arXiv research

The paper proves a criterion for virtual Euler class one in hyperbolic 3-manifolds.

The constructions of the virtual Euler (or moduli) cycles and their properties are explained and developed systematically in the general abstract settings.

We show that many 3-manifold groups have no nonabelian surface subgroups. For example, any link of an isolated complex surface singularity has this property. In fact, we determine the exact class of closed graph-manifolds which have no immersed pi_1-injective surface of negative Euler characteristic. We also determine …

New polynomial for checkerboard-colorable 4-valent virtual graphs.

Study shows virtually abelian subgroups have commensurable counterparts in mapping class groups.

Virtual invariants defined from sheaves on surfaces.

We conjecture a formula for the generating function of virtual $χ_y$-genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with holomorphic 2-form. Specializing the conjecture to minimal surfaces of general type and to virtual Euler characteristics, we recover (part of) a formula of C. Vafa and E. Witten. The…

We show that, if g is more than or equal to 2, the virtual cohomological dimension of the mapping class group of a 3-dimensional handlebody of genus g is equal to 4g-5 and the Euler number of it is equal to 0.

Study non-vanishing $\ell^2$-Betti numbers for specific groups.

Outer automorphism group of hyperbolic groups is HHG under certain conditions.

New theory for unoriented virtual links, extending classical invariants.

Analyzes a finite set of metrics and functions to determine manifold torsion.

Decides undecidability of equations and first-order theory for Seifert 3-manifold groups.

In recent years, the RFRS condition has been used to analyze virtual fibering in 3-manifold topology. Agol's work shows that any 3-manifold with zero Euler characteristic satisfying the RFRS condition on its fundamental group virtually fibers over the circle. In this note we will show that a finitely generated nilpoten…

Constructs 4-manifolds with positive Euler characteristic proving a conjecture.

The article calculates the minimal model dimensions for classifying spaces of surface braid groups.

New evidence supports the Euler class one conjecture for tight contact structures.

Cieliebak, Mundet i Riera and Salamon recently formulated a definition of branched submanifold of Euclidean space in connection with their discussion of multivalued sections and the Euler class. This note proposes an intrinsic definition of a weighted branched manifold Z that is obtained from the usual definition of or…

In this paper, we study Euler classes in groups of homeomorphisms of Seifert fibered 3-manifolds. We show that, in contrast to the familiar Euler class for $\mathrm{Homeo}_0(S^1)^δ$, these Euler classes for $\mathrm{Homeo}_0(M^3)^δ$ are unbounded classes. In fact, we give examples of flat topological M bundles over a g…

The study enumerates virtual quandles up to isomorphism.

A virtual $n$-string is a chord diagram with $n$ core circles and a collection of arrows between core circles. We consider virtual $n$-strings up to virtual homotopy, compositions of flat virtual Reidemeister moves on chord diagrams. Given a virtual 1-string $α$, Turaev associated a based matrix that encodes invariants…

This is the first of two articles in which we give a proof - for a broad class of four-manifolds - of Witten's conjecture that the Donaldson and Seiberg-Witten series coincide, at least through terms of degree less than or equal to c-2, where c is a linear combination of the Euler characteristic and signature of the fo…

Let $D$ be an oriented classical or virtual link diagram with directed universe $\vec{U}$. Let $C$ denote a set of directed Euler circuits, one in each connected component of $U$. There is then an associated looped interlacement graph $L(D,C)$ whose construction involves very little geometric information about the way …

Minimal crossing virtual links have minimal supporting genus.

Paper shows Euler class vanishes in certain subgroup of mapping class group.

Paper constructs a cohomology class related to McDuff's secondary class, proving it transgresses to the Euler class of foliated sphere bundles.

The paper extends Euler class theory to measurable cocycles.

Study virtual fundamental classes of derived manifolds, proving invariant vanishes.

Extends Euler class result to symplectic group.

Generalizes meander diagrams to virtual knots and introduces new invariants.

In this article we give a sharp upper bound on the possible values of the Euler characteristic for a minimal symplectic filling of a tight contact structure on a lens space. This estimate is obtained by looking at the topology of the spaces involved, extending this way what we already knew from the universally tight ca…

Study shows torsion grows subexponentially in book of I-bundles but can grow exponentially in non-regular covers.

In this paper, we discuss filamentations on oriented chord diagrams. When a filamentation cannot be realized on an oriented chord diagram, then the corresponding flat virtual knot is non-trivial. If a flat knot diagram is non-trivial, then any virtual diagram whose shadow is the flat diagram must also be non-trivial. W…

We calculate the virtually-cyclic dimension of the mapping class group of a sphere with at most six punctures. As an immediate consequence, we obtain the virtually-cyclic dimension of the mapping class group of the twice-holed torus and of the closed genus-two surface.

Continues work on derived manifolds and symplectic schemes, constructing virtual classes.

Handlebody groups are virtual duality groups in positive genus.

Proves Alexander and Markov theorems for higher genus virtual doodles.

We define the virtual bridge number $vb(K)$ and the virtual unknotting number $vu(K)$ invariants for virtual knots. For ordinary knots $K$ they are closely related to the bridge number $b(K)$ and the unknotting number $u(K)$ and we have $vu(K)\leq u(K), vb(K)\leq b(K).$ There are no ordinary knots $K$ with $b(K)=1.$ We…

Study Euler class of surface bundles with nontrivial results.

The paper disproves a conjecture about 3D manifolds using even lattice points.

Let $({\bf X},ω_{\bf X}^*)$ be a separated, $-2$-shifted symplectic derived $\mathbb C$-scheme, in the sense of Pantev, Toen, Vezzosi and Vaquie arXiv:1111.3209, of complex virtual dimension ${\rm vdim}_{\mathbb C}{\bf X}=n\in\mathbb Z$, and $X_{\rm an}$ the underlying complex analytic topological space. We prove that …

We find two new classes of virtually fibered classic Montesinos links of type $\widetilde{SL_2}$.

This paper proves a converse to Thurston's 1976 observation for taut foliations on hyperbolic 3-manifolds.

This paper studies symplectic structures on elliptic surfaces with positive Euler number.

The paper defines and proves non-triviality of volume and Euler classes in bounded cohomology of transformation groups.

Virtual knots can be transformed by $2k$-moves, affecting their writhes.

Study on Euler class and flux homomorphisms for non-orientable surfaces.

A virtual link is a generalization of a classical link that is defined as an equivalence class of certain diagrams, called virtual link diagrams. It is further generalized to a twisted link. Twisted links are in one-to-one correspondence with stable equivalence classes of links in oriented thickenings of (possibly non-…