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Yuli B. Rudyak

Publications and source records attributed to Yuli B. Rudyak.

25 records · Page 2Linked to original sources
arXiv2004

Lusternik-Schnirelmann category and systolic category of low dimensional manifolds

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 existence of systolic inequalities satisfied by every metric g, as initiated by C. Loewner and later developed by M. Gromov. We compare the two categories. In all our examples, the inequality cat_{sys}(M) \le cat_{LS}(M) is satisfied, which typically turns out to be an equality, e.g. in dimension 3. We show that a number of existing systolic inequalities can be reinterpreted as special cases of such equality, and that both categories are sensitive to Massey products. The comparison with the value of cat(M) leads us to prove or conjecture new systolic inequalities on M.

math.DG↗
arXiv2004

Affine Linking Numbers and Causality Relations for Wave Fronts

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 other front before the other front appeared. Clearly, if $CR(W_1, W_2)\neq 0$, then $W_1$ and $W_2$ are causally related. If $CR(W_1, W_2)= 0$, then we generally can not make any conclusion about fronts being causally related. However we show that for front propagation given by a complete Riemannian metric of non-positive sectional curvature, $CR(W_1, W_2)\neq 0$ if and only if the two fronts are causally related. The models where the law of propagation is given by a metric of constant sectional curvature are the famous Friedmann Cosmology models. The classical linking number $lk$ is a $Z$-valued invariant of two zero homologous submanifolds. We construct the affine linking number generalization $AL$ of the $lk$ invariant to the case of linked $(n-1) $-spheres in the total space of the unit sphere tangent bundle $(STM)^{2n-1}\to M^n$. For all $M$, except of odd-dimensional rational homology spheres, $AL$ allows one to calculate the value of $CR(W_1, W_2)$ from the picture of the two wave fronts at a certain moment. This calculation is done without the knowledge of the front propagation law and of their points and times of birth. Moreover, in fact we even do not need to know the topology of $M$ outside of a part $\bar M$ of $M$ such that $W_1$ and $W_2$ are null-homotopic in $\bar M$.

math.GT↗
arXiv2003

Examples of non-formal closed $(k-1)$-connected manifolds of dimensions $4k-1$ and more

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.

math.AT↗
arXiv2002

On sequences cat (f\sp k)

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.

math.AT↗
arXiv1998

On strict category weight and Arnold conjecture

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.

math.DG↗
arXiv1997

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

We prove that the Lusternik-Schnirelmann category $cat(M)$ of a closed symplectic manifold $(M, ω)$ equals the dimension $dim(M)$ provided that the symplectic cohomology class vanishes on the image of the Hurewicz homomorphism. This holds, in particular, when $π_2(M)=0$. The Arnold conjecture asserts that the number of fixed points of a Hamiltonian symplectomorphism of $M$ is greater than or equal to the number of critical points of some function on $M$. A modified form of the conjecture, replacing the latter quantity (via Lusternik-Schnirelmann theory) by $cup(M) + 1$, has been proved recently by various authors using techniques of Floer. The first author has also recently shown that the original form of the conjecture holds when $cat(M) =dim(M)$. Thus, this paper completes the proof of the original Arnold conjecture for closed symplectic manifolds with, for example, $π_2(M)=0$.

dg-ga↗