arXiv research

Study Euler obstruction of 1-forms on determinantal singularities.

Study the topological information of map germs using Euler obstruction.

Applying a local Gauss-Bonnet formula for closed subanalytic sets to the complex analytic case, we obtain characterizations of the Euler obstruction of a complex analytic germ in terms of the Lipschitz-Killing curvatures and the Chern forms of its regular part. We also prove analogous results for the global Euler obstr…

We study the Euler obstruction of essentially isolated determinantal singularities (EIDS). The EIDS were defined by W. Ebeling and S. Gusein-Zade, as a generalization of isolated singularity. We obtain some formulas to calculate the Euler obstruction for the determinantal varieties with singular set an ICIS.

Proves Massey's theorems on complex structure obstructions.

We introduce a complete obstruction to the existence of nonvanishing vector fields on a closed orbifold $Q$. Motivated by the inertia orbifold, the space of multi-sectors, and the generalized orbifold Euler characteristics, we construct for each finitely generated group $Γ$ an orbifold called the space of $Γ$-sectors o…

New invariant $c_{sm}$ simplifies computing geometric invariants of recursive group orbits.

We present an avatar of the Euler obstruction to foliated structures on certain non-metric surfaces. This adumbrates (at least for the simplest 2D-configurations) that the standard mechanism---to the effect that the devil of algebra sometimes barricades the existence of angelic geometric structures (obstruction theory …

New obstruction found for embedding Riemannian manifolds into Euclidean spaces.

The Whitehead link exterior lacks most Euler class taut foliations.

This article is concerned with the question: For which pairs of hyperbolic Euler-Lagrange systems in the plane does there exist a rank-$1$ Bäcklund transformation relating them? We express some obstructions to such existence in terms of the local invariants of the Euler-Lagrange systems. In addition, we discover a clas…

Using the equivalence between the renormalized Euler characteristic of Ozsvath and Szabo, and the Turaev torsion normalized by the Casson-Walker invariant, we make calculations for $S^3_{p/q}(K)$. An alternative proof of a theorem by Ozsváth and Szabó on $L$-space surgery obstructions is provided.

The purpose of this paper is to study finite dimensional equivariant moduli problems from the viewpoint of stratification theory. We show that there exists a stratified obstruction system for a finite dimensional equivariant moduli problem. In addition, we define a coindex for a G-vector bundle which is determined by t…

The paper introduces new invariants to study topological properties of map germs.

Defines a new conformally invariant Yang-Mills type energy for 6-manifolds.

We determine several necessary and sufficient conditions for a closed almost-complex orbifold $Q$ with cyclic local groups to admit a nonvanishing vector field. These conditions are stated separately in terms of the orbifold Euler-Satake characteristics of $Q$ and its sectors, the Euler characteristics of the underlyin…

Manifolds admitting positive sectional curvature are conjectured to have rigid homotopical structure and, in particular, comparatively small Euler charateristics. In this article, we obtain upper bounds for the Euler characteristic of a positively curved Riemannian manifold that admits a large isometric torus action. W…

We present an equivalent criterion for the global existence of Euler's multiplier for an integrable one-form taking into account the corresponding codim-1-foliation. In particular, the impact of inseparable leaves is considered. Here, we suppose that the foliation can be reduced to a graph; we also discuss obstructions…

Decouples moduli groups in heterotic string theory cohomology.

We solve Euler equations on graph manifolds, classifying steady flows with Morse-Bott Bernoulli functions.

Let $M^n$ be a compact K$\ddot{a}$hler manifold with almost nonnegative Ricci curvature and nonzero first Betti number. We show that the holomorphic Euler number of $M^n$ vanishes, which gives a new obstruction for compact complex manifolds admitting K$\ddot{a}$hler metrics with almost nonnegative Ricci curvature. A cr…

We prove that the norm of the Euler class E for flat vector bundles is $2^{-n}$ (in even dimension $n$, since it vanishes in odd dimension). This shows that the Sullivan--Smillie bound considered by Gromov and Ivanov--Turaev is sharp. We construct a new cocycle representing E and taking only the two values $\pm 2^{-n}$…

Non-trivial obstructions found for topological solitons in Yang-Mills-Chern-Simons theories.

For every orientable surface of finite negative Euler characteristic, we find a right-angled Artin group of cohomological dimension two which does not embed into the associated mapping class group. For a right-angled Artin group on a graph $\gam$ to embed into the mapping class group of a surface $S$, we show that the …

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

An $n$-dimensional manifold $M$ ($n\ge 3$) is called {\it generalized graph manifold} if it is glued of blocks that are trivial bundles of $(n-2)$-tori over compact surfaces (of negative Euler characteristic) with boundary. In this paper two obstructions for generalized graph manifold to be nonpositively curved are des…

Let M be a complete orientable manifold of bounded geometry. Suppose that M has finitely many ends, each having a neighborhood quasi-isometric to a neighborhood of an end of an infinite cyclic covering of a compact manifold. We consider a class of exponentially weighted inner products (\cdot ,\cdot)_k on forms, indexed…

Study on knot classification using 3-braid closures and ribbon surfaces.

Study on curvature functions for compact manifolds with boundary.

We propose and study the following Mirror Principle: certain sequences of multiplicative equivariant characteristic classes on Kontsevich's stable map moduli spaces can be computed in terms of certain hypergeometric type classes. As applications, we compute the equivariant Euler classes of obstruction bundles induced b…

Study on Legendrian knots and their non-orientable Lagrangian fillings.

Geometrically computes cohomotopy groups for higher-dimensional manifolds.

We compute explicit transgression forms for the Euler and Pontrjagin classes of a Riemannian manifold $M$ of dimension 4 under a conformal change of the metric, or a change to a Riemannian connection with torsion. These formulae describe the singular set of some connections with singularities on compact manifolds as a …

We consider closed manifolds that admit a metric locally isometric to a product of symmetric planes. For such manifolds, we prove that the Euler characteristic is an obstruction to the existence of flat structures, confirming an old conjecture proved by Milnor in dimension 2. In particular, the Chern conjecture follows…

Study shows non-orientable manifolds restrict signature-changing metrics globally.

Einstein metrics are blocked by manifold features and group growth.

We show that if a compact, oriented 4-manifold admits a coassociative-free immersion into the Euclidean 7-space then its Euler characteristic and signature vanish. Moreover, in the spin case the Gauss map is contractible, so that the immersed manifold is parallelizable. The proof makes use of homotopy theory in particu…

The Weyl functional is analyzed on 4-manifolds with positive scalar curvature.

New method shows nonorientable surfaces in 4D are topologically unknotted.

On a polarised surface, solutions of the Vafa-Witten equations correspond to certain polystable Higgs pairs. When stability and semistability coincide, the moduli space admits a symmetric obstruction theory and a $\mathbb C^*$ action with compact fixed locus. Applying virtual localisation we define invariants constant …

We show that every bad orbifold vector bundle can be realized as the restriction of a good orbifold vector bundle to a suborbifold of the base space. We give an explicit construction of this result in which the Chen-Ruan orbifold cohomology of the two base spaces are isomorphic (as additive groups). This construction i…

In this paper we study an energy of maps between almost Hermitian manifolds for which pseudo-holomorphic maps are global minimizers. We derive its Euler-Lagrange equation, the $\bar{\partial}$-harmonic map equation, and show that it coincides with the harmonic map equation up to first order terms. We prove results anal…

This paper is devoted to the existence of contact forms of prescribed Webster scalar curvature on a $3-$dimensional CR compact manifold locally conformally CR equivalent to the unit sphere $\mathbb{S}^{3}$ of $\mathbb{C}^{2}$. Due to Kazdan-Warner type obstructions, conditions on the function $H$ to be realized as a We…

This paper studies torsion obstructions to complex sections on manifolds.

The aim of this paper is to use mapping class group relations to approach the `geography' problem for Stein fillings of a contact 3-manifold. In particular, we adapt a formula of Endo and Nagami so as to calculate the signature of such fillings as a sum of the signatures of basic relations in the monodromy of a related…

We study type one generalized complex and generalized Calabi--Yau manifolds. We introduce a cohomology class that obstructs the existence of a globally defined, closed 2-form which agrees with the symplectic form on the leaves of the generalized complex structure, the twisting class. We prove that in a compact, type on…

For every connected manifold with corners we use a homology theory called conormal homology, defined in terms of faces and incidences and whose cycles correspond geometrically to corner's cycles. Its Euler characteristic (over the rationals, dimension of the total even space minus the dimension of the total odd space),…

New Euler characteristics for groupoids generalize orbifold Euler characteristics.