arXiv research

Study shows bounds on symplectic fillings for spinal open book decompositions.

New geometric constructions for contact manifolds, not symplectically fillable.

The paper shows conditions under which certain 4-manifolds have no smooth spines.

This paper concerns thin presentations of knots K in closed 3-manifolds M^3 which produce S^3 by Dehn surgery, for some slope gamma. If M does not have a lens space as a connected summand, we first prove that all such thin presentations, with respect to any spine of M have only local maxima. If M is a lens space and K …

Paper proves uniqueness of bridge multisections for surfaces in 4-space.

Let $X$ be a connected compact 3-manifold with non-empty boundary. Consider the boundary $M$ of $X\times D^2$. $M$ is a 4-dimensional closed manifold and has the same fundamental group as $X$. Various examples of $X$ are known for which a certain assembly map $A:H_4(X;L)\to L_4(π_1(X))$ is injective. For such an $X$ an…

Algorithm detects surgical site infections from hospital data.

Let K be a connected finite complex. This paper studies the problem of whether one can attach a cell to some iterated suspension S^j K so that the resulting space satisfies Poincare duality. When this is possible, we say that S^j K is a spine. We introduce the notion of quadratic self duality and show that if K is quad…

New spines found in complex 4D shapes.

Paper proves unique contact structure supported by positive flow-spines.

Nonpositive towers property in 3-manifolds spines.

In this paper we raise the question whether every closed Riemannian manifold has a spine of minimal area, and we answer it affirmatively in the surface case. On constant curvature surfaces we introduce the spine systole, a continuous real function on moduli space that measures the minimal length of a spine in each surf…

Paper removes singularities from compact area minimizers in positive scalar curvature manifolds.

Unique simple spines of homotopy 2-spheres are shown to be ambiently isotopic.

Bing's house-like spines approximate all PL manifolds.

New method proves cosmetic surgery conjecture for certain knots.

Corrects a 1-off error in Harer's spine dimension calculation for decorated Teichmüller spaces.

Constructs Teichmüller curve to study Thurston spine structure.

This paper proves a map from flow-spines to contact structures is surjective.

The Thurston spine's properties are studied in relation to Morse-Smale complexes.

In this article we establish the relation between the spines of 3-manifolds and the polyhedra with identified faces. We do this by showing that the spines of the closed, connected, orientable 3-manifolds can be presented through polyhedra with identified faces in a very natural way. We also prove the equivalence betwee…

In this paper we develop an approach to conformal geometry of piecewise flat metrics on manifolds. In particular, we formulate the combinatorial Yamabe problem for piecewise flat metrics. In the case of surfaces, we define the combinatorial Yamabe flow on the space of all piecewise flat metrics associated to a triangul…

A special spine of a three-manifold is said to be poor if it does not contain proper simple subpolyhedra. Using the Turaev-Viro invariants, we establish that every compact three-dimensional manifold M with connected nonempty boundary has a finite number of poor special spines. Moreover, all poor special spines of the m…

The abstract extends Reidemeister theorem to 3-manifolds using diagrams of links and bands.

New geometric spine for Artin groups defined by cube complexes.

Paper shows mapping class group-equivariant Teichmüller space deformation to Thurston spine.

Thurston's spine dimension exceeds virtual cohomological dimension.

A spine is constructed for a non-orientable surface's decorated Teichmüller space.

New 3-manifold spines with unique Whitehead graphs identified.

Theory of link projections to 3-manifold spines, proving combinatorial moves for isotopic links.

Any bounding compact smooth manifold bounds a compact manifold with a spine consisting of transversely intersecting codimension one submanifolds. This paper provides details for a picture proof given in previous papers with S. Akbulut.

Simplified combinatorial descriptions of branched spines for 3-manifolds using primary MP move and sliding moves.

Two codimension-one submanifolds are cobordant if they have the same homology class.

Outer space and Teichmüller space fail well-rounded retract analogy.

Geometrical spines are defined for 3-manifolds with natural metrics, in particular, for lens manifolds. We show that any spine of L(p,q) close enough to its geometrical spine (i.e., to the cut locus with respect to the standard metric) contains at least E(p,q)-3 vertices, which is exactly the conjectured value for Matv…

$2$-stratifolds are a generalization of $2$-manifolds in that there are disjoint simple closed branch curves. We obtain a list of all closed $3$-manifolds that have a $2$-stratifold as a spine.

In this paper we analyze the long-time behavior of 3 dimensional Ricci flows with surgery. Our main result is that if the surgeries are performed correctly, then only finitely many surgeries occur and after some time the curvature is bounded by $C t^{-1}$. This result confirms a conjecture of Perelman. In the course of…

We extend Matveev's complexity of 3-manifolds to PL compact manifolds of arbitrary dimension, and we study its properties. The complexity of a manifold is the minimum number of vertices in a simple spine. We study how this quantity changes under the most common topological operations (handle additions, finite coverings…

New proof confirms surfaces can be divided into polygons.

Continuum-wise hyperbolicity is exactly the pseudo-Anosov dynamics with spine singularities.

We establish a calculus for branched spines of 3-manifolds by means of branched Matveev-Piergallini moves and branched bubble-moves. We briefly indicate some of its possible applications in the study and definition of State-Sum Quantum Invariants.

This note popularizes a proof for 3-manifold triangulations and spines.

We introduce and analyze the characteristic foliation induced by a contact structure on a branched surface, in particular a branched standard spine of a 3-manifold. We extend to (fairly general) singular foliations of branched surfaces the local existence and uniqueness results which hold for genuine surfaces. Moreover…

The paper studies curves in surfaces using flow-spines and apparent contours.

In this paper we develop a new theory of infinitesimal harmonic deformations for compact hyperbolic 3-manifolds with ``tubular boundary''. In particular, this applies to complements of tubes of radius at least $R_0 = \arctanh(1/\sqrt{3}) \approx 0.65848$ around the singular set of hyperbolic cone manifolds, removing th…

New contractible complex shows virtual cohomological dimension of RAAGs.

For a 3-dimensional manifold $M^3$, its complexity $c(M^3)$, introduced by S.Matveev, is the minimal number of vertices of an almost simple spine of $M^3$; in many cases it is equal to the minimal number of tetrahedra in a singular triangulation of $M^3$. An approach to estimating $c(M^3)$ from below for total spaces o…

We show that under reasonable conditions, the spines of the handlebodies of a strongly irreducible Heegaard splitting will intersect a closed ball in a graph which is isotopic into the boundary of the ball. This is in some sense a generalization of the results by Scharlemann on how a strongly irreducible Heegaard split…