Extends Euler class result to symplectic group.
problem Relationship between bounded Euler class and symplectic rotation number.
method Extends Ghys's result to symplectic group.
result Establishes relationship between bounded Euler class and symplectic rotation number.
Study Euler and Betti numbers of homology groups for a specific type of superalgebra.
problem Calculating Euler and Betti numbers for homology groups of pre Lie superalgebras.
method Introduced double weighted chain spaces to analyze pre Lie superalgebras of multi-vector fields with polynomial coefficients. Calculated Euler and Betti numbers for these homology groups.
result Derived formulas for Euler and Betti numbers of homology groups of pre Lie superalgebras.
The paper derives a local formula for the Euler number of circle bundles.
problem Calculating the Euler number of circle bundles over surfaces.
method Derives a local formula for the Euler number using quasisections and singularities.
result The Euler number of the bundle equals the sum of weights of singularities of a quasisection.
A rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers. In dimension at least three only multiples of the top Chern number, which is the Euler characteristic, are invarian…
Study the topological information of map germs using Euler obstruction.
problem Capturing topological information from map germs with complex singular spaces.
method Investigates the Euler obstruction and its relation to local Euler obstruction and Brasselet number.
result Relates Chern number to the number of cusps in a perturbed map-germ.
This paper studies symplectic structures on elliptic surfaces with positive Euler number.
problem Determining symplectic representatives for cohomology classes on elliptic surfaces.
method Analyzes the symplectic cone for elliptic surfaces with positive Euler number.
result Characterizes the symplectic cone for elliptic surfaces with positive Euler number.
Holomorphic Euler number vanishes for certain Kähler manifolds.
problem Finding obstructions for Kähler manifolds with specific curvature properties.
method Vanishing theorem of Dolbeault-Morse-Novikov cohomology.
result Holomorphic Euler number of Kähler manifolds with almost nonnegative Ricci curvature vanishes.
Study finds Euler number inequality for certain Kähler manifolds.
problem Euler number of locally conformally Kähler manifolds.
method Analyzes properties of locally conformally Kähler manifolds with non-positive or negative sectional curvature.
result Euler number inequality for homeomorphic LCK manifolds.
The Euler number of special symplectic hyperbolic manifolds is positive.
problem Understanding the Euler number of symplectic hyperbolic manifolds.
method Study L2-harmonic forms on the universal covering space and prove the Singer conjecture. result The Euler number of a special symplectic manifold satisfies (−1)nχ(X)>0. The paper investigates the relationship between curvature operator and Euler number on manifolds.
problem Relationship between curvature operator and Euler number on manifolds.
method Analysis based on vanishing theorems for a Dirac operator associated with a smooth 1-form.
result The Euler number of a compact 2m-dimensional manifold with ANCO and nontrivial first de Rham cohomology group vanishes.
Everyone knows that the Euler characteristic of a combinatorial manifold is given by the alternating sum of its numbers of simplices. It is shown that there are other linear combinations of the numbers of simplices which are combinatorial invariants, but that all such invariants are multiples of the Euler characteristi…
The paper shows how to unknot certain nonorientable surfaces in 4-dimensional spaces.
problem Tackles the unknotting of nonorientable surfaces in 4-spheres and 4-balls.
method Uses topological isotopy and properties of knot groups and normal Euler numbers.
result Proves that certain nonorientable surfaces are topologically unknotted under specific conditions.
Researchers found two types of graphs for 6D torus manifolds with Euler number 6.
problem Identifying and constructing 6D almost complex torus manifolds with specific Euler numbers.
method Examined labeled directed graphs associated with fixed points and isotropy spheres, used to construct manifolds and determine Chern numbers.
result Proved the existence of two types of 6D almost complex torus manifolds with Euler number 6.
Study affinely transverse foliations in sphere bundles, finding bounds and vanishing conditions.
problem Understanding affine transverse foliations in sphere bundles and their properties.
method Provided upper bounds for the Euler number and a new proof for vanishing conditions under amenable fundamental group.
result Upper bounds and vanishing conditions for the Euler number of sphere bundles.
The paper proves a criterion for virtual Euler class one in hyperbolic 3-manifolds.
problem Determining the virtual Euler class one in hyperbolic 3-manifolds.
method Analyzing Alexander polynomials and constructing taut foliations.
result Constructing examples of hyperbolic 3-manifolds with virtual Euler class one.
We use some basic properties of binomial and Stirling numbers to prove that the Euler characteristic is, essentially, the unique numerical topological invariant for compact polyhedra which can be expressed as a linear combination of the numbers of faces of triangulations. We obtain this result converting it into an eig…
In the note, we give a proof, based on the Generalized Thom Conjecture, of Bennequin's Theorem on upper bound for the Euler number of a link which is considered as a closed braid. A lower bound for the Euler number of a link is also given.
Formula for manifold Euler characteristic using even faces.
problem Calculating Euler characteristic of triangulated manifolds.
method Formula based on even-dimensional faces.
result Universal coefficients for Euler characteristic.
Euler's theorem extended to complex structures.
problem Generalizing Euler's theorem to complex structures.
method Analyzing strongly connected, pure n-dimensional regular CW-complexes. result Evenness of cells is equivalent to generalized cycle decomposition and traversability.
We show how in many cases the algebraic number of immersed hyperspheres of constant (and prescribed) curvature may be related to the Euler Characteristic of the ambient space.
If a real value invariant of compact combinatorial manifolds (with or without boundary) depends only on the number of simplices in each dimension on the manifold, then the invariant is completely determined by Euler characteristics of the manifold and its boundary. So essentially, Euler characteristic is the unique inv…
The paper explores actions of surface mapping class groups on 3-manifolds.
problem Understanding when the natural surjection from homeomorphisms to mapping class groups splits.
method Analyzing circle bundles and their properties over surfaces.
result The homomorphism does not split in many cases where the Euler characteristic divides the Euler number.
We prove that a rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers. In dimension at least three we prove that only multiples of the top Chern number, which is the Euler …
Study non-vanishing ℓ2-Betti numbers for specific groups.
problem Calculating non-vanishing ℓ2-Betti numbers for certain groups. method Using Euler characteristics, higher Kazhdan projections, and Baum-Connes assembly map.
result Non-vanishing calculations for delocalised ℓ2-Betti numbers. The paper explores existence of specific almost complex manifolds with unique Betti numbers.
problem Existence of n=4k dimensional simply-connected closed almost complex manifolds with specific Betti numbers. method Characterization of rational cohomology rings, application of Sullivan's rational surgery realization theorem, and computation of Riemann-Roch integrality relations.
result Necessary and sufficient conditions for realizing a prescribed rational cohomology ring by a simply connected almost complex manifold.
Generalizes Thorpe's inequality for 4k-manifolds.
problem Euler characteristic and Pontryagin numbers of 4k-manifolds.
method Generalization and correction of Thorpe's original work.
result Corrected and generalized Thorpe's inequality.
Researchers refine the non-orientable 4-genus of torus knots using Batson's surfaces.
problem Finding the minimum non-orientable 4-genus for torus knots. method Developed and analyzed Batson's non-orientable spanning surfaces in B4. result Batson's surfaces minimize the non-orientable 4-genus among certain surfaces. New cohomology groups generalize Euler number for Lie superalgebras.
problem Generalizing cohomology groups for Lie superalgebras.
method Abstracted Poisson cohomology groups to Poisson-like cohomology groups for general Lie superalgebras.
result De Rham cohomology groups match Poisson-like cohomology groups for differential forms.
We discuss analogies between number theory and the theory of dynamical systems on spaces with a one-codimensional foliation. The emphasis is on comparing the "explicit formulas" of analytic number theory with certain dynamical Lefschetz trace formulas. We also point out a possible relation between an Arakelov-Euler cha…
We introduce the Γ-Euler-Satake characteristics of a general orbifold Q presented by an orbifold groupoid G, generalizing to orbifolds that are not necessarily global quotients the generalized orbifold Euler characteristics of Bryan-Fulman and Tamanoi. Each of these Euler characteristics is defined as t…
We show that when the genus and punctures of a surface are directly proportional by some rational number the minimal asymptotic translation length in the curve complex has behavior inverse to the square of the Euler characteristic. We also show that when the genus is fixed and the number of punctures varies the behavio…
In this note, we prove that if a compact even dimensional manifold Mn with negative sectional curvature is homotopic to some compact space-like manifold Nn, then the Euler characteristic number of Mn satisfies (−1)2nχ(Mn)>0. We also show that the minimal volume conjecture of Gromov is tr…
For M and N closed oriented connected smooth manifolds of the same dimension, we consider the mapping space Map(M,N;f) of continuous maps homotopic to f:M--> N.We show that the evaluation map from the space of maps to the manifold N induces a nontrivial homomorphism on the fundamental group only if the self coincidence…
Let M be a compact 3-manifold with a triangulation τ. We give an inequality relating the Euler characteristic of a surface F normally embedded in M with the number of normal quadrilaterals in F. This gives a relation between a topological invariant of the surface and a quantity derived from its combinatorial …
Study topological invariants of complexes for Riemannian manifolds.
problem Understanding topological properties of Riemannian manifolds.
method Analyzing Betti numbers and Euler characteristic of Vietoris-Rips and Čech complexes.
result Betti curve converges to manifold's Betti number within a scale parameter interval.
Study on invariants of complex hyperbolic disc bundles over surfaces, proving a conjecture.
problem Investigating relationships between three invariants of complex hyperbolic disc orbibundles.
method Analyzing Euler characteristic, Euler number, and Toledo invariant of disc orbibundles over 2-orbifolds.
result Proved that -3|τ| = 2e + 2χ holds for certain complex hyperbolic disc orbibundles.
Polynomial bound on surfaces in hyperbolic 3-manifolds.
problem Bounding the number of surfaces in hyperbolic 3-manifolds.
method Using polynomial functions of the volume of the manifold and the Euler characteristic.
result An upper bound for the number of compact essential surfaces is a polynomial function of the volume of the manifold.
We determine the expected curvature polynomial of random real projective varieties given as the zero set of independent random polynomials with Gaussian distribution, whose distribution is invariant under the action of the orthogonal group. In particular, the expected Euler characteristic of such random real projective…
The study bounds the excess of disjoint nonorientable surfaces in a 4-manifold.
problem Bounding the excess of disjoint nonorientable surfaces in a 4-manifold.
method Combining tubing construction with signature and Euler-characteristic formulas for 2-fold branched covers.
result The normal-Euler excess is bounded by a constant depending only on the ambient 4-manifold.
The group of bordism classes of unoriented surfaces in 4-space is determined. The bordism classes are characterized by normal Euler numbers,double linking numbers, and triple linking numbers.
New volume invariant for cocycles of hyperbolic lattices, proving rigidity results.
problem Volume calculation for cocycles of hyperbolic lattices.
method Introducing a new volume invariant and proving Milnor-Wood type inequalities.
result Characterization of maximal cocycles and proof of rigidity results.
The paper explores the topology of polygonal meshes and their properties.
problem Understanding the topological properties of polygonal meshes.
method Overview of topological concepts, definitions of intrinsic and extrinsic topology, proofs of Euler and Euler-Poincaré formulas, and discussion on cutting meshes.
result Detailed understanding and definitions of polygonal mesh topology, including intrinsic and extrinsic properties.
The paper computes various invariants of Legendrian knots in open book presentations.
problem Computing numerical invariants of Legendrian knots in contact manifolds.
method Front projections, rotation numbers, intersection numbers, Euler class, Lagrangian projections.
result Explicit formulas for computing invariants from front projections.
Given a flexible n-gon with generic side lengths, the moduli space of its configurations in R2 as well as in R3 is a smooth manifold. It is equipped with n \textit{tautological} line bundles whose definition is motivated by M. Kontsevich's tautological bundles over M0,n. We st…
Proving NP-hardness of unknotting and related link problems.
problem Determining if a knot can be untangled with a limited number of moves.
method Proving NP-hardness through reductions to known hard problems.
result Several link problems are proven to be NP-hard.
Study finds symplectic fillings' properties for specific contact covers.
problem Determining symplectic fillings' Euler characteristics and signatures.
method Analyzing exact symplectic fillings of contact branched covers.
result Identified links' symplectic fillings' properties.
A new homomorphism connects group actions on circles to Euler classes.
problem Understanding group actions on circles and their implications.
method Using crossed homomorphisms and Poincaré translation numbers.
result Relates the Euler class of actions to a specific homomorphism.
For a one parameter family of Calabi-Yau threefolds, Green, Griffiths and Kerr have expressed the total singularities in terms of the degrees of Hodge bundles and Euler number of the general fiber. In this paper, we show that the total singularities can be expressed by the sum of asymptotic values of BCOV invariants, s…