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?     

A geometric classification of immersions of 3-manifolds into 5-space

In this paper we define two regular homotopy invariants c and i for immersions of oriented 3-manifolds into ⁵ in a geometric manner. The pair (c(f), i(f)) completely describes the regular homotopy class of the immersion f. The invariant i corresponds to the 3-dimensional obstruction that arises from Hirsch-Smale theory and extends the one defined in [10] for immersions with trivial normal bundle.Mathematics Subject Classification (2000): 57N35, 57R45, 57R42     

Lifting the 3-dimensional invariant of 2-plane fields on 3-manifolds

Homotopy classes of plane fields on 3-manifolds have been classified using a 2-dimensional invariant Γ and a 3-dimensional invariant θ by R. Gompf. Under regular covering maps, Γ lifts in the natural way. The lifting property of θ remained unresolved. In this paper, we present the lifting property of θ together with applications to Lens spaces. The applications help in specifying the liftings of the contact structures of the Lens space L(p,1) when lifted to S³.     

Isotopy invariants for smooth tori in 4-manifolds

In this paper we define invariants under smooth isotopy for certain two-dimensional knots using some refined Cerf theory. One of the invariants is the knot type of some classical knot generalizing the string number of closed braids. The other invariant is a generalization of the unique invariant of degree 1 for classical knots in 3-manifolds. Possibly, these invariants can be used to distinguish smooth embeddings of tori in some 4-manifolds but which are equivalent as topological embeddings.     

Configurations of few lines in 3-space. Isotopy,chirality and planar layouts

We give a characterization of the planar layouts of configurations with at most five lines. From this we obtain a new proof of Viro's theorem that the isotopy type of such configurations is completely determined by chirality. We extend this result to labelled configurations. We also give an infinite family of non-realizable line diagrams, called alternatingC-angles, not containing non-realizable subdiagrams.     

Parallelisability of 3-manifolds via surgery

We present two proofs that all closed, orientable 3-manifolds are parallelisable. Both are based on the Lickorish–Wallace surgery presentation; one proof uses a refinement of this presentation due to Kaplan and some basic contact geometry. This complements a recent paper by Benedetti–Lisca.     

Some classification of surfaces of revolution in Minkowski 3-space

We study the surfaces of revolution with the non-degenerate second fundamental form in Minkowski 3-space. In particular, we investigate the surfaces of revolution satisfying an equation in terms of the position vector field and the 2nd-Laplacian in Minkowski 3-space. As a result, we give some new examples of the surfaces of revolution with light-like axis in Minkowski 3-space.     

Topological classification of gradient-like diffeomorphisms on 3-manifolds

We give a complete invariant, called global scheme, of topological conjugacy classes of gradient-like diffeomorphisms, on compact 3-manifolds. Conversely, we can realize any abstract global scheme by such a diffeomorphism.     

Skein related links in 3-manifolds

This paper addresses two problems in the skein theory of homotopy spheres first posed by P. Traczyk. Solutions to both problems are obtained for a large class of manifolds and, since one of the basic techniques used requires the first homology group of the ambient manifold to be torsion free, the extent to which this hypothesis is actually necessary is further explored.     

Cartesian product stabilization of 3-manifolds

The Cartesian product of a closed, orientable prime geometric 3-manifold and a closed orientable surface is unique except for the case of the Cartesian product of a special class of Seifert manifolds and a torus. The same type of uniqueness holds for stabilization of 3-manifolds by an n-dimensional torus. Cartesian squares of Seifert fibered 3-manifolds are completely classified.     

Time-like Willmore surfaces in Lorentzian 3-space

Let R13 be the Lorentzian 3-space with inner product (, ). Let Q3 be the conformal compactification of R13, obtained by attaching a light-cone C∞ to R13 in infinity. Then Q3 has a standard conformal Lorentzian structure with the conformal transformation group O(3,2)/{±1}. In this paper, we study local conformal invariants of time-like surfaces in Q3 and dual theorem for Willmore surfaces in Q3. Let M (?) R13 be a time-like surface. Let n be the unit normal and H the mean curvature of the surface M. For any p ∈ M we define S12(p) = {X ∈ R13 (X - c(P),X - c(p)) = 1/H(p)2} with c(p) = P 1/H(p)n(P) ∈ R13. Then S12 (p) is a one-sheet-hyperboloid in R3, which has the same tangent plane and mean curvature as M at the point p. We show that the family {S12(p),p ∈ M} of hyperboloid in R13 defines in general two different enveloping surfaces, one is M itself, another is denoted by M (may be degenerate), and called the associated surface of M. We show that (i) if M is a time-like Willmore surface in Q3 with non-degenerate associated surface M, then M is also a time-like Willmore surface in Q3 satisfying M = M; (ii) if M is a single point, then M is conformally equivalent to a minimal surface in R13.     

Incompressible Surfaces in Handlebodies and Closed 3-manifolds

in this paper we prove that for any positive integer n, 1) a handlebody of genus 2contains a separating incompressible surface of genus n, and 2) there exists a closed 3manifold of heegaard genus 2 which contains a separating incompressible surface of genus n.     

Geodesic knots in closed hyperbolic 3-manifolds

We consider the existence of simple closed geodesics or “geodesic knots” in finite volume orientable hyperbolic 3-manifolds. Every such manifold contains at least one geodesic knot by results of Adams, Hass and Scott in (Adams et al. Bull. London Math. Soc. 31: 81–86, 1999). In (Kuhlmann Algebr. Geom. Topol. 6: 2151–2162, 2006) we showed that every cusped orientable hyperbolic 3-manifold in fact contains infinitely many geodesic knots. In this paper we consider the closed manifold case, and show that if a closed orientable hyperbolic 3-manifold satisfies certain geometric and arithmetic conditions, then it contains infinitely many geodesic knots. The conditions on the manifold can be checked computationally, and have been verified for many manifolds in the Hodgson-Weeks census of closed hyperbolic 3-manifolds. Our proof is constructive, and the infinite family of geodesic knots spiral around a short simple closed geodesic in the manifold.      

Invariants of piecewise-linear 3-manifolds

This paper presents an algebraic framework for constructing invariants of closed oriented 3-manifolds by taking a state sum model on a triangulation. This algebraic framework consists of a tensor category with a condition on the duals which we have called a spherical category. A significant feature is that the tensor category is not required to be braided. The main examples are constructed from the categories of representations of involutive Hopf algebras and of quantised enveloping algebras at a root of unity.

     

A census of genus-two 3-manifolds up to 42 coloured tetrahedra

We improve and extend to the non-orientable case a recent result of Karábaš, Mali?ký and Nedela concerning the classification of all orientable prime 3-manifolds of Heegaard genus two, triangulated with at most 42 coloured tetrahedra.     

Curvature loci of 3-manifolds

We refine the affine classification of real nets of quadrics in order to obtain generic curvature loci of regular 3-manifolds in ${\mathbb{R}}^6$ and singular corank one 3-manifolds in ${\mathbb{R}}^5$. For this, we characterize the type of the curvature locus by the number and type of solutions of a system of equations given by four ternary cubics (which is a determinantal variety in some cases). We also study how singularities of the curvature locus of a regular 3-manifold can go to infinity when the manifold is projected orthogonally in a tangent direction.