Related papers: Handlebody decompositions of 3-manifolds and polyc…

This is the last of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold.

Let M_1 and M_2 be compact, orientable 3-manifolds, and M the manifold obtained by gluing some component F of \bdy M_1 to some component of \bdy M_2 by a homeomorphism \phi. We show that when \phi is "sufficiently complicated" then (1) the…

We present a new proof of Reidemeister and Singer's Theorem that any two Heegaard splittings of the same 3-manifold have a common stabilization. The proof leads to an upper bound on the minimal genus of a common stabilization in terms of…

Let $M$ be a 3-manifold with torus boundary components $T_1$ and $T_2$. Let $\phi \colon T_1 \to T_2$ be a homeomorphism, $M_\phi$ the manifold obtained from $M$ by gluing $T_1$ to $T_2$ via the map $\phi$, and $T$ the image of $T_1$ in…

In this paper, we give infinitely many non-Haken hyperbolic genus three 3-manifolds each of which has a finite cover whose induced Heegaard surface from some genus three Heegaard surface of the base manifold is reducible but can be…

Let M be a closed, connected and oriented 3-manifold. This article is the first of a five part series that constructs an isomorphism between the Heegaard Floer homology groups of M and the corresponding Seiberg-Witten Floer homology groups…

We study the isosceles three-body problem with Manev interaction. Using a McGehee-type technique, we blow up the triple collision singularity into an invariant manifold, called the collision manifold, pasted into the phase space for all…

The main result is a short effective proof of Tao Li's theorem that a closed non Haken hyperbolic 3-manifold N has at most finitely many irreducible Heegaard splittings.

We propose a way to derive polynomial invariants of closed, orientable $3$-manifolds from Heegaard diagrams via cellularly embedded graphs. Given a Heegaard diagram of an irreducible $3$-manifold $M$, we associate a Heegaard graph $G\subset…

In this paper we generalize previous work on decomposition in three-dimensional orbifolds by 2-groups realized as analogues of central extensions, to orbifolds by more general 2-groups. We describe the computation of such orbifolds in…

We examine three key conjectures in 3-manifold theory: the virtually Haken conjecture, the positive virtual b_1 conjecture and the virtually fibred conjecture. We explore the interaction of these conjectures with the following seemingly…

This note surveys recent progress toward the profinite rigidity of orientable finite-volume hyperbolic 3-manifolds. Beginning in a brief review of some basic settings of profinite completion and rigidity of general groups, we state the…

In this paper, we describe the relation between the study of closed connected surfaces embedded in $S^3$ and the theory of handlebody-knots. By Fox's theorem, a pair of handlebody-knots is associated to a closed connected surface embedded…

Bordered Heegaard Floer homology is an invariant for 3-manifolds, which associates to a surface F an algebra A(Z), and to a 3-manifold Y with boundary, together with an orientation-preserving diffeomorphism \phi from F to \bdy Y, a module…

We give the rectangle condition for strong irreducibility of Heegaard splittings of $3$-manifolds with non-empty boundary. We apply this to a generalized Heegaard splitting of a $2$-fold covering of $S^3$ branched along a link. The…

Trisections of closed 4-manifolds, first defined and studied by Gay and Kirby, have proved to be a useful tool in the systematic analysis of 4-manifolds via handlebodies. Subsequent work of Abrams, Gay, and Kirby established a connection…

Associated to an embedded surface in the $3$-sphere, we construct a diagram of fundamental groups, and prove that it is a complete invariant, wherefrom we deduce complete invariants of handlebody links, tunnels of handlebody links, and…

Let $\F$ be a compact surface and let $I$ be the unit interval. This paper gives a standard form for all 2-sided incompressible surfaces in the 3-manifold $\F \times I$. Since $\F \times I$ is a handlebody when $\F$ has boundary, this…

Using basic properties of one-sided Heegaard splittings, a direct proof that geometrically compressible one-sided splittings of RP^3 are stabilised is given. The argument is modelled on that used by Waldhausen to show that two-sided…

A Heegaard diagram for a 3-manifold is regarded as a pair of simplexes in the complex of curves on a surface and a Heegaard splitting as a pair of subcomplexes generated by the equivalent diagrams. We relate geometric and combinatorial…