arXiv research

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

For a manifold with boundary, the restriction of Chern's transgression form of the Euler curvature form over the boundary is closed. Its cohomology class is called the secondary Chern-Euler class and used by Sha to formulate a relative Poincaré-Hopf theorem, under the condition that the metric on the manifold is locall…

Extends Euler class formula to general connections with metric.

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

New representations preserve hyperbolicity but not Fuchsian property.

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…

Characterizes components of representations space for punctured surfaces.

We consider a class of stochastic path-dependent volatility models where the stochastic volatility, whose square follows the Cox-Ingersoll-Ross model, is multiplied by a (leverage) function of the spot price, its running maximum, and time. We propose a Monte Carlo simulation scheme which combines a log-Euler scheme for…

We define the tangent Euler top in General Relativity through a constrained Lagrangian on the orthonormal frame bundle. The corresponding motions are studied to various degrees of approximation, the lowest of which is shown to yield the Mathisson-Papapetrou equations.

One-parameter hyperbolic planar motion was first studied by S. Y$\ddot{\texttt{u}}$ce and N. Kuruo$\tilde{\texttt{g}}$lu. Moreover, they analyzed the relationships between the absolute, relative and sliding velocities of one-parameter hyperbolic planar motion as well as the related pole curves, \cite{Yuc}. One-paramete…

Let $D$ be a 2-dimensional closed unit disk and $\rm{Symp}(D,0)_{\rm{rel}}$ the group of symplectomorphisms preserving the origin and the boundary $\partial D$ pointwise. We consider the $\mathbb{R}$-valued flux homomorphism on $\rm{Symp}(D,0)_{\rm{rel}}$ and define the central $\mathbb{R}$-extension called the $\mathb…

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

In this paper, it is shown that every closed hyperbolic 3-manifold contains an immersed quasi-Fuchsian closed subsurface of odd Euler characteristic. The construction adopts the good pants method, and the primary new ingredient is an enhanced version of the connection principle, which allows one to connect any two fram…

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…

This paper provides a precise sense in which the time t map for the Euler equations of an ideal fluid in a region in R^n (or a smooth compact n-manifold with boundary) is a Poisson map relative to the Lie-Poisson bracket associated with the group of volume preserving diffeomorphism group. This is interesting and nontri…

The paper shows how to unknot certain nonorientable surfaces in 4-dimensional spaces.

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

New group found not satisfying quasi-isometric triviality property.

The paper extends Euler class theory to measurable cocycles.

Extends Euler class result to symplectic group.

In "The Gel'fand-Kalinin-Fuks class and characteristic classes of transversely symplectic foliations", arXiv:0910.3414, (October 2009) by D.Kotschick and S.Morita, the relative Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields without constant vector fields on 2n-plane were characterized b…

Study Euler class of surface bundles with nontrivial results.

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

Let Y be a closed oriented 3-manifold with a contact form such that all Reeb orbits are nondegenerate. The embedded contact homology (ECH) index associates an integer to each relative 2-dimensional homology class of surfaces whose boundary is the difference between two unions of Reeb orbits. This integer determines the…

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.

Combinatorial transgressions are secondary invariants of a space admitting triangulations. They arise from subdivisions and are analogous to transgressive forms such as those arising in Chern-Weil theory. Unlike combinatorial characteristic classes, combinatorial transgressions have not been previously studied. First, …

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

We study local, global and local-to-global properties of threefolds with certain singularities. We prove criteria for these threefolds to be rational homology manifolds and conditions for threefolds to satisfy rational Poincaré duality. We relate the topological Euler characteristic of elliptic Calabi-Yau threefolds wi…

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.

We examine open books with powers of fibered Dehn twists as monodromy. The resulting contact manifolds can be thought of as Boothby-Wang orbibundles over symplectic orbifolds. Using the mean Euler characteristic of equivariant symplectic homology we can distinguish these contact manifolds and hence show that some fiber…

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

Formula for Euler characteristic of moduli spaces of Abelian differentials.

Let $(X,ω)$ be a symplectic rational 4 manifold. We study the space of tamed almost complex structures $\mathcal{J}_ω$ using a fine decomposition via smooth rational curves and a relative version of the infinite-dimensional Alexander duality. This decomposition provides new understandings of both the variation and stab…

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 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…

We give counterexamples to a question of Bowditch that if a non-elementary type-preserving representation $ρ:π_1(Σ_{g,n})\rightarrow PSL(2;\mathbb R)$ of a punctured surface group sends every non-peripheral simple closed curve to a hyperbolic element, then must $ρ$ be Fuchsian. The counterexamples come from relative Eu…

We give the definition of the Seiberg-Witten-Floer homology group for a homology 3-sphere. Its Euler characteristic number is a Casson-type invariant. For a four-manifold with boundary a homology sphere, a relative Seiberg-Witten invariant is defined taking values in the Seiberg-Witten-Floer homology group, these relat…

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.

It is well-known that a knot in a contact manifold $(M,C)$ transverse to a trivialized contact structure possesses the natural framing given by the first of the trivialization vectors along the knot. If the Euler class $e_C\in H^2(M)$ of $C$ is nonzero, then $C$ is nontrvivializable and the natural framing of transvers…