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Egor Shelukhin

Publications and source records attributed to Egor Shelukhin.

At least 37 records · Page 2Linked to original sources
arXiv2021

The Schwarz-Milnor lemma for braids and area-preserving diffeomorphisms

We prove a number of new results on the large-scale geometry of the $L^p$-metrics on the group of area-preserving diffeomorphisms of each orientable surface. Our proofs use in a key way the Fulton-MacPherson type compactification of the configuration space of $n$ points on the surface due to Axelrod-Singer and Kontsevich. This allows us to apply the Schwarz-Milnor lemma to configuration spaces, a natural approach which we carry out successfully for the first time. As sample results, we prove that all right-angled Artin groups admit quasi-isometric embeddings into the group of area-preserving diffeomorphisms endowed with the $L^p$-metric, and that all Gambaudo-Ghys quasi-morphisms on this metric group coming from the braid group on $n$ strands are Lipschitz. This was conjectured to hold, yet proven only for low values of $n$ and the genus $g$ of the surface.

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arXiv2020

The $\mathbb{Z}/p \mathbb{Z}$-equivariant product-isomorphism in fixed point Floer cohomology

Let $p \geq 2$ be a prime, and $\mathbb{F}_p$ be the field with $p$ elements. Extending a result of Seidel for $p=2,$ we construct an isomorphism between the Floer cohomology of an exact or Hamiltonian symplectomorphism $ϕ,$ with $\mathbb{F}_p$ coefficients, and the $\mathbb{Z}/p \mathbb{Z}$-equivariant Tate Floer cohomology of its $p$-th power $ϕ^p.$ The construction involves a Kaledin-type quasi-Frobenius map, as well as a $\mathbb{Z}/p \mathbb{Z}$-equivariant pants product: an equivariant operation with $p$ inputs and $1$ output. Our method of proof involves a spectral sequence for the action filtration, and a local $\mathbb{Z}/p \mathbb{Z}$-equivariant coproduct providing an inverse on the $E^2$-page. This strategy has the advantage of accurately describing the effect of the isomorphism on filtration levels. We describe applications to the symplectic mapping class group, as well as develop Smith theory for the persistence module of a Hamiltonian diffeomorphism $ϕ$ on symplectically aspherical symplectic manifolds. We illustrate the latter by giving a new proof of the celebrated no-torsion theorem of Polterovich, and by relating the growth rate of the number of periodic points of the $p^k$-th iteration of $ϕ$ and its distance to the identity. Along the way, we prove a sharpening of the classical Smith inequality for actions of $\mathbb{Z}/p \mathbb{Z}.$

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arXiv2020

Hamiltonian no-torsion

In 2002 Polterovich has notably established that on closed aspherical symplectic manifolds, Hamiltonian diffeomorphisms of finite order, which we call Hamiltonian torsion, must in fact be trivial. In this paper we prove the first higher-dimensional Hamiltonian no-torsion theorems beyond the symplectically aspherical case. We start by showing that closed symplectic Calabi-Yau and negative monotone symplectic manifolds do not admit Hamiltonian torsion. Going still beyond topological constraints, we prove that every closed positive monotone symplectic manifold $(M,ω)$ admitting Hamiltonian torsion is geometrically uniruled by holomorphic spheres for every $ω$-compatible almost complex structure, partially answering a question of McDuff-Salamon. This provides many additional no-torsion results, and as a corollary yields the geometric uniruledness of monotone Hamiltonian $S^1$-manifolds, a fact closely related to a celebrated result of McDuff from 2009. Moreover, the non-existence of Hamiltonian torsion implies the triviality of Hamiltonian actions of lattices like $SL(k,\mathbb{Z})$ for $k \geq 2,$ as well as those of compact Lie groups. Finally, for monotone symplectic manifolds admitting Hamiltonian torsion, we prove an analogue of Newman's theorem on finite transformation groups for several natural norms on the Hamiltonian group: such subgroups cannot be contained in arbitrarily small neighborhoods of the identity. Our arguments rely on generalized Morse-Bott methods, as well as on quantum Steenrod powers and Smith theory in filtered Floer homology.

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arXiv2020

Symplectic cohomology and a conjecture of Viterbo

We identify a new class of closed smooth manifolds for which there exists a uniform bound on the Lagrangian spectral norm of Hamiltonian deformations of the zero section in a unit cotangent disk bundle, settling a well-known conjecture of Viterbo from 2007 as the special case of $T^n.$ This class of manifolds is defined in topological terms involving the Chas-Sullivan algebra and the BV-operator on the homology of the free loop space, contains spheres and is closed under products. We discuss generalizations and various applications.

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arXiv2020

Pseudo-rotations and Steenrod squares

In this note we prove that if a closed monotone symplectic manifold $M$ of dimension $2n,$ satisfying a homological condition that holds in particular when the minimal Chern number is $N>n,$ admits a Hamiltonian pseudo-rotation, then the quantum Steenrod square of the point class must be deformed. This gives restrictions on the existence of pseudo-rotations. Our methods rest on previous work of the author, Zhao, and Wilkins, going back to the equivariant pair-of-pants product-isomorphism of Seidel.

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arXiv2019

On the Hofer-Zehnder conjecture

We prove that if a Hamiltonian diffeomorphism of a closed monotone symplectic manifold with semisimple quantum homology has more contractible fixed points, counted homologically, than the total dimension of the homology of the manifold, then it must have an infinite number of contractible periodic points. This constitutes a higher-dimensional homological generalization of a celebrated result of Franks from 1992, as conjectured by Hofer and Zehnder in 1994.

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arXiv2019

Pseudo-rotations and Steenrod squares revisited

In this note we prove that if a closed monotone symplectic manifold admits a Hamiltonian pseudo-rotation, which may be degenerate, then the quantum Steenrod square of the cohomology class Poincaré dual to the point must be deformed. This result gives restrictions on the existence of pseudo-rotations, implying a form of uniruledness by pseudo-holomorphic spheres, and generalizes a recent result of the author. The new component in the proof consists in an elementary calculation with capped periodic orbits.

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arXiv2019

On the entropy norm on $Ham(S^2)$

In this note we prove that for each positive integer $m$ there exists a bi-Lipschitz embedding $Z^m\to Ham(S^2)$, where $Ham(S^2)$ is equipped with the entropy metric. In particular, the same result holds when the entropy metric is substituted with the autonomous metric.

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arXiv2019

K-theoretic invariants of Hamiltonian fibrations

We introduce new invariants of Hamiltonian fibrations with values in the suitably twisted K-theory of the base. Inspired by techniques of geometric quantization, our invariants arise from the family analytic index of a family of natural $Spin^c$-Dirac operators. As an application we give new examples of non-trivial Hamiltonian fibrations, that have not been previously detected by other methods. As one crucial ingredient we construct a potentially new homotopy equivalence map, with a certain naturality property, from $BU$ to the space of index $0$ Fredholm operators on a Hilbert space, using elements of modern theory of homotopy colimits.

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arXiv2018

Viterbo conjecture for Zoll symmetric spaces

We prove a conjecture of Viterbo from 2007 on the existence of a uniform bound on the Lagrangian spectral norm of Hamiltonian deformations of the zero section in unit cotangent disk bundles, for bases given by compact rank one symmetric spaces $S^n, \mathbb{R} P^n, \mathbb{C} P^n, \mathbb{H} P^n,$ $n\geq 1.$ We discuss generalizations and give applications, in particular to $C^0$ symplectic topology. Our key methods, which are of independent interest, consist of a reinterpretation of the spectral norm via the asymptotic behavior of a family of cones of filtered morphisms, and a quantitative deformation argument for Floer persistence modules, that allows to excise a divisor.

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arXiv2018

Bounds on spectral norms and barcodes

We investigate the relations between algebraic structures, spectral invariants, and persistence modules, in the context of monotone Lagrangian Floer homology with Hamiltonian term. Firstly, we use the newly introduced method of filtered continuation elements to prove that the Lagrangian spectral norm controls the barcode of the Hamiltonian perturbation of the Lagrangian submanifold, up to shift, in the bottleneck distance. Moreover, we show that it satisfies Chekanov type low-energy intersection phenomena, and non-degeneracy theorems. Secondly, we introduce a new averaging method for bounding the spectral norm from above, and apply it to produce precise uniform bounds on the Lagrangian spectral norm in certain closed symplectic manifolds. Finally, by using the theory of persistence modules, we prove that our bounds are in fact sharp in some cases. Along the way we produce a new calculation of the Lagrangian quantum homology of certain Lagrangian submanifolds, and answer a question of Usher.

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arXiv2018

Lagrangian Shadows and Triangulated Categories

Under certain assumptions (such as weak exacteness or monotonicity) we show that splitting Lagrangians through cobordism has an energy cost and, from this cost being smaller than certain explicit bounds, we deduce some strong forms of rigidity of Lagrangian intersections. As a consequence, we construct some new pseudo-metrics and metrics on certain classes of Lagrangian submanifolds. We also fit these constructions in a more general setting, independent of Lagrangian cobordism. As a main technical tool, we develop aspects of the theory of (weakly) filtered A-infinity categories.

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arXiv2017

Persistence modules with operators in Morse and Floer theory

We introduce a new notion of persistence modules endowed with operators. It encapsulates the additional structure on Floer-type persistence modules coming from the intersection product with classes in the ambient (quantum) homology, along with a few other geometric situations. We provide sample applications to the $C^0$-geometry of Morse functions and to Hofer's geometry of Hamiltonian diffeomorphisms, that go beyond spectral invariants and traditional persistent homology.

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arXiv2016

On a uniform estimate for the quaternionic Calabi problem

We establish a C^0 a priori bound on the solutions of the quaternionic Calabi-Yau equation (of Monge-Ampere type) on compact HKT manifolds with a locally flat hypercomplex structure. As an intermediate step, we prove a quaternionic version of the Gauduchon theorem.

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arXiv2016

The $L^p$-diameter of the group of area-preserving diffeomorphisms of $S^2$

We show that for each $p \geq 1,$ the $L^p$-metric on the group of area-preserving diffeomorphisms of the two-sphere has infinite diameter. This solves the last open case of a conjecture of Shnirelman from 1985. Our methods extend to yield stronger results on the large-scale geometry of the corresponding metric space, completing an answer to a question of Kapovich from 2012. Our proof uses configuration spaces of points on the two-sphere, quasi-morphisms, optimally chosen braid diagrams, and, as a key element, the cross-ratio map $X_4(\mathbb{C} P^1) \to \mathcal{M}_{0,4} \cong \mathbb{C} P^1 \setminus \{\infty,0,1\}$ from the configuration space of $4$ points on $\mathbb{C} P^1$ to the moduli space of complex rational curves with $4$ marked points.

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arXiv2015

Lagrangian cobordism and metric invariants

We introduce new pseudo-metrics on spaces of Lagrangian submanifolds of a symplectic manifold $(M,ω)$ by considering areas associated to projecting Lagrangian cobordisms in $\mathbb{C} \times M$ to the "time-energy plane" $\mathbb{C}$. We investigate the non-degeneracy properties of these pseudo-metrics, reflecting the rigidity and flexibility aspects of Lagrangian cobordisms.

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