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     

Finite order q-invariants of immersions of surfaces into 3-space

Given a surface F, we are interested in valued invariants of immersions of F into , which are constant on each connected component of the complement of the quadruple point discriminant in . Such invariants will be called “q-invariants.” Given a regular homotopy class , we denote by the space of all q-invariants on A of order . We show that ifF is orientable, then for each regular homotopy class A and each n, $\dim (V_n (A) / V_{n-1}(A) ) \leq 1$. Received June 15, 1999; in final form September 22, 1999 / Published online October 30, 2000     

Order one invariants of immersions of surfaces into 3-space

We classify all order one invariants of immersions of a closed orientable surface F into ³, with values in an arbitrary Abelian group . We show that for any F and and any regular homotopy class of immersions of F into ³, the group of all order one invariants on is isomorphic to is the group of all functions from a set of cardinality . Our work includes foundations for the study of finite order invariants of immersions of a closed orientable surface into ³, analogous to chord diagrams and the 1-term and 4-term relations of knot theory.Partially supported by the Minerva FoundationMathamatics Subject Classification (2000):57M, 57R42     

On 4-manifolds homotopy equivalent to circle bundles over 3-manifolds

We give criteria for a closed 4-manifold to be homotopy equivalent to the total space of an S¹-bundle over a closed 3-manifold. In the aspherical case the conditions are that the Euler characteristic be 0 and that the fundamental group have an infinite cyclic normal subgroup such that the quotient group has one end and finite cohomological dimension. Under further assumptions on this quotient group we characterize the total spaces of such bundles over -or H² × E¹-manifolds and over E³-, Nil³- or Sol³-manifolds up to s-cobordism and homeomorphism respectively.     

Regular homeomorphisms of connected 3-manifolds

We establish some equivalent conditions to the Hilbert–Smith conjecture on connected n-manifolds M. For n = 3; we show that regularly almost periodic homeomorphisms of M are periodic; this result extends Theorem 5.34 of Gottschalk and Hedlund (Topological Dynamics. American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, 1956). For the special case of \({M = \mathbb{R}^3}\) , we extend the result of Brechner (Pac J Math 59(2):367–374, 1975) saying that “almost periodic homeomorphisms of the plane are periodic” to \({\mathbb{R}^3}\) , and we show that any compact abelian group of homeomorphisms of \({\mathbb{R}^3}\) is either finite or topologically equivalent to a subgroup of the orthogonal group O(3).     

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     

Topological invariants of stable immersions of oriented 3-manifolds in R4

We show that the $\mathbb{Z}$-module of first order local Vassiliev type invariants of stable immersions of oriented 3-manifolds into ${\mathbb{R}}^4$ is generated by 3 topological invariants: The number of pairs of quadruple points and the positive and negative linking invariants ${?}^{+}$ and ${?}^{?}$ introduced by V. Goryunov (1997) [7]. We obtain the expression for the Euler characteristic of the immersed 3-manifold in terms of these invariants. We also prove that the total number of connected components of the triple points curve is a non-local Vassiliev type invariant.     

Two classes of conformally flat contact metric 3-manifolds

We give a classification of 3—dimensional conformally flat contact metric manifolds satisfying: =0(=L g) orR(Y, Z)=k[(Z)Y–(Y)Z]+[(Z)hY]–(Y)hZ] wherek and are functions. It is proved that they are flat (the non-Sasakian case) or of constant curvature 1 (the Sasakian case).     

A homotopy classification of certain 7-manifolds

This paper gives a homotopy classification of Wallach spaces and a partial homotopy classification of closely related spaces obtained by free -actions on and on .