifolds into (2m-1)-manifolds. Received: 3 December 1999 / Revised version: 10 October 2000     

Some involutory similarities

Matrices A,B over an arbitrary field F, when given to be similar to each other, are shown to be involutorily similar (over F) to each other (i.e.B = CAC^-1 for some C = C^-1 over F) in the following cases: (1)B= aI ? Afor some a ε F and (2) B = A^-1 . Result (2) for the cases where char F ≠ 2 is essentially a 1966 result of Wonenburger.     

The amalgamation of high distance Heegaard splittings is always efficient

Let M be a compact orientable manifold, and F be an essential closed surface which cuts M into two 3-manifolds M ₁ and M ₂. Let be a Heegaard splitting for i = 1, 2. We denote by d(S _i ) the distance of . If d(S ₁), d(S ₂) ≥ 2(g(M ₁) + g(M ₂) − g(F)), then M has a unique minimal Heegaard splitting up to isotopy, i.e. the amalgamation of and . Ruifeng Qiu is supported by NSFC(10625102).     

Guts of surfaces in punctured-torus bundles

Let M be a hyperbolic manifold of finite volume which fibers over the circle with fiber a once punctured torus, and let S be an arbitrary incompressible surface in M. We determine the characteristic Jaco-Shalen-Johannson-submanifold of M−S and show, in particular, that Guts (M, S) is empty. Received: 02 April 2004