arxiv: 2605.00715 · v3 · submitted 2026-05-01 · 🧮 math.SG · math.AG· math.RT

Recognition: 2 theorem links

· Lean Theorem

Curves on surfaces and moduli of associative algebras

Yanki Lekili

Authors on Pith no claims yet

Pith reviewed 2026-05-12 05:00 UTC · model grok-4.3

classification 🧮 math.SG math.AGmath.RT

keywords associative algebrasFukaya categoriesLagrangian immersionsA-infinity structurespunctured surfacesGauss wordsbounding cochainsmoduli of algebras

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The pith

Lagrangian circle immersions in punctured surfaces realize all but one associative algebra of dimension at most four as endomorphism algebras in relative Fukaya categories.

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

The paper gives an explicit finite procedure to compute the full A-infinity algebra of an immersed circle in a punctured surface from its signed Gauss word and the visible polygons it bounds. Applying the procedure to curves with at most three double points shows that, over an algebraically closed field, every associative algebra of dimension four or less except one arises as the degree-zero endomorphism algebra of such an immersed circle equipped with a bounding cochain in a relative Fukaya category F(Σ,D). The same method also proves that every finite-dimensional algebra with radical square zero is realized in the ordinary Fukaya category of some punctured surface. This supplies a geometric source for the classification of small algebras.

Core claim

Given an immersion of a circle in a punctured surface Σ, the A∞-algebra of the curve as an object in the relative Fukaya category F(Σ,D) is computed explicitly and finitely from the signed Gauss word of its double points and the visible polygons bounded by the curve. This computation proves that all associative algebras of dimension ≤4 except one are realized as the degree 0 endomorphism algebra of some such immersed circle with bounding cochain, and that radical-square-zero algebras arise in F(Σ) for some Σ.

What carries the argument

Signed Gauss word of double points together with visible polygons, which determine all higher A∞-products for the immersed curve.

If this is right

Explicit A∞-products are now known for every immersion with at most three self-intersections.
Every finite-dimensional algebra whose radical squares to zero appears as the endomorphism algebra of an object in the Fukaya category of some punctured surface.
The classification of associative algebras up to dimension four is reduced to the enumeration of suitable immersed circles and their bounding cochains on punctured surfaces.

Where Pith is reading between the lines

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

The same geometric data may classify additional families of algebras once immersions with four or more double points are treated.
Deformations of the surface or the immersion could correspond to deformations of the resulting algebra.
One could check whether the single exceptional algebra of dimension four can be realized by allowing non-orientable surfaces or more general bounding cochains.

Load-bearing premise

The formulas extracted from the signed Gauss word and visible polygons correctly reproduce the structure maps of the relative Fukaya category.

What would settle it

An explicit A∞-structure computation for a specific immersion with two or three double points that fails to match the algebra predicted by the Gauss-word formula, or the discovery of a four-dimensional associative algebra (other than the known exception) that cannot be obtained from any circle immersion plus bounding cochain.

read the original abstract

Given an immersion of a circle in a punctured surface $\Sigma$, we give an explicit (and finite) computation of the $A_\infty$-algebra associated with this curve when viewed as an object in a (relative) Fukaya category of $\Sigma$ in terms of the signed Gauss word recording the double points in a traversal of the curve and the visible polygons that it bounds in $\Sigma$. We illustrate our computational technique by fully determining the $A_\infty$-products for immersions with up to three self-intersections. In particular, it is proved that, over an algebraically closed field, all associative algebras of dimension $\leq 4$, with one exception, can be realized as the (degree 0) endomorphism algebra of some Lagrangian immersion of a circle equipped with a bounding cochain computed in some relative Fukaya category $\mathcal{F}(\Sigma,D)$. We also note that any finite-dimensional algebra with radical square zero arises as the (degree 0) endomorphism algebra of an object in the Fukaya category $\mathcal{F}(\Sigma)$ of some punctured surface $\Sigma$.

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

Lekili supplies an explicit combinatorial recipe for A∞-products from the signed Gauss word and visible polygons of a circle immersion, then uses it to realize nearly all associative algebras of dimension ≤4 in relative Fukaya categories.

read the letter

This paper's main contribution is a finite, explicit way to calculate the A∞-products for the endomorphism algebra of a circle immersed in a punctured surface, using only the signed Gauss word of its double points and the visible polygons it bounds. It then applies this to realize all associative algebras of dimension 4 or less, except one, as such endomorphism algebras in a relative Fukaya category, after twisting by a bounding cochain. The new part is the computational technique itself. Earlier work on Fukaya categories of surfaces often left the structure maps as abstract counts of holomorphic disks. Here, Lekili turns that into a recipe based on the immersion data that can be read off directly. For curves with at most three self-intersections, the paper carries out the full calculation and lists the resulting algebras. This lets them match against the classification of low-dimensional associative algebras and hit all but one case. The separate remark that any algebra with radical square zero appears in some Fukaya category of a punctured surface is a nice, broader observation that follows from similar ideas. On the soft spots, the central one is the match between the combinatorial formulas and the actual structure maps in the relative Fukaya category. The stress test raises whether non-visible polygons or choices of almost complex structure could add extra terms. The paper works only with small numbers of intersections and claims the realizations work, so presumably the visible polygons capture everything in these examples. That seems plausible for low complexity, but the justification should be spelled out clearly so readers can see why no other contributions arise. The exception for one algebra is noted without much comment, which is fine if the list is complete. This is for researchers in symplectic geometry who care about explicit descriptions of Fukaya categories and how they relate to algebra. It will be useful to anyone trying to build concrete models or check small cases. The work shows clear engagement with both the geometric and algebraic sides, so it deserves a serious referee who can verify the computations and the completeness of the realization list. I recommend putting it through peer review. The explicit nature of the results makes it refereeable, and the technique could be of use beyond this paper once confirmed.

Referee Report

1 major / 2 minor

Summary. The paper develops an explicit combinatorial formula for the A_∞-algebra structure associated to an immersed circle in a punctured surface Σ, interpreted as an object in the relative Fukaya category F(Σ, D), expressed in terms of the signed Gauss word of the immersion and the visible polygons it bounds. The authors fully compute the A_∞-products for all such immersions with at most three self-intersections. They then apply this to prove that, over an algebraically closed field, every associative algebra of dimension at most 4 except one can be realized as the degree-zero endomorphism algebra of such an immersed circle equipped with a suitable bounding cochain in some relative Fukaya category. Additionally, they show that any finite-dimensional algebra with radical square zero arises as the degree-zero endomorphism algebra of an object in the Fukaya category of some punctured surface.

Significance. If the explicit combinatorial computations accurately capture the geometric A_∞-structure maps in the Fukaya category, this provides a valuable bridge between combinatorial topology of curves on surfaces and the algebraic structures arising in symplectic geometry. The realization results offer concrete geometric models for a large class of low-dimensional associative algebras, which could facilitate further study of their moduli spaces and deformations. The finite and explicit nature of the computations for small numbers of self-intersections is a notable strength, as it allows direct verification and construction of examples.

major comments (1)

The realization theorem (abstract and the section applying the computations to algebras of dimension ≤4) asserts that immersions with ≤3 self-intersections suffice to realize all but one such algebra. However, the manuscript establishes the combinatorial formulas only for these small cases and does not supply a case-by-case or general argument that non-visible holomorphic polygons or almost-complex-structure-dependent contributions are absent in the specific Σ and D chosen for each realization; if any such term appears after twisting by the bounding cochain, the resulting degree-0 algebra would differ from the claimed associative algebra. This is load-bearing for the central claim.

minor comments (2)

The introduction could state explicitly which algebra of dimension 4 is the exception and briefly indicate why it cannot be realized with the given method.
Notation for the signed Gauss word and the enumeration of visible polygons would benefit from an early illustrative example with a figure before the general formulas.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying this important point concerning the rigor of the realization theorem. We address the comment directly below.

read point-by-point responses

Referee: The realization theorem (abstract and the section applying the computations to algebras of dimension ≤4) asserts that immersions with ≤3 self-intersections suffice to realize all but one such algebra. However, the manuscript establishes the combinatorial formulas only for these small cases and does not supply a case-by-case or general argument that non-visible holomorphic polygons or almost-complex-structure-dependent contributions are absent in the specific Σ and D chosen for each realization; if any such term appears after twisting by the bounding cochain, the resulting degree-0 algebra would differ from the claimed associative algebra. This is load-bearing for the central claim.

Authors: We agree that an explicit argument is needed to confirm that only the visible polygons contribute in the specific surfaces used for the realizations. In the constructions, each Σ is built as a minimal punctured surface containing the immersed curve, with punctures placed precisely so that the complement of the curve consists solely of the visible polygonal regions together with punctured components that cannot support additional holomorphic polygons with boundary on the curve (any such polygon would necessarily intersect a puncture, which is excluded in the relative Fukaya category). The almost complex structure is chosen to be standard and integrable in a neighborhood of the curve and cylindrical near the punctures, ensuring that any non-visible region has no holomorphic representatives. These choices are uniform across the low-dimensional cases and are compatible with the bounding cochain, so that the twisted degree-zero endomorphism algebra remains exactly the combinatorial one computed from the visible polygons. We will add a short clarifying subsection (or appendix) detailing this reasoning for each algebra of dimension ≤4. This does not alter the results but strengthens the exposition. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit geometric construction determines the algebras

full rationale

The paper starts from an immersion of a circle in a punctured surface, records its signed Gauss word and visible polygons, and gives an explicit finite formula for the A∞-products in the relative Fukaya category. It then enumerates all such immersions with at most three self-intersections and exhibits bounding cochains whose degree-0 endomorphism algebras realize every associative algebra of dimension ≤4 except one. These steps are self-contained: the input data are independent geometric objects, the output algebras are computed directly from them, and no equation or claim reduces a prediction to a fitted parameter, a self-definition, or a load-bearing self-citation. The construction therefore supplies concrete realizations rather than tautologically recovering its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard axioms of A_∞-categories and the definition of relative Fukaya categories; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math The A_∞-structure maps on the Fukaya category satisfy the standard A_∞ relations.
Invoked when the endomorphism algebra is extracted from the immersed curve.
domain assumption The relative Fukaya category F(Σ,D) is defined and its objects include Lagrangian immersions equipped with bounding cochains.
Required for the realization statement in the relative setting.

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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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
explicit computation of the A∞-algebra ... in terms of the signed Gauss word ... and the visible polygons ... all associative algebras of dimension ≤4, with one exception, can be realized as the (degree 0) endomorphism algebra of some Lagrangian immersion ... in some relative Fukaya category F(Σ,D)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Definition 3 ... m3 products ... from subintervals of the signed Gauss word

Reference graph

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