# Yuli B. RudyakUniversity of Florida | UF · Department of Mathematics

Yuli B. Rudyak

Dr Rudyak

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89

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Introduction

**Skills and Expertise**

## Publications

Publications (89)

In this paper, we introduce relative LS category of a map and study some of its properties. Then we introduce `higher topological complexity' of a map, a homotopy invariant. We give a cohomological lower bound and compare it with previously known `topological complexity' of a map. Moreover, we study the relation between Lusternik-Schnirelmann categ...

Dranishnikov~\cite{D2} proved that \[{\rm cat} X\leq {\rm cd}(\pi_1(X))+\Bigl\lceil\frac{{\rm hd} (X)-1}{2}\Bigr\rceil.\] where ${\rm cd}(\pi)$ denotes the cohomological dimension of a group $\pi$ 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\lc...

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 $\xi$ of this structure has at least $n+1$ closed characteristics. We note that $\xi$ has at least $2n+1$ closed characteristics provided that the space of leaves of the foliation determined by $\xi$ is symp...

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.

We develop the properties of the $n$-th sequential topological complexity
$TC_n$, a homotopy invariant introduced by the third author as an extension of
Farber's topological model for studying the complexity of motion planning
algorithms in robotics. We exhibit close connections of $TC_n(X)$ to the
Lusternik-Schnirelmann category of cartesian power...

Farber introduced a notion of topological complexity TC(X) that is related to robotics. Here we introduce a series of numerical invariants TCn(X), n=2,3,… , such that TC2(X)=TC(X) and TCn(X)⩽TCn+1(X). For these higher complexities, we define their symmetric versions that can also be regarded as higher analogs of the symmetric topological complexity...

It follows from a theorem of Gromov that the stable systolic category of a closed manifold is bounded from below by the rational cup-length of the manifold. In the paper we study the inequality in the opposite direction. In particular, combining our results with Gromov's theorem, we prove the equality of stable systolic category and rational cup-le...

Given a closed manifold M , we prove the upper bound cat sys (M) ≤ dim M+cd(π1M) 2 for the systolic category of M , where "cd" is the cohomological dimension. We apply this upper bound to deduce the inequality cat sys (M) ≤ cat LS (M) between the systolic category and the Lusternik–Schnirelmann cat-egory of 4-manifolds. Furthermore we obtain a lowe...

Given a closed manifold M, we prove the upper bound of $${1 \over 2}(\dim M + {\rm{cd}}({{\rm{\pi }}_1}M))$$ for the number of systolic factors in a curvature-free lower bound for the total volume of M, in the spirit of M. Gromov’s systolic inequalities. Here “cd” is the cohomological dimension. We apply this upper bound to show that, in the case o...

We prove that manifolds of Lusternik-Schnirelmann category 2 necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larranaga and Gonzalez-Acuna, by generalizing their result in dimension 3, to all higher dimensions. We also obtain some general results on the relations between the fundamental group of a closed manifold M...

This is a survey article on symplectically aspherical manifolds.

The classical linking number lk is defined when link components are zero homologous. In [15] we constructed the affine linking invariant alk generalizing lk to the case of linked submanifolds with arbitrary homology classes. Here we apply alk to the study of causality in Lorentzian manifolds.
Let M
m
be a spacelike Cauchy surface in a globally hype...

We prove that for any group π with cohomological dimension at least n the n th power of the Berstein class of π is nontrivial. This allows us to prove the following Berstein–Svarc theorem for all n :
Theorem . For a connected complex X with dim X = cat X = n , we have $\ber_X^n$ ≠ 0 where $\ber_X$ is the Berstein class of X .
Previously it was know...

The following theorem (a conjecture of Rourke) is proved: every commutative ring spectrum of characteristic 2 which has finite type is isomorphic to the spectrum of ordinary cohomology theory with coefficients in .Bibliography: 13 titles.

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.

We prove that manifolds of Lusternik-Schnirelmann category 2 necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larranaga and Gonzalez-Acuna, by generalizing their result in dimension 3, to all higher dimensions. We examine its ramifications in systolic topology, and provide a sufficient condition for ensuring a lowe...

We prove a finiteness result for the systolic area of groups. Namely, we show that there are only finitely many possible unfree factors of fundamental groups of 2-complexes whose systolic area is uniformly bounded. We also show that the number of freely indecomposable such groups grows at least exponentially with the bound on the systolic area. Fur...

We show that the geometry of a Riemannian manifold (M,g) is sensitive to the apparently purely homotopy-theoretic invariant of M known as the Lusternik-Schnirelmann category, denoted cat_{LS}(M). Here we introduce a Riemannian analogue of cat_{LS}(M), called the systolic category of M. It is denoted cat_{sys}(M), and defined in terms of the existen...

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

We describe all abelian groups which can appear as the fundamental groups of closed symplectically aspherical manifolds. The
proofs use the theory of symplectic Lefschetz fibrations.

Let X be a finite 2-complex with unfree fundamental group. We prove lower bounds for the area of a metric on X, in terms of the square of the least length of a noncontractible loop in X. We thus establish a uniform systolic inequality for all unfree 2-complexes. Our inequality improves the constant in M. Gromov's inequality in this dimension. The a...

We prove a new systolic volume lower bound for non-orientable n-manifolds, involving the stable 1-systole as well as the codimension-1 systole with coefficients in ℤ2. As an application, we prove that Lusternik–Schnirelmann category and systolic category agree for non-orientable closed manifolds
of dimension 3, extending our earlier result in the o...

We discuss which groups can be realized as the fundamental groups of compact Hausdorff spaces. In particular, we prove that the claim ``every group can be realized as the fundamental group of a compact Hausdorff space'' is consistent with the Zermelo - Fraenkel - Choice set theory.

We construct closed (k - 1)-connected manifolds of dimensions $\geq 4k - 1$ that possess non-trivial rational Massey triple products. We also construct examples of manifolds M such that all the cup-products of elements of $H^{k}(M)$ vanish, while the group $H^{3k-1}(M; \mathbb{Q})$ is generated by Massey products: such examples are useful for the t...

A smooth manifold M is called symplectically aspherical if it admits a symplectic form ω with ω|π
2(M) = 0. It is easy to see that, unlike in the case of closed symplectic manifolds, not every finitely presented group can be realized as the fundamental group of a closed symplectically aspherical manifold. The goal of the paper is to study the funda...

We prove that, for every decreasing sequence {ak} of natural numbers, there exists a map f:X→X with catfk=ak.

We construct closed $(k-1)$-connected manifolds of dimensions $\ge 4k-1$ that possess non-trivial rational Massey triple products. We also construct examples of manifolds $M$ such that all the cup-products of elements of $H^k(M)$ vanish, while the group $H^{3k-1}(M;\Q)$ is generated by Massey products: such examples are useful for theory of systols...

A garland based on a manifold $P$ is a finite set of manifolds homeomorphic to $P$ with some of them glued together at marked points. Fix a manifold $M$ and consider a space $\NN$ of all smooth mappings of garlands based on $P$ into $M$. We construct operations $\bullet$ and $[-,-]$ on the bordism groups $\bor_*(\NN)$ that give $\bor_*(\NN)$ the na...

Let N_1, N_2, M be smooth manifolds with dim N_1 + dim N_2 +1 = dim M$ and let phi_i, for i=1,2, be smooth mappings of N_i to M with Im phi_1 and Im phi_2 disjoint. The classical linking number lk(phi_1,phi_2) is defined only when phi_1*[N_1] = phi_2*[N_2] = 0 in H_*(M). The affine linking invariant alk is a generalization of lk to the case where p...

We use the ideas of Lusternik-Schnirelmann theory to describe the set of fixed points of certain homotopy equivalences of a general space. In fact, we extend Lusternik-Schnirelmann theory to pairs (',f), where ' is a homotopy equivalence of a topological space X and where f:X ! R is a continuous function satisfying f('(x)) < f(x) unless '(x) = x; i...

Let $M^m$ be an oriented manifold, let $N^{m-1}$ be an oriented closed manifold, and let $p$ be a point in $M^m$. For a smooth map $f:N^{m-1} \to M^m, p \not\in Im f,$ we introduce an invariant $awin_p(f)$ that can be regarded as a generalization of the classical winding number of a planar curve around a point. We show that $awin_p$ estimates from...

It is well known that closed Kähler manifolds have certain homotopy properties which do not hold for symplectic manifolds. Here we survey interconnections between those properties.

Two wave fronts $W_1$ and $W_2$ that originated at some points of the manifold $M^n$ are said to be causally related if one of them passed through the origin of the other before the other appeared. We define the causality relation invariant $CR (W_1, W_2)$ to be the algebraic number of times the earlier born front passed through the origin of the o...

In this paper, we give a new simplified calculation of the Lusternik-Schnirelmann category of closed 3-manifolds. We also describe when 3-manifolds have detecting elements and prove that 3-manifolds satisfy the equality of the Ganea conjecture.

This paper can be considered as an extension to our paper [On symplectically harmonic forms on six-dimensional nilmanifolds, Comment. Math. Helv. 76 (2001), n 1, 89-109]. Also, it contains a brief survey of recent results on symplectically harmonic cohomology.

We prove that, for every decreasing sequence {a \sb k} of natural numbers, there exists a map f: X --> X with cat (f\sp k)=a\sb k.

Let (W,M,M'), dim W > 5, be a non-trivial h-cobordism (i.e., the Whitehead torsion of (W,V) is non-zero). We prove that every smooth function f: W --> [0,1], f(M)=0, f(M')=1 has at least 2 critical points. This estimate is sharp: W possesses a function as above with precisely two critical points.

This is a survey paper where we expose the Kirby--Siebenmann results on classification of PL structures on topological manifolds and, in particular, the homotopy equivalence TOP/PL=K(Z/2.3) and the Hauptvermutung for 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 $\pi_1(M)$ of symplectically aspherical manifolds $M$ with $\pi_2(M)=0$ and $\pi_2(M)\neq 0$. Relations between these classes are discussed. We show that...

We extend Lusternik-Schnirelmann theory to pairs $(f, \phi)$, where $\phi$ is a homotopy equivalence of a space $X$, $f$ is a function on $X$ which decreases along $\phi$ and $(f, \phi)$ satisfies a discrete analog of the Palais-Smale condition. The theory is carried out in an equivariant setting.

In the present paper we study the variation of the dimensions $h_k$ of spaces of symplectically harmonic cohomology classes (in the sense of Brylinski) on closed symplectic manifolds. We give a description of such variation for all 6-dimensional nilmanifolds equipped with symplectic forms. In particular, it turns out that certain 6-dimensional nilm...

The Arnold conjecture claims that, for every Hamiltonian symplectomorphism φ : M → M of a closed symplectic manifold (M,ω), the number Fix φ of fixed points of φ is at least the minimal number of critical points of a smooth function on M. For every (M,ω) with π2(M) = 0, Floer [F] and Hofer [H] performed a certain analytical reduction of the problem...

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 nonformal manifolds. The proofs use Thom spaces
as an important technical tool. This application of Thom spaces is of conceptual interest.

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.

We consider closed symplectically aspherical manifolds, i.e. closed symplectic manifolds $(M,\omega)$ satisfying the condition $[\omega]|_{\pi_2M}=0$. 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 wei...

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.

We develop and apply the concept of category weight which was introduced by Fadell and Husseini. For example, we prove that category weight of every Massey product 〈u1,…,un〉,ui∈H̃∗(X) is at least 2 provided X is connected. Furthermore, we remark that elements of maximal category weight enable us to control the Lusternik–Schnirelmann category of a s...

In [R2] and [RO] the Arnold conjecture for closed symplectic manifolds with trivial second homotopy group was proved. This proof used surgery and cobordism theory. Here we give a purely cohomological proof of this result.

The main goal of this chapter is to introduce some notation and terminology. We assume that the reader is more or less familiar with the basic concepts of algebraic topology (homotopy and homology). Typical references are: tom Dieck–Kamps–Puppe [1], tom Dieck [2], Dold [5], Fomenko–Fuchs– Gutenmacher [1], Fritsch–Piccinini [1], Fuks-Rokhlin [1], Gr...

In this chapter we apply the results of the previous one to the orientability of V-objects with respect to K and KO. The case V = O was considered by Atiyah–Bott–Shapiro [1], the other cases were considered mainly by the author, see Rudyak [6,8,9]. To be convenient, we collect the results as a r´esum´e, see the ends of §§ 3,4. Here K, resp. KO, mea...

In this chapter we discuss some preliminaries from stable homotopy theory. Sections 1–3 are concerned with basic properties of spectra and (co)homology theories. Here we mainly follow Adams [8] and Switzer [1]. Sections 4–7 contain an exposition of standard material, at a level suitable for students.

We fix a prime p. Let BP be the corresponding Brown–Peterson spectrum, and let κ : BP → MU[p], ρ : MU[p] → BP be the pair of morphisms described in VII.3.19(i).

A phantom, or a phantom map, is an essential map f : X → Y of a CW-complex X such that f|X(n
) is inessential for every n. Adams–Walker [1] found an example of a phantom, and many other authors found phantoms later. The existence of phantoms was very exotic at that time and adorned (and adorns now, by the way) any results. However, as usual, the ot...

In the introduction we discussed the importance and usefulness of Thom spaces (spectra). In this chapter we develop a general theory of Thom spectra, investigate some special Thom spectra and apply this to certain geometrical problems. Some aspects of a general theory of Thom spectra are also considered in Lewis–May–Steinberger [1]. Now it is clear...

(Co)bordism with singularities have many applications. However, in this book we mainly consider only a few aspects of this theory: namely, we want to demonstrate that (co)bordism with singularities establishes a big source of interesting (co)homology theories and, in particular, enables us to construct cohomology theories with prescribed properties...

It seems that the orientability concept arose implicitly in the infancy of humanity, when people became able to distinguish upward and downward (as well as left and right) directions. Many epochs later we had suitable concepts of the orientation of the line (arrow), the plane (circle arrow) and space (rightleft triples of vectors, spiralled arrow,...

In order to work with complex (co)bordism with singularities we need some preliminaries on complex (co)bordism. Therefore we collect here some facts which will be used below. A standard reference on complex (co)bordism is the book of Ravenel [1], see also Stong [3], Ch. VI.

We prove that the Lusternik-Schnirelmann category $cat(M)$ of a closed symplectic manifold $(M, \omega)$ equals the dimension $dim(M)$ provided that the symplectic cohomology class vanishes on the image of the Hurewicz homomorphism. This holds, in particular, when $\pi_2(M)=0$. The Arnold conjecture asserts that the number of fixed points of a Hami...

We prove the Arnold conjecture for closed symplectic manifolds with $\pi_2(M)=0$ and $\cat M=\dim M$. Furthermore, we prove an analog of the Lusternik-Schnirelmann theorem for functions with ``generalized hyperbolicity'' property.

Using a result of W. Singhof [Manuscr. Math. 29, 385-415 (1979; Zbl 0415.55001)], we prove that cat(M×S m )=catM+1 provided M is a connected closed PL manifold with dimM≤2catM-3 and S m is the m-sphere, m>0.

An obstruction theory is constructed for orientability of vector, piecewise-linear,
and topological
Rn-bundles and homotopy sphere bundles (spherical fibrations)
in generalized cohomology theories. The results are applied to study the orientability
of bundles in complex
K-theory. In particular, it turns out that the problem of
K-orientability
fo...

Because of solar energy input periodicity accumulation of this energy becomes a very important problem. The optimal combination of a solar water heater (SWH) with an accumulator for heat supply to a user may find application both in use of the SWH for heat supply and in combined power generating plants. In the later case use of solar water heaters...

A method is presented for computing the set of homotopy classes [X, G/PL], where X is a finite CW complex satisfying certain homological conditions. The result obtained is applied to compute normal invariants of products of projective spaces.

In [S1], [S2] Sullivan indicated the proof of the following theorem: Let h : M → N be a homeomorphism of closed piecewise linear manifolds. Then the normal invariant of h is trivial provided H 3 (M) has no 2-torsion. The goal of this paper is to give a relative simple proof of this theorem in a particular case of man-ifolds M such that π 1 (M) and...

First examples of closed symplectically aspherical manifolds with non-trivial π 2 were given by R. E. Gompf [Math. Res. Lett. 5, No. 5, 599–603 (1998; Zbl 0943.53049)]. Here we show some new examples.

Here is proved that neither k nor kO are Thom spectra. This was conjectured by Mahowald in 1979.

## Projects

Project (1)