Yuli B. Rudyak

ON CATEGORY WEIGHT AND ITS APPLICATIONS

Abstract 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 〈u 1 , …, u n 〉, u i ∈ H ∗ (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 space. For example, we prove that if f: N→M is a map of degree 1 of closed stable parallelizable manifolds and dim M⩽2 cat M−4 then cat N⩾ cat M . We also prove that if M is a closed manifold with dim M⩽2 cat M−3 then cat (M×S m 1 ×⋯×S m n )= cat M+n , i.e., the Ganea conjecture holds for M.

On the Lusternik–Schnirelmann category of symplectic manifolds and the Arnold conjecture

In [Arn, Appendix 9] Arnold proposed a beautiful conjecture concerning the relation between the number of fixed points of certain (i.e., exact or Hamiltonian) selfdiffeomorphisms of a closed symplectic manifold (M,ω) and the minimum number of critical points of any smooth (= C∞) function on M . The first author succeeded in proving this form of the Arnold conjecture [R2] under the hypothesis that ω and c1 vanish on all spherical homology classes and that there is equality between the Lusternik–Schnirelmann category of M and the dimension of M . In this paper, we use a fundamental property of category weight to show that, for any closed symplectic manifold whose symplectic form vanishes on the image of the Hurewicz map, the required equality holds. Thus, we show that the original form of the Arnold Conjecture holds for all symplectic manifolds having ω|π2(M) = 0 = c1|π2(M).

Higher topological complexity and its symmetrization

We develop the properties of the n th sequential topological complexity TCn , 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 TCn.X/ to the Lusternik‐Schnirelmann category of cartesian powers of X , to the cup length of the diagonal embedding X ,! X n , and to the ratio between homotopy dimension and connectivity of X . We fully compute the numerical value of TCn for products of spheres, closed 1‐connected symplectic manifolds and quaternionic projective spaces. Our study includes two symmetrized versions of TCn.X/. The first one, unlike Farber and Grant’s symmetric topological complexity, turns out to be a homotopy invariant of X ; the second one is closely tied to the homotopical properties of the configuration space of cardinality-n subsets of X . Special attention is given to the case of spheres. 55M30; 55R80

On Thom spaces, Massey products, and nonformal symplectic manifolds

We suggest a simple general method of constructing of non-formal man- ifolds. In particular, we construct a large family of non-formal symplectic manifolds. Here we detect non-formality via non-triviality of rational Massey products. In fact, we analyze the behaviour of Massey products of closed manifolds under the blow-up construction. In this context Thom spaces play the role of a technical tool which allows us to construct non-trivial Massey products in an elegant way.

ON THE BERSTEIN-SVARC THEOREM IN DIMENSION 2

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 ≠ 0 where is the Berstein class of X . Previously it was known for n ≥ 3. We also prove that, for every map f : M → N of degree ±1 of closed orientable manifolds, the fundamental group of N is free provided that the fundamental group of M is.

On certain geometric and homotopy properties of closed symplectic manifolds

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

On analytical applications of stable homotopy (the Arnold conjecture, critical points)

We prove the Arnold conjecture for closed symplectic manifolds with π2(M) = 0 and catM = dimM . Furthermore, we prove an analog of the Lusternik– Schnirelmann theorem for functions with “generalized hyperbolicity” property.

Linking and Causality in Globally Hyperbolic Space-times

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 Mm be a spacelike Cauchy surface in a globally hyperbolic space-time (Xm+1, g). The spherical cotangent bundle ST*M is identified with the space

MINIMAL ATLASES OF CLOSED SYMPLECTIC MANIFOLDS

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