arXiv research

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…

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.

Extends Euler class result to symplectic group.

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.

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

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 Euler class of taut foliations on Dehn fillings is studied, leading to new left-orderable slopes.

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.

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 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…

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 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.

It is well-known that odd-dimensional manifolds have Euler characteristic zero. Furthemore orientable manifolds have an even Euler characteristic unless the dimension is a multiple of $4$. We prove here a generalisation of these statements: a $k$-orientable manifold (or more generally Poincaré complex) has even Euler c…

A classical result says that a free action of the circle $\Bbb{S}^1$ on a topological space $X$ is geometrically classified by the orbit space $B$ and by a cohomological class ${H}^{^{2}}{(B,\Bbb{Z})}$, the Euler class. When the action is not free we have a difficult open question: $Π$ : "Is the space $X$ determined by…

Develops a new theory of localization in algebraic geometry.

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…

This note proves that, as K-theory elements, the symbol classes of the de Rham operator and the signature operator on a closed manifold of even dimension are congruent mod 2. An equivariant generalization is given pertaining to the equivariant Euler characteristic and the multi-signature.

Estimates the dual Thurston norm for foliations on negative curvature 3-manifolds.

Study characteristic classes of a specific type of determinantal varieties.

We prove the following generalization of the classical Lichnerowicz vanishing theorem: if $F$ is an oriented flat vector bundle over a closed spin manifold $M$ such that $TM$ carries a metric of positive scalar curvature, then $<\widehat A(TM)e(F),[M]>=0$, where $e(F)$ is the Euler class of $F$.

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…

Proves Massey's theorems on complex structure obstructions.

Study on group cocycles for volume-preserving diffeomorphisms.

New Euler characteristics for groupoids generalize orbifold Euler characteristics.