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Yuli Rudyak

Publications and source records attributed to Yuli Rudyak.

13 recordsLinked to original sources
arXiv2024

On spaces of minimal higher topological complexity

Let TC$_n$(X) denote the n-th topological complexity of a topological space X. It is known that TC$_n$(X) does not exceed n-1 for non-contractible X, and so it makes sense to describe spaces X with TC$_n$(X) =n-1. Grant--Lupton--Oprea proved the following: If X is a nilpotent space with TC$_n$(X)=n-1 then X is homotopy equivalent to an odd-dimensional sphere. Here we made an attempt to get rid of nilpotency condition and prove the following: If TC$_n$(X) =n-1 then either X is homotopy equivalent to a sphere of odd dimension or is a homology circle with the infinite cyclic fundamental group.

math.AT↗
arXiv2017

Cohomological Dimension, Connectivity, and Lusternik--Schnirelmann category

Dranishnikov~\cite{D2} proved that \[{\rm cat} X\leq {\rm cd}(π_1(X))+\Bigl\lceil\frac{{\rm hd} (X)-1}{2}\Bigr\rceil.\] where ${\rm cd}(π)$ denotes the cohomological dimension of a group $π$ and ${\rm hd}(X)$ denotes the homotopy dimension of $X$. Furthermore, there is a well-known inequality of Grossman,~\cite{G}: \[ {\rm cat} X\leq \Bigl\lceil\frac{{\rm hd} (X)}{k+1}\Bigr\rceil \text{ if } π_i(X)=0 \text{ for } i\leq k. \] We make a synthesis and generalization of both of these results, by demonstrating the main result: \[ {\rm cat}\leq {\rm cd}(π_1(X))+\Bigl\lceil\frac{{\rm hd} (X)-1}{k+1}\Bigr\rceil \text { if }π_i(X)=0 \text{ for } i=2, \ldots, k. \] The proof of the main theorem uses the Oprea--Strom inequality ${\rm cat} X\leq {\rm hd} (Bπ_1(X))+{\rm cat}^1X$, \cite{OS} where ${\rm cat}^1$ is the Clapp-Puppe ${\rm cat} \mathcal{A}$ with $\mathcal{A}$ the class of 1-dimensional CW complexes. The inequality clarified the Dranishnikov inequality.

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arXiv2016

Symplectic Asphericity, Category Weight, and Closed Characteristics of K-Contact Manifolds

Let $M$ be a closed K-contact $(2n+1)$-manifold equipped with a quasi-regular K-contact structure. Rukimbira proved that the Reeb vector field $ξ$ of this structure has at least $n+1$ closed characteristics. We note that $ξ$ has at least $2n+1$ closed characteristics provided that the space of leaves of the foliation determined by $ξ$ is symplectically aspherical.

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arXiv2016

Maps of Degree 1 and Lusternik--Schnirelmann Category

Given a map $f: M \to N$ of degree 1 of closed manifolds. Is it true that the Lusternik--Schnirelmann category of the range of the map is not more that the category of the domain? We discuss this and some related questions.

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arXiv2014

On topological complexity of Eilenberg-MacLane spaces

We note that, for any natural $k$ and every natural $l$ between $k$ and $2k$, there exists a group $π$ with $\cat K(π,1)=k$ and $\TC(K(π,1))=l$. Because of this, we can set up a problem of searching of purely group-theoretical description of $\TC(K(π,1))$ as an invariant of $π$.

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arXiv2007

Symplectically aspherical manifolds

This is a survey article on symplectically aspherical manifolds. The paper contains a discussion on constructions of symplectically aspherical manifolds, their topological properties and the role of this class in symplectic topology. Research perspectives are discussed.

math.SG↗
arXiv2006

Minimal atlases of closed symplectic manifolds

We study the number of Darboux charts needed to cover a closed connected symplectic manifold $(M,ω)$, and effectively estimate this number from below and from above in terms of the Lusternik--Schnirelmann category of $M$ and the Gromov width of $(M,ω)$.

math.SG↗
arXiv2003

Symplectically aspherical manifolds

The main subjects of the paper is studying the fundamental groups of closed symplectically aspherical manifolds. Motivated by some results of Gompf, we introduce two classes of fundamental groups $π_1(M)$ of symplectically aspherical manifolds $M$ with $π_2(M)=0$ and $π_2(M)\neq 0$. Relations between these classes are discussed. We show that several important classes of groups can be realized in both classes, while some of groups can be realized in the first class but not in the second one. Also, we notice that there are some interesting dimensional phenomena in the realization problem. The above results are framed by a general research of symplectically aspherical manifolds. For example, we find some conditions which imply that the Gompf sum of symplectically aspherical manifolds is symplectically aspherical, or that a total space of a bundle is symplectically aspherical, etc.

math.SG↗
arXiv2000

On Thom spaces, Massey products and non-formal symplectic manifolds

In this work we analyze the behavior of Massey products of closed manifolds under the blow-up construction. The results obtained in the article are applied to the problem of constructing closed symplectic non-formal manifolds. The proofs use Thom spaces as an important technical tool. This application of Thom spaces is of conceptual interest.

math.DG↗
arXiv2000

On certain geometric and homotopy properties of closed symplectic manifolds

The paper deals with relations between the Hard Lefschetz property, (non)vanishing of Massey products and the evenness of odd-degree Betti numbers of closed symplectic manifolds. It is known that closed symplectic manifolds can violate all these properties (in contrast with the case of Kaehler manifolds). However, the relations between such homotopy properties seem to be not analyzed. This analysis may shed a new light on topology of symplectic manifolds. In the paper, we summarize our knowledge in tables (different in the simply-connected and in symplectically aspherical cases). Also, we discuss the variation of symplectically harmonic Betti numbers on some 6-dimensional manifolds.

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arXiv1999

A remark on fixed point sets of gradient-like flows

Let $S$ be a set of critical points of a smooth real-valued function on a closed manifold $M$. Generalizing a well-known result of Lusternik--Schnirelmann, Reeken~[R] proved that $\cat S \geq \cat M$. Here we prove a generalization of Reeken"s inequality for gradient-like flows on compact spaces.

math.DG↗
arXiv1999

On symplectic manifolds with aspherical symplectic form

We consider closed symplectically aspherical manifolds, i.e. closed symplectic manifolds $(M,ω)$ satisfying the condition $[ω]|_{π_2M}=0$. Rudyak and Oprea [RO] remarked that such manifolds have nice and controllable homotopy properties. Now it is clear that these properties are mostly determined by the fact that the strict category weight of $[ω]$ equals 2. We apply the theory of strict category weight to the problem of estimating the number of closed orbits of charged particles in symplectic magnetic fields. In case of symplectically aspherical manifolds our theory enables us to improve some known estimations.

math.DG↗