arXiv research

In this paper we consider type-preserving representations of the fundamental group of the three--holed projective plane into $\mathrm{PGL}(2, \R) =\mathrm{Isom}(\HH^2)$ and study the connected components with non-maximal euler class. We show that in euler class zero for all such representations there is a one simple cl…

Holomorphic connections found on Riemann surfaces with Fuchsian monodromy.

As for the theory of maximal representations, we introduce the volume of a Zimmer's cocycle $Γ\times X \rightarrow \mbox{PO}^\circ(n, 1)$, where $Γ$ is a torsion-free (non-)uniform lattice in $\mbox{PO}^\circ(n, 1)$, with $n \geq 3$, and $X$ is a suitable standard Borel probability $Γ$-space. Our numerical invariant ex…

We study the (relative) SL(2,C) character varieties of the four-holed sphere and the action of the mapping class group on it. We describe a domain of discontinuity for this action, and, in the case of real characters, show that this domain of discontinuity may be non-empty on the components where the relative euler cla…

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

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 height function of various surfaces decomposes into finite sums of scaled and translated versions of itself.

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.

Extends Euler class result to symplectic group.

In this article, we study the maximal length of positive Dehn twist factorizations of surface mapping classes. In connection to fundamental questions regarding the uniform topology of symplectic 4-manifolds and Stein fillings of contact 3-manifolds coming from the topology of supporting Lefschetz pencils and open books…

Extends Euler's problem to Lorentz-Minkowski plane.

Study Euler class of surface bundles with nontrivial results.

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

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

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.

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

The study counts 23 maximal 1-systems on a torus with 2 punctures.

A formula calculates the Euler class of foliations using dual graphs.

The paper studies spectral properties of Jacobi operator for surfaces with nonpositive Euler characteristic.

Toolbox for stochastic Euler equations using Ebin-Marsden theory.

Extends Euler class formula to general connections with metric.

The Whitehead link exterior lacks most Euler class taut foliations.

The paper bounds the $L^2$-norm of Euler class for foliations on 3-manifolds.

The Euler class conjecture links geometric structures to integral points on the Thurston norm ball.

The paper constructs 3-manifolds with co-orientable taut foliations but no foliations with vanishing Euler class.

The authors give a short survey of previous results on $δ$-homogeneous Riemannian manifolds, forming a new proper subclass of geodesic orbit spaces with non-negative sectional curvature, which properly includes the class of all normal homogeneous Riemannian manifolds. As a continuation and an application of these resul…

Formula for Euler characteristic of moduli spaces of Abelian differentials.

The paper derives a local formula for the Euler number of circle bundles.

A new homomorphism connects group actions on circles to Euler classes.

We study groups of C^1 orientation-preserving homeomorphisms of the plane, and pursue analogies between such groups and circularly-orderable groups. We show that every such group with a bounded orbit is circularly-orderable, and show that certain generalized braid groups are circularly-orderable. We also show that the …

We exhibit a cocycle in the simplicial de Rham complex which represents the Euler class. As an application, we construct a Lie algebra cocycle on $L\mathfrak{so}(4)$.

The paper conjectures a 4D characterization of tight contact structures and proves it for certain cases.

The paper proves non-triviality of certain classes in sphere bundle cohomology.

In 1976, Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that conversely, any integral second cohomology class with norm equal to one is the Euler class of a taut foliation. This is the first from a series of two papers that together give a neg…

In this paper, we use a result of Dahmani to show that the Euler class of some power subgroup (the subgroup normally generated by a fixed power of Dehn twist about a non-separating curve) is trivial inside the mapping class group of once punctured surface.

Schwarzian derivative connects to Euler-Lagrange equations in variational calculus.

We prove that the mean Euler characteristic of a Gorenstein toric contact manifold, i.e. a good toric contact manifold with zero first Chern class, is equal to half the normalized volume of the corresponding toric diagram and give some applications. A particularly interesting one, obtained using a result of Batyrev and…

We give a summary of known results on the maximal distances between Dehn fillings on a hyperbolic 3-manifold that yield 3-manifolds containing a surface of non-negative Euler characteristic that is either essential or Heegaard.

We define and study the secondary Chern-Euler class for a general submanifold of a Riemannian manifold. Using this class, we define and study index for a vector field with non-isolated singularities on a submanifold. As an application, our studies give conceptual proofs of a classical result of Chern.

Apparently a lost theorem of Thurston states that the cube of the Euler class $e^3\in H^6(BDiff^δ_ω(S^1);\mathbb{Q})$ is zero where $Diff^δ_ω(S^1)$ is the analytic orientation preserving diffeomorphisms of the circle with the discrete topology. This is in contrast with Morita's theorem that the powers of the Euler clas…

Let $e$ denote the Euler class on the space $Hom(Γ_g, PSL(2,\mathbb R))$ of representations of the fundamental group $Γ_g$ of the closed surface $Σ_g$ of genus $g$. Goldman showed that the connected components of $Hom(Γ_g, PSL(2,\mathbb R))$ are precisely the inverse images $e^{-1}(k)$, for $2-2g\leq k\leq 2g-2$, and t…

The paper explores actions of surface mapping class groups on 3-manifolds.

The Euler characteristic is the only additive topological invariant for spaces of certain sort, in particular, for manifolds with some finiteness properties. A generalization of the notion of a manifold is the notion of a V-manifold. Here we discuss a universal additive topological invariant of V-manifolds: the univers…

A 4-manifold is parallelizable if its Stiefel-Whitney and Pontryagin classes vanish.

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