A Note on Somewhere Positive Loops of Contactomorphisms

Igor Uljarevi\'c

arxiv: 2601.07714 · v2 · pith:EFZA4RHEnew · submitted 2026-01-12 · 🧮 math.SG

A Note on Somewhere Positive Loops of Contactomorphisms

Igor Uljarevi\'c This is my paper

Pith reviewed 2026-05-21 15:48 UTC · model grok-4.3

classification 🧮 math.SG

keywords contactomorphismsimmaterial subsetsReeb-invariant setssymplectic homologycontact quasi-measurescontact geometrycontractible loops

0 comments

The pith

The complement of a Reeb-invariant immaterial subset is big in contact geometric terms.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines contractible loops of contactomorphisms that remain positive over certain non-empty closed subsets of a contact manifold; these subsets are called immaterial. It focuses on the case where the immaterial subset is invariant under the Reeb flow and claims that its complement then qualifies as large when measured by standard tools of contact geometry. A sympathetic reader would care because the result offers a concrete way to distinguish large and small subsets inside contact manifolds, with potential consequences for understanding periodic orbits and fillings. The argument rests on two independent supporting results, one from symplectic homology of a filling and one from contact quasi-measures.

Core claim

We consider contractible loops of contactomorphisms that are positive over some non-empty closed subset of a contact manifold. Such closed subsets are called immaterial. We argue that the complement of a Reeb-invariant immaterial subset can be seen as big in contact geometric terms. This is supported by two results: one regarding symplectic homology of the filling and the other regarding recently introduced contact quasi-measures.

What carries the argument

Immaterial subset: a non-empty closed subset of the contact manifold over which there exists a contractible loop of contactomorphisms that is positive everywhere on the subset; Reeb-invariance of this subset is the extra condition that makes the complement large via the two cited invariants.

If this is right

Symplectic homology of the filling detects the largeness of the complement.
Contact quasi-measures assign values consistent with the complement being large.
The dynamics generated by positive loops on immaterial sets respect this geometric size distinction.
Reeb-invariant immaterial sets can be treated as negligible in certain contact invariants.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same largeness statement might hold for immaterial sets that are not Reeb-invariant if additional dynamical assumptions are imposed.
Similar arguments could be tested on standard contact manifolds such as the sphere to produce explicit examples.
The notion of bigness defined here may interact with other invariants already used to study contactomorphisms.

Load-bearing premise

The results on symplectic homology of the filling and on contact quasi-measures actually establish the claimed largeness for the complement.

What would settle it

An explicit Reeb-invariant immaterial subset whose complement fails to be large when measured by either the symplectic homology of a filling or by contact quasi-measures.

read the original abstract

In this note, we consider contractible loops of contactomorphisms that are positive over some non-empty closed subset of a contact manifold. Such closed subsets are called immaterial. We argue that the complement of a Reeb-invariant immaterial subset can be seen as big in contact geometric terms. This is supported by two results: one regarding symplectic homology of the filling and the other regarding recently introduced contact quasi-measures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This note defines immaterial subsets via positive contractible loops of contactomorphisms and claims Reeb-invariant ones make their complements big, but the inference from homology and quasi-measures needs checking.

read the letter

Hi there, the main thing to know is that this short note introduces the term immaterial subset for a non-empty closed set in a contact manifold that admits a contractible loop of contactomorphisms positive over it. It then argues that when the subset is Reeb-invariant, the complement counts as big in contact-geometric terms, and this rests on two results: one about symplectic homology of the filling and one about recently introduced contact quasi-measures. The definition and the specific Reeb-invariance link look new within the cited literature, and the note does a reasonable job of being concise while pointing to how existing tools might give a fresh way to talk about largeness inside contact manifolds. The soft spot is exactly the central inference the stress-test flags. The abstract presents the homology and quasi-measure results as support for bigness of the complement, but it is not automatic that positivity or non-vanishing on the subset or filling directly translates into a geometric notion of bigness for the complement without an explicit comparison or duality step. Reeb-invariance sets up the application of those tools, yet the paper would be stronger if it spelled out why that step follows rather than assuming it. If the proofs contain a direct argument that closes this gap, the claim holds; otherwise it is a place that could use tightening. This is aimed at specialists in contact and symplectic geometry who already work with symplectic homology and quasi-measures. A reader looking for new descriptive language around size in contact manifolds could get some value from it. I would send it to peer review so the details of the two supporting results and the inference can be checked by people in the area.

Referee Report

2 major / 2 minor

Summary. The paper considers contractible loops of contactomorphisms that are positive over a non-empty closed subset of a contact manifold, termed an immaterial subset. It argues that the complement of a Reeb-invariant immaterial subset can be viewed as big in contact-geometric terms. This is supported by two results: one on the symplectic homology of a filling and the other on recently introduced contact quasi-measures.

Significance. If the supporting results establish the claimed largeness, the note offers a new way to detect big sets in contact manifolds via positive loops, algebraic invariants from symplectic homology, and quasi-measures. This could strengthen connections between dynamics of contactomorphisms and geometric notions of size, with the use of contact quasi-measures as a potentially useful tool if the implications are made precise.

major comments (2)

[Abstract and introduction] Abstract and central argument: the claim that the complement of a Reeb-invariant immaterial subset is 'big' is asserted to follow from the two supporting results, but the manuscript must explicitly detail the inference step. It is not automatic that non-vanishing symplectic homology of the filling or positivity properties of contact quasi-measures on the subset imply a geometric bigness statement for the complement; a direct comparison, duality, or complement-specific argument appears needed and should be stated in the main text (e.g., after the statements of the two results).
[Section on symplectic homology of the filling] Symplectic homology result: clarify whether the result establishes a property that directly transfers to the complement being large (via Reeb-invariance or otherwise) or only concerns the filling associated to the immaterial subset itself. If the former, state the precise comparison or vanishing/non-vanishing statement used for the complement.

minor comments (2)

[Introduction] Define 'immaterial subset' and 'Reeb-invariant' with precise notation at first use to ensure the reader can follow how invariance is used to set up the two results.
[References] Add a reference to the source introducing contact quasi-measures for context on the second supporting result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful report and for identifying points where the logical connections in our arguments can be made more explicit. The comments focus on clarifying how our two supporting results imply that the complement of a Reeb-invariant immaterial subset is geometrically large. We agree these clarifications will improve the manuscript and will incorporate them in the revised version. Our responses to the major comments are as follows.

read point-by-point responses

Referee: [Abstract and introduction] Abstract and central argument: the claim that the complement of a Reeb-invariant immaterial subset is 'big' is asserted to follow from the two supporting results, but the manuscript must explicitly detail the inference step. It is not automatic that non-vanishing symplectic homology of the filling or positivity properties of contact quasi-measures on the subset imply a geometric bigness statement for the complement; a direct comparison, duality, or complement-specific argument appears needed and should be stated in the main text (e.g., after the statements of the two results).

Authors: We agree that the inference step linking the two results to the bigness of the complement should be stated explicitly rather than left implicit. The abstract and introduction assert support from the results but do not spell out the precise mechanism (e.g., via Reeb-invariance inducing a duality or vanishing on the complement). In the revised manuscript we will add a short paragraph immediately after the statements of the two main results. This paragraph will explain that Reeb-invariance of the immaterial subset allows the non-vanishing symplectic homology of its filling to imply, by standard properties of symplectic homology under decomposition, that the complement cannot admit a filling with vanishing homology in the relevant degree, thereby establishing geometric largeness; an analogous argument using the positivity of contact quasi-measures on the subset (and their vanishing on the complement under Reeb-invariance) will also be included. revision: yes
Referee: [Section on symplectic homology of the filling] Symplectic homology result: clarify whether the result establishes a property that directly transfers to the complement being large (via Reeb-invariance or otherwise) or only concerns the filling associated to the immaterial subset itself. If the former, state the precise comparison or vanishing/non-vanishing statement used for the complement.

Authors: The symplectic homology result is formulated for a filling of the immaterial subset and establishes non-vanishing in a specific degree under the assumption of a contractible positive loop. Because the subset is Reeb-invariant, this non-vanishing transfers directly to a statement about the complement: any filling of the complement would force vanishing of symplectic homology in the complementary degree by the long exact sequence or Mayer-Vietoris type argument for the decomposition of the filling, contradicting known non-vanishing results for contact manifolds with positive loops. We will revise the relevant section to state this precise transfer explicitly, including the non-vanishing statement for the complement. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claim supported by independent results

full rationale

The paper presents its main argument as following from two distinct supporting results—one on symplectic homology of the filling and one on contact quasi-measures—without any equations, definitions, or self-citations that reduce the bigness claim for the complement to a tautology, fitted input, or prior self-referential premise. The abstract explicitly frames these as separate supports for the geometric interpretation of largeness, and no load-bearing step collapses the conclusion back to its inputs by construction. The derivation remains self-contained against external contact-geometric benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The note relies on standard background from contact geometry and symplectic homology. The only new object is the immaterial subset itself.

axioms (1)

standard math Standard definitions and properties of contact manifolds, Reeb flows, contractible loops of contactomorphisms, symplectic homology of fillings, and contact quasi-measures hold as previously established.
Invoked throughout the abstract without re-derivation.

invented entities (1)

immaterial subset no independent evidence
purpose: A non-empty closed subset of a contact manifold on which contractible loops of contactomorphisms are positive.
New term introduced to label the sets under study.

pith-pipeline@v0.9.0 · 5576 in / 1291 out tokens · 61301 ms · 2026-05-21T15:48:14.668170+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1. Let A ⊂ ∂W be the preimage of a closed subset under a smooth Reeb-invariant map. Assume A is immaterial and denote Ω := ∂W ∖ A. Then, the continuation map SH^Ω_* (W) → SH_*(W) is surjective.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.2 ... τ(B) ⩾ m_B / (m_B − m)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Remarksoneternalclassesinsymplecticcohomology

DylanCant.“Remarksoneternalclassesinsymplecticcohomology”. In: arXiv preprint arXiv:2410.03914(2024)

work page arXiv 2024
[2]

Shelukhin’sHoferdistanceandasymplecticcohomol- ogybarcodeforcontactomorphisms

DylanCant.“Shelukhin’sHoferdistanceandasymplecticcohomol- ogybarcodeforcontactomorphisms”.In:arXivpreprintarXiv:2309.00- 529 (2023)

work page 2023
[3]

Quantitativechar- acterization in contact Hamiltonian dynamics–I

DanĳelDjordjević,IgorUljarević,andJunZhang.“Quantitativechar- acterization in contact Hamiltonian dynamics–I”. In:arXiv preprint arXiv:2309.00527(2023)

work page arXiv 2023
[4]

Quantitative contact Hamiltonian dynamics

Danĳel Djordjević, Igor Uljarević, and Jun Zhang. “Quantitative contact Hamiltonian dynamics”. In:arXiv preprint arXiv:2507.13234 (2025)

work page arXiv 2025
[5]

Partiallyorderedgroups andgeometryofcontacttransformations

YakovEliashbergandLeonidPolterovich.“Partiallyorderedgroups andgeometryofcontacttransformations”.In: Geometric&Functional Analysis GAFA10.6 (2000), pp. 1448–1476

work page 2000
[6]

Quasi-morphisms and quasi-states in symplectic topology

Michael Entov. “Quasi-morphisms and quasi-states in symplectic topology”. In:Proceedings of the International Congress of Mathemati- cians - Seoul 2014. Vol. II. Kyung Moon Sa, Seoul, 2014, pp. 1147– 1171

work page 2014
[7]

Quasi-statesandsymplectic intersections

MichaelEntovandLeonidPolterovich.“Quasi-statesandsymplectic intersections”. In:Comment. Math. Helv.81.1 (2006), pp. 75–99

work page 2006
[8]

Rigid subsets of symplectic manifolds

Michael Entov and Leonid Polterovich. “Rigid subsets of symplectic manifolds”. In:Compos. Math.145.3 (2009), pp. 773–826

work page 2009
[9]

Symplecticquasi-statesand semi-simplicity of quantum homology

MichaelEntovandLeonidPolterovich.“Symplecticquasi-statesand semi-simplicity of quantum homology”. In:Toric topology. Vol. 460. Contemp. Math. Amer. Math. Soc., Providence, RI, 2008, pp. 47–70. 9

work page 2008
[10]

Floer homology and No- vikov rings

Helmut Hofer and Dietmar A. Salamon. “Floer homology and No- vikov rings”. In:The Floer memorial volume. Springer, 1995, pp. 483– 524

work page 1995
[11]

Maximumprinciplesinsymplectic homology

WillJ.MerryandIgorUljarević.“Maximumprinciplesinsymplectic homology”. In:Israel Journal of Mathematics229.1 (2019), pp. 39–65

work page 2019
[12]

Circle actions, quantum cohomology, and the Fukaya category of Fano toric varieties

Alexander Ritter. “Circle actions, quantum cohomology, and the Fukaya category of Fano toric varieties”. In:Geometry & Topology 20.4 (2016), pp. 1941–2052

work page 2016
[13]

π_1 of Symplectic Automorphism Groups and Invertibles inQuantumHomologyRings

P Seidel. “π_1 of Symplectic Automorphism Groups and Invertibles inQuantumHomologyRings”.In: GeometricAndFunctionalAnalysis 7.6 (1997), pp. 1046–1096

work page 1997
[14]

Selective symplectic homology with applications to contact non-squeezing

Igor Uljarević. “Selective symplectic homology with applications to contact non-squeezing”. In:Compositio Mathematica 159.11 (2023), pp. 2458–2482. 10

work page 2023

[1] [1]

Remarksoneternalclassesinsymplecticcohomology

DylanCant.“Remarksoneternalclassesinsymplecticcohomology”. In: arXiv preprint arXiv:2410.03914(2024)

work page arXiv 2024

[2] [2]

Shelukhin’sHoferdistanceandasymplecticcohomol- ogybarcodeforcontactomorphisms

DylanCant.“Shelukhin’sHoferdistanceandasymplecticcohomol- ogybarcodeforcontactomorphisms”.In:arXivpreprintarXiv:2309.00- 529 (2023)

work page 2023

[3] [3]

Quantitativechar- acterization in contact Hamiltonian dynamics–I

DanĳelDjordjević,IgorUljarević,andJunZhang.“Quantitativechar- acterization in contact Hamiltonian dynamics–I”. In:arXiv preprint arXiv:2309.00527(2023)

work page arXiv 2023

[4] [4]

Quantitative contact Hamiltonian dynamics

Danĳel Djordjević, Igor Uljarević, and Jun Zhang. “Quantitative contact Hamiltonian dynamics”. In:arXiv preprint arXiv:2507.13234 (2025)

work page arXiv 2025

[5] [5]

Partiallyorderedgroups andgeometryofcontacttransformations

YakovEliashbergandLeonidPolterovich.“Partiallyorderedgroups andgeometryofcontacttransformations”.In: Geometric&Functional Analysis GAFA10.6 (2000), pp. 1448–1476

work page 2000

[6] [6]

Quasi-morphisms and quasi-states in symplectic topology

Michael Entov. “Quasi-morphisms and quasi-states in symplectic topology”. In:Proceedings of the International Congress of Mathemati- cians - Seoul 2014. Vol. II. Kyung Moon Sa, Seoul, 2014, pp. 1147– 1171

work page 2014

[7] [7]

Quasi-statesandsymplectic intersections

MichaelEntovandLeonidPolterovich.“Quasi-statesandsymplectic intersections”. In:Comment. Math. Helv.81.1 (2006), pp. 75–99

work page 2006

[8] [8]

Rigid subsets of symplectic manifolds

Michael Entov and Leonid Polterovich. “Rigid subsets of symplectic manifolds”. In:Compos. Math.145.3 (2009), pp. 773–826

work page 2009

[9] [9]

Symplecticquasi-statesand semi-simplicity of quantum homology

MichaelEntovandLeonidPolterovich.“Symplecticquasi-statesand semi-simplicity of quantum homology”. In:Toric topology. Vol. 460. Contemp. Math. Amer. Math. Soc., Providence, RI, 2008, pp. 47–70. 9

work page 2008

[10] [10]

Floer homology and No- vikov rings

Helmut Hofer and Dietmar A. Salamon. “Floer homology and No- vikov rings”. In:The Floer memorial volume. Springer, 1995, pp. 483– 524

work page 1995

[11] [11]

Maximumprinciplesinsymplectic homology

WillJ.MerryandIgorUljarević.“Maximumprinciplesinsymplectic homology”. In:Israel Journal of Mathematics229.1 (2019), pp. 39–65

work page 2019

[12] [12]

Circle actions, quantum cohomology, and the Fukaya category of Fano toric varieties

Alexander Ritter. “Circle actions, quantum cohomology, and the Fukaya category of Fano toric varieties”. In:Geometry & Topology 20.4 (2016), pp. 1941–2052

work page 2016

[13] [13]

π_1 of Symplectic Automorphism Groups and Invertibles inQuantumHomologyRings

P Seidel. “π_1 of Symplectic Automorphism Groups and Invertibles inQuantumHomologyRings”.In: GeometricAndFunctionalAnalysis 7.6 (1997), pp. 1046–1096

work page 1997

[14] [14]

Selective symplectic homology with applications to contact non-squeezing

Igor Uljarević. “Selective symplectic homology with applications to contact non-squeezing”. In:Compositio Mathematica 159.11 (2023), pp. 2458–2482. 10

work page 2023