arXiv subjects

Frol Zapolsky

Publications and source records attributed to Frol Zapolsky.

At least 19 recordsLinked to original sources

arXiv2025

Constraints on symplectic quasi-states

We prove that given a closed connected symplectic manifold equipped with a Borel probability measure, an arbitrarily large portion of the measure can be covered by a symplectically embedded polydisk, generalizing a result of Schlenk. We apply this to constraints on symplectic quasi-states. Quasi-states are a certain class of not necessarily linear functionals on the algebra of continuous functions of a compact space. When the space is a symplectic manifold, a more restrictive subclass of symplectic quasi-states was introduced by Entov--Polterovich. We use our embedding result to prove that a certain `soft' construction of quasi-states, which is due to Aarnes, cannot yield nonlinear symplectic quasi-states in dimension at least four.

math.SG↗

arXiv2025

Hofer-continuous quasi-morphisms on Liouville manifolds

We show that, given a complete Liouville manifold, any homogeneous quasi-morphism on its Hamiltonian group, which satisfies a strengthened version of Hofer continuity called stability, must vanish. This partially addresses a conjecture due to L. Polterovich.

math.SG↗

arXiv2025

On the Lagrangian Hofer geometry of Clifford tori

We show that the space of Lagrangians which are Hamiltonian isotopic to the Clifford torus in a complex projective space or in the four-dimensional quadric, taken with Chekanov's Lagrangian Hofer metric, contains a quasi-isometric copy of the real line, and in particular has infinite diameter.

math.SG↗

arXiv2024

Symplectic topology and ideal-valued measures

We adapt Gromov's notion of ideal-valued measures to symplectic topology, and use it for proving new results on symplectic rigidity and symplectic intersections. Furthermore, it enables us to discuss three "big fiber theorems", the Centerpoint Theorem in combinatorial geometry, the Maximal Fiber Inequality in topology, and the Non-displaceable Fiber Theorem in symplectic topology, from a unified viewpoint. Our main technical tool is an enhancement of the symplectic cohomology theory recently developed by Varolgunes.

math.SG↗

arXiv2019

Quasi-morphisms on contactomorphism groups and Grassmannians of 2-planes

We construct a natural prequantization space over a monotone product of a toric manifold and an arbitrary number of complex Grassmannians of 2-planes in even-dimensional complex spaces, and prove that the universal cover of the identity component of the contactomorphism group of its total space carries a nonzero homogeneous quasi-morphism. The construction uses Givental's nonlinear Maslov index and a reduction theorem for quasi-morphisms on contactomorphism groups previously established together with M. Strom Borman. We explore applications to metrics on this group and to symplectic and contact rigidity. In particular we obtain a new proof that the quaternionic projective space, naturally embedded in the Grassmannian of 2-planes in a 2n-dimensional complex space as a Lagrangian, cannot be displaced from the real part of the complex Grassmannian by a Hamiltonian isotopy.

math.SG↗

arXiv2018

Approximation of quasi-states on manifolds

Quasi-states are certain not necessarily linear functionals on the space of continuous functions on a compact Hausdorff space. They were discovered as a part of an attempt to understand the axioms of quantum mechanics due to von Neumann. A very interesting and fundamental example is given by the so-called median quasi-state on the 2-sphere. In this paper we present an algorithm which numerically computes it to any specified accuracy. The error estimate of the algorithm crucially relies on metric continuity properties of a map, which constructs quasi-states from probability measures, with respect to appropriate Wasserstein metrics. We close with non-approximation results, particularly for symplectic quasi-states.

math.FA↗

arXiv2016

Rigidity of the L^p norm of the Poisson bracket on surfaces

For a symplectic manifold $M$ let $\{\cdot,\cdot\}$ be the corresponding Poisson bracket. In this note we prove that the functional $(F,G) \mapsto \|\{F,G\}\|_{L^p(M)}$ is lower-semicontinuous with respect to the $C^0$-norm on $C^\infty_c(M)$ when $\dim M = 2$ and $p < \infty$, extending previous rigidity results for $p = \infty$ in arbitrary dimension.

math.SG↗

arXiv2016

On the contact mapping class group of the contactization of the $A_m$- Milnor fiber

We construct an embedding of the full braid group on $m+1$ strands $B_{m+1}$, $m \geq 1$, into the contact mapping class group of the contactization $Q \times S^1$ of the $A_m$-Milnor fiber $Q$. The construction uses the embedding of $B_{m+1}$ into the symplectic mapping class group of $Q$ due to Khovanov and Seidel, and a natural lifting homomorphism. In order to show that the composed homomorphism is still injective, we use a partially linearized variant of the Chekanov--Eliashberg dga for Legendrians which lie above one another in $Q \times \mathbb{R}$, reducing the proof to Floer homology. As corollaries we obtain a contribution to the contact isotopy problem for $Q \times S^1$, as well as the fact that in dimension $4$, the lifting homomorphism embeds the symplectic mapping class group of $Q$ into the contact mapping class group of $Q \times S^1$.

math.SG↗

arXiv2015

Spectral invariants for monotone Lagrangians

Since spectral invariants were introduced in cotangent bundles via generating functions by Viterbo in the seminal paper "Symplectic topology as the geometry of generating functions," they have been defined in various contexts, mainly via Floer homology theories, and then used in a great variety of applications. In this paper we extend their definition to monotone Lagrangians, which is so far the most general case for which a "classical" Floer theory has been developed. Then, we gather and prove the properties satisfied by these invariants, and which are crucial for their applications. Finally, as a demonstration, we apply these new invariants to symplectic rigidity of some specific monotone Lagrangians.

math.SG↗

arXiv2015

The Lagrangian Floer-quantum-PSS package and canonical orientations in Floer theory

The purpose of this paper is to extend the construction of the PSS-type isomorphism between the Floer homology and the quantum homology of a monotone Lagrangian submanifold $L$ of a symplectic manifold $M$, provided that the minimal Maslov number of $L$ is at least two, to arbitrary coefficients. We provide a proof, again over arbitrary coefficients, that this isomorphism respects the natural algebraic structures on both sides, such as the quantum product and the quantum module action. This isomorphism serves as the technical foundation for the construction of Lagrangian spectral invariants in a joint paper with Remi Leclercq (arXiv:1505:07430). Our constructions work when the second Stiefel--Whitney class of $L$ vanishes on the image of the boundary homomorphism $π_3(M,L) \to π_2(L)$, a condition strictly weaker than being relatively Pin; in particular we do not require $L$ to be orientable. The constructions are done using canonical orientations, and require no further choices such as relative Pin-structures. Such structures do however play a significant role when endowing the various complexes and homologies with structures of modules over Novikov rings, and in calculations.

math.SG↗

arXiv2014

Quasi-morphisms on contactomorphism groups and contact rigidity

We build homogeneous quasi-morphisms on the universal cover of the contactomorphism group for certain prequantizations of monotone symplectic toric manifolds. This is done using Givental's nonlinear Maslov index and a contact reduction technique for quasi-morphisms. We show how these quasi-morphisms lead to a hierarchy of rigid subsets of contact manifolds. We also show that the nonlinear Maslov index has a vanishing property, which plays a key role in our proofs. Finally we present applications to orderability of contact manifolds and Sandon-type metrics on contactomorphism groups.

math.SG↗

arXiv2012

Geometric structures on contactomorphism groups and contact rigidity in jet spaces

For a closed connected manifold N, we establish the existence of geometric structures on various subgroups of the contactomorphism group of the standard contact jet space J^1N, as well as on the group of contactomorphisms of the standard contact T*N \times S^1 generated by compactly supported contact vector fields. The geometric structures are biinvariant partial orders (for J^1N and T*N \times S^1) and biinvariant integer-valued metrics (T*N\times S^1 only). Also we prove some forms of contact rigidity in T*N \times S^1, namely that certain (possibly singular) subsets of the form X \times S^1 cannot be disjoined from the zero section by a contact isotopy, and in addition that there are restrictions on the kind of contactomorphisms of T*N\times S^1 which are products of pairwise commuting contactomorphisms generated by vector fields supported in sets of the form U \times S^1 with U \subset T*N Hamiltonian displaceable. The method is that of generating functions for Legendrians in jet spaces.

math.SG↗

arXiv2012

On the Lagrangian Hofer geometry in symplectically aspherical manifolds

We use spectral invariants in Lagrangian Floer theory in order to show that there exist \emph{isometric} embeddings of normed linear spaces (finite or infinite dimensional, depending on the case) into the space of Hamiltonian deformations of certain Lagrangian submanifolds in tame symplectically aspherical manifolds. In addition to providing a new class of examples in which the Lagrangian Hofer metric can be computed explicitly, we refine and generalize some known results about it.

math.SG↗

arXiv2011

Partial quasi-morphisms and quasi-states on cotangent bundles, and symplectic homogenization

For a closed connected manifold N, we construct a family of functions on the Hamiltonian group G of the cotangent bundle T^*N, and a family of functions on the space of smooth functions with compact support on T^*N. These satisfy properties analogous to those of partial quasi-morphisms and quasi-states of Entov and Polterovich. The families are parametrized by the first real cohomology of N. In the case N=T^n the family of functions on G coincides with Viterbo's symplectic homogenization operator. These functions have applications to the algebraic and geometric structure of G, to Aubry-Mather theory, to restrictions on Poisson brackets, and to symplectic rigidity.

math.SG↗

arXiv2011

Quasi-morphisms on cotangent bundles and symplectic homogenization

For a class of closed manifolds N, we construct a family of functions on the Hamiltonian group G of the cotangent bundle T*N. These restrict to homogeneous quasi-morphisms on the subgroup generated by Hamiltonians with support in a given cotangent ball bundle. The family is parametrized by the first real cohomology of N, and in the case N=T^n, it coincides with Viterbo's symplectic homogenization operator. These functions have applications to the algebraic and geometric structure of G and its subgroups, to symplectic rigidity, and to Aubry-Mather and weak KAM theory.

math.SG↗

arXiv2010

A comparison of symplectic homogenization and Calabi quasi-states

We compare two functionals defined on the space of continuous functions with compact support in an open neighborhood of the zero section of the cotangent bundle of a torus. One comes from Viterbo's symplectic homogenization while the other from the Calabi quasi-states due to Entov and Polterovich. In dimension 2 we are able to say when these two functionals are equal. A partial result in higher dimensions is presented. We also give a link to asymptotic Hofer geometry on T^*S^1. Proofs are based on the theory of quasi-integrals and topological measures on locally compact spaces.

math.SG↗

arXiv2010

On almost Poisson commutativity in dimension two

Consider the following question: given two functions on a symplectic manifold whose Poisson bracket is small, is it possible to approximate them in the $C^0$ norm by commuting functions? We give a positive answer in dimension two, as a particular case of a more general statement which applies to functions on a manifold with a volume form. This result is based on a lemma in the spirit of geometric measure theory. We give some immediate applications to function theory and the theory of quasi-states on surfaces with area forms.

math.SG↗

arXiv2009

Functoriality in Morse theory on closed manifolds

We develop functoriality for Morse theory, namely, to a pair of Morse-Smale systems and a generic smooth map between the underlying manifolds we associate a chain map between the corresponding Morse complexes, which descends to the correct map on homology. This association does not in general respect composition. We give sufficient conditions under which composition is preserved. As an application we provide a new proof that the cup product as defined in Morse theory on the chain level agrees with the cup product in singular cohomology. In appendices we present a proof (due to Paul Biran) that the unstable manifolds of a Morse-Smale system are the open cells of a CW structure on the underlying manifold, and also we show that the Morse complex of the triple is canonically isomorphic to the cellular complex of the CW structure. This gives a new proof that the Morse complex is actually a complex and that it computes the homology of the manifold.

math.DG↗