arXiv research

The non-vanishing conjecture implies the abundance conjecture in certain cases.

We shall prove a new non-vanishing theorem for the stable cohomotopy Seiberg-Witten invariant of connected sums of 4-manifolds with positive first Betti number. The non-vanishing theorem enables us to find many new examples of 4-manifolds with non-trivial stable cohomotopy Seiberg-Witten invariants and it also gives a …

Study non-vanishing $\ell^2$-Betti numbers for specific groups.

Extending work of Chen, we prove the Weinstein conjecture in dimension three for strongly fillable contact structures with either non-vanishing first Chern class or with strong and exact filling having non-trivial canonical bundle. This implies the Weinstein conjecture for certain Stein fillable contact structures obta…

We show the non-vanishing of cohomology groups of sufficiently small congruence lattices in $SL(1,D)$, where $D$ is a quaternion division algebras defined over a number field $E$ contained inside a solvable extension of a totally real number field. As a corollary, we obtain new examples of compact, arithmetic, hyperbol…

New proof confirms periodic orbit conjecture for Eulerisable flows.

We show that the ungraded ruling invariants of a Legendrian link can be realized as certain coefficients of the Kauffman polynomial which are non-vanishing if and only if the upper bound for the Bennequin number given by the Kauffman polynomial is sharp. This resolves positively a conjecture of Fuchs. Using similar met…

Proves simple type conjecture for mod 2 Seiberg-Witten invariants.

Harer, Kas and Kirby have conjectured that every handle decomposition of the elliptic surface $E(1)_{2,3}$ requires both 1- and 3-handles. In this article, we construct a smooth 4-manifold which has the same Seiberg-Witten invariant as $E(1)_{2,3}$ and admits neither 1- nor 3-handles, by using rational blow-downs and K…

We give a new proof of Witten asymptotic conjecture for Seifert manifolds with non vanishing Euler class and one exceptional fiber. Our method is based on semiclassical analysis on a two dimensional phase space torus. We prove that the Witten-Reshetikhin-Turaev invariant of a Seifert manifold is the scalar product of t…

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

The study explores Legendrian invariants and half Giroux torsion in contact structures.

We prove the non-vanishing of the CMC flux of the boundaries of certain Riemannian manifolds with constant mean curvature.

Polynomial-time algorithm matches correlated random graphs with non-vanishing correlation.

The paper constructs metrics on compact manifolds using Aubin's deformations.

This article analyzes the interplay between symplectic geometry in dimension four and the invariants for smooth four-manifolds constructed using holomorphic triangles introduced in math.SG/0110169. Specifically, we establish a non-vanishing result for the invariants of symplectic four-manifolds, which leads to new proo…

We prove that if $X$ is a compact, oriented, connected $4$-dimensional smooth manifold, possibly with boundary, satisfying $χ(X)\neq 0$, then there exists an integer $C\geq 1$ such that any finite group $G$ acting smoothly and effectively on $X$ has an abelian subgroup $A$ satisfying $[G:A]\leq C$, $χ(X^A)=χ(X)$, and $…

New criterion for Ricci-flat manifolds with non-vanishing Rosenberg index.

Study shows non-vanishing Stiefel-Whitney classes and absence of spin^C structures in certain hyperbolic manifolds.

Paper proves non-extendability of quasimorphism and non-vanishing of Reznikov's class.

The main result of this paper is non-vanishing of the image of the index map from the $G$-equivariant $K$-homology of a proper $G$-compact $G$-manifold $X$ to the $K$-theory of the $C^{*}$-algebra of the group $G$. Under the assumption that the Kronecker pairing of a $K$-homology class with a low-dimensional cohomology…

Non-vanishing steady Euler flows and Beltrami fields found in high dimensions.

In his book "Differential Geometry of Spray and Finsler spaces", page 177, Zhongmin Shen asks "wether or not there always exist non-trivial Funk functions on a spray space". In this note, we will prove that the answer is negative for the geodesic spray of a finslerian function of non-vanishing scalar flag curvature.

Study confirms optimal bounds for group cohomology of Lie groups.

Paper answers Gromov's compactness question on noncompact manifolds.

We use Donaldson invariants of regular surfaces with p_g >0 to make quantitative statements about modulispaces of stable rank 2 sheaves. We give two examples: a quantitative existence theorem for stable bundles, and a computation of the rank of the canonical holomorphic two forms on the moduli space. The results are in…

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

New examples show contact invariants can be non-zero even with half Giroux torsion.

We use SO(3) gauge theory to define a functor from a category of unoriented webs and foams to the category of finite-dimensional vector spaces over the field of two elements. We prove a non-vanishing theorem for this SO(3) instanton homology of webs, using Gabai's sutured manifold theory. It is hoped that the non-vanis…

Proves Birkhoff-Poritsky conjecture for centrally-symmetric billiards.

Study shows non-abelian free groups' 4th cohomology is non-zero.

The problem on the minimal number (with respect to deformation) of intersection points of two closed curves on a surface is solved. Following the Nielsen approach, we define classes of intersection points and essential classes of intersection points, which "are preserved under deformation" and whose total number is cal…

We study the first uniformly finite homology group of Block and Weinberger for uniformly locally finite graphs, with coefficients in $\mathbb{Z}$ and $\mathbb{Z}_2$. When the graph is a tree, or coefficients are in $\mathbb{Z}_2$, a characterisation of the group is obtained. In the general case, we describe three pheno…

Study of instantons and knot representations in 3-manifolds.

This paper proves positivity of Riemann-Roch polynomials for hyperkähler manifolds.

We consider Milnor invariants for certain covering links as a generalization of covering linkage invariants formulated by R. Hartley and K. Murasugi. A set of Milnor invariants for covering links is a cobordism invariant of a link, and that this invariant can distinguish some links for which the ordinary Milnor invaria…

We show that surface bundles over surfaces with base and fiber of genus at least 2 have non-vanishing simplicial volume.

Embedding calculus invariants solve knot connectivity and grope cobordism problems.

We state and prove a generalization of the Poincaré-Hopf index theorem for manifolds with boundary. We then apply this result to non-vanishing complex vector fields.

We prove a non-vanishing result for the $L_{q,p}$-cohomology of complete simply-connected Riemannian manifolds with pinched negative curvature.

Let $M$ be a closed connected spin manifold such that its spinor Dirac operator has non-vanishing (Rosenberg) index. We prove that for any Riemannian metric on $V = M \times [-1,1]$ with scalar curvature bounded below by $σ> 0$, the distance between the boundary components of $V$ is at most $C_n/\sqrtσ$, where $C_n = \…

The study finds hyperbolic manifolds without spin^c structures in dimensions 5 and above.

We determine the closed, oriented Seifert fibered 3-manifolds which carry positive tight contact structures. Our main tool is a new non-vanishing criterion for the contact Ozsvath-Szabo invariant.

We determine the isomorphism classes of symmetric symplectic manifolds of dimension at least 4 which are connected, simply-connected and have a curvature tensor which has only one non-vanishing irreducible component -- the Ricci tensor.

The existence of a nowhere zero real vector field implies a well-known restriction on a compact manifold. But all manifolds admit nowhere zero complex vector fields. The relation between these observations is clarified.

The affine-additive group is hyperbolic with a non-vanishing 4-capacity.

We prove that the mean curvature $τ$ of the slices given by a constant mean curvature foliation can be used as a time function, i.e. $τ$ is smooth with non-vanishing gradient.

We construct examples of $S^1$-manifolds with finite second homotopy group and non-vanishing $\hat A$-genus. This is related to the classification of positive quaternionic Kaehler manifolds.