Koszul duality and the link surgery formula

Ian Zemke

REVIEW 4 major objections 4 minor 5 references

Koszul duality and the link surgery formula

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper establishes a Koszul duality between the surgery algebra K and a curved dg-algebra K!, proving that the relevant module categories are equivalent via dualizing bimodules.

desk verdict Strong, explicit construction of a Koszul dual surgery algebra with real computational payoff, but the main category equivalence rests on an unproved curved-dg associativity claim. read the letter →

arxiv 2507.09964 v1 pith:DBWEDDLT submitted 2025-07-14 math.GT math.RA

Ian Zemke This is my paper · ORCID

classification math.GTmath.RA MSC 57R5816E4557K18

keywords KoszuldualitylinksurgeryformulaHeegaardFloerhomologyborderedtheorycurveddg-algebraA-infinitymodulestype-DL-spacelinks

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that the surgery algebra K, whose A∞-module categories encode Heegaard Floer Dehn surgery formulas for links, admits a Koszul dual curved dg-algebra K! that carries equivalent information in a smaller algebraic package. The author constructs dualizing bimodules that translate modules between the two sides, and proves that the category of bonsai, regularly U-adic type-A modules over K is equivalent to the category of cobonsai, regularly U-adic type-D modules over K!. If correct, computations with link surgery bimodules—often infinite-dimensional objects—can be performed on the K! side, where data is encoded by a single differential rather than an infinite family of higher actions, and then translated back without loss. This is motivated by practical computation of Heegaard Floer invariants of 3-manifolds and of satellite operators.

What carries the argument

The machinery is Koszul duality realized through bimodules in bordered Floer theory: the curved dg-algebra $K!$ and the dualizing pair of bimodules $_{K!}[\mathrm{Co}]_K$ and $_K[\mathrm{Tr}]_{K!}$. The trace bimodule $_K[\mathrm{Tr}]_{K!}$ is built using the homological perturbation lemma from a box tensor product, and the load-bearing identity is the pair of box-tensor equalities in Theorem 3.17. The category-level equivalence then follows by showing that these bimodules preserve the 'bonsai' and 'cobonsai' finiteness conditions as well as the 'regularly U-adic' filtration conditions, so that the functors land in the right categories and are mutually inverse up to homotopy.

What would settle it

Take the transformer bimodule $K[T]_K$ introduced in Section 5.2, tensor it with the Koszul dual bimodules in the two orders, and compute the associativity constraint from [LOT15, Lemma 2.3.14(3)] in the presence of the curvature $\mu_0 = \varphi_+\varphi_- + \varphi_-\varphi_+$; if any non-zero curvature term appears in the reassociation, the natural transformations in Proposition 4.11 are not well-defined.

Watch

Extended reading notes

Core claim

The central discovery is that the surgery algebra K has a curved dg-algebra dual K! generated over idempotents by $w,z,\theta$, $\varphi_+,\varphi_-,\theta$, and cross-elements $s,t$, with differential $\mu_1(\theta)=wz+zw$ and curvature $\mu_0=\varphi_+\varphi_-+\varphi_-\varphi_+$. The paper constructs a DD-bimodule $_{K!}[\mathrm{Co}]_K$ and an AA-bimodule $_K[\mathrm{Tr}]_{K!}$ whose box tensor products are the identity bimodules, so $[\mathrm{Co}]\boxtimes[\mathrm{Tr}]$ and $[\mathrm{Tr}]\boxtimes[\mathrm{Co}]$ equal the identity bimodules of $K!$ and $K$. Tensoring with these bimodules then gives an equivalence between the categories $_{K!}\mathrm{Mod}_{(U),b}$ and $_K\mathrm{Mod}_{(U),b}$, meaning the surgery formula's algebraic data can be moved to the dual algebra and recorded by a single differential instead of a whole A∞-module structure.

Load-bearing premise

The proof of the category equivalence assumes that a strict associativity result for box tensor products, proven for uncurved dg-algebras, carries over to curved dg-algebras such as K!; the paper states without proof that the same argument works.

Editorial extensions

If this is right

Any DA-bimodule for a 2-component link can be converted to a DD-bimodule over $K!$, whose data is finite and frequently small; the original DA-bimodule is recovered by tensoring with the trace bimodule.
Computations of satellite operators, such as those for L-space links, have a natural formulation on the $K!$ side, potentially extending algorithmic methods to broader families of links.
The category equivalence means that a type-A module over $K$ and a type-D module over $K!$ carry identical information, so one may choose whichever side is easier to encode.
Since $K!$ is a curved dg-algebra, the translation justifies working with type-D modules on that side, avoiding the poorly behaved A∞-module categories of curved algebras.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the stated extension of strict associativity to curved dg-algebras fails, the bimodule-level identities of Theorem 3.17 might still hold because they are proven by direct computation; the category equivalence is the part most sensitive to that gap.
The 'cobonsai' condition introduced here is a new finiteness condition tailored to curved dg-algebras; it could be extracted and applied to other Koszul-duality settings where curvature appears.
The same scheme should apply to any quadratic-linear-scalar algebra whose Koszul dual is curved, since the bimodule construction is essentially algorithmic; other surgery-type algebras might admit duals with equally small presentations.
A direct check of the curved associativity extension on the transformer bimodule example in Section 5.2 would provide a quick test of the equivalence.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Here is my read of arXiv:2507.09964. Zemke constructs a Koszul dual algebra K! to the surgery algebra K, builds explicit dualizing bimodules, and proves equivalences of categories. The construction is genuinely new and concrete, not an abstract existence result. The algebraic grading on K! does real work, and the examples—especially the L-space link bimodules and the transformer bimodule—show that the machinery has practical computational teeth. Theorem 1.3, the bimodule-level Koszul duality, is proved by explicit homological perturbation with many checks left as direct but plausible computations. I buy the overall shape of the paper.

The soft spots are the ones you flagged. The proof of Theorem 1.4 (via Proposition 4.11) quotes [LOT15, Lemma 2.3.14(3)] for associativity of box tensor products and adds a parenthetical that the same proof works for curved dg-algebras. That is not an argument. The curvature in K! contributes extra terms to the differentials of box products, and strict associativity is a coherence statement that needs proof, not an automatic extension. Second, the claim that the triple box product K![Co]K ⊠ K L K! ⊠ K![Co]K is separated is asserted without proof. Mixed terms like θ|U and s|σ make this non-obvious. These are load-bearing: the natural transformations in the equivalence are constructed through exactly this associativity step. If either claim fails, the equivalence is not established. I do not think these are fatal to the program, but they are real gaps, not cosmetic.

Minor but annoying: Remark 3.2 invokes a relation ZW=U in K to justify the quadratic-linear-scalar perspective. There is no such relation in the surgery algebra as defined; U lives in a different idempotent. The explicit definition of K! does not depend on this remark, but the remark is wrong and should be fixed.

I also want to note that several key verifications—the strong deformation retraction in Lemma 3.9, the direct computations in Lemma 3.22, and parts of Proposition 3.18—are delegated to the reader. That is normal for a paper of this kind, but combined with the unproved associativity claim it makes the conditional verdict honest.

Who should read this: anyone working on bordered Heegaard Floer homology, link surgery formulas, or low-dimensional topological computations. It deserves a serious referee, not a desk reject. I would send it to an expert in bordered Floer homology with enough patience to check the curved-dg associativity and the separatedness of the triple box product. Those are the two things I want nailed down before I would rely on Theorem 1.4. Engage with it, but require the fixes.

Referee Report

4 major / 4 minor

Summary. This paper develops a Koszul dual description of the surgery algebra K introduced by the author in earlier work. The author defines a curved dg-algebra K!, constructs explicit dualizing bimodules K![Co]K and K[Tr]K!, proves that their box tensor products give the identity bimodules (Theorem 1.3), and uses this to prove an equivalence of certain module categories (Theorem 1.4, proved in Proposition 4.11). The final sections give applications to bimodules for 2-component links and to L-space links.

Significance. The main contribution is a concrete and explicit Koszul dual algebra for the surgery algebra, together with a duality theorem at the level of modules. If the proof is completed, the result has clear computational value: it provides a smaller, often finite model for bimodules in the link surgery formula, as illustrated in Section 5. The paper makes a genuine effort to give explicit formulas for the bimodules and to verify the duality directly. The main theorems are not circular; they are supported by direct constructions. However, the proof of the category equivalence rests on nontrivial technical assertions that are currently not proved in the manuscript.

major comments (4)

[§4.4, Proposition 4.11] The proof of Proposition 4.11 invokes [LOT15, Lemma 2.3.14(3)] for strict associativity of box tensor products with a separated type-DD bimodule, with the parenthetical that 'the same proof works for curved dg-algebras.' This is an unproved extension: K! is a curved dg-algebra (Section 3.1, with curvature µ0 = φ+φ− + φ−φ+), and box tensor products over curved algebras involve additional µ0 terms in differentials and morphism complexes. The coherence needed for the strict associativity isomorphism is exactly what the omitted proof would establish. Since the natural transformation comparing F[Tr]∘F[Co] with the identity is constructed through this associativity step, this gap is load-bearing for Theorem 1.4.
[§4.4, Proposition 4.11] The proof asserts without proof that K![Co]K ⊠ K L K! ⊠ K![Co]K is a separated type-DD structure. This is not immediate: K![Co]K itself contains mixed terms such as θ|U and s|σ (Section 3.4), so the separatedness hypothesis of [LOT15, Lemma 2.3.14(3)] is not automatic. The separatedness must be verified explicitly; otherwise the quoted lemma cannot be applied. This is a second load-bearing gap in the proof of the main equivalence.
[§3.4, Lemma 3.9] The proof of Lemma 3.9, which constructs K[Tr]K! via the homological perturbation lemma, leaves several essential computations to the reader: that the diagrammatically defined d and H form a strong deformation retraction, that H∘α^{µ2}_{0|1|0}∘H = 0, and that H∘α^{µ1}_{0|1|0}∘H∘α^{µ1}_{0|1|0}∘H = 0. The same pattern appears in Lemma 3.3 and in the direct computations in Proposition 3.18 and Lemma 3.20. Because the bimodule K[Tr]K! and the duality statements in Theorem 1.3 rest on these identifications, the omitted verifications should be supplied or at least outlined in sufficient detail for the reader to reproduce them.
[§3.1, Remark 3.2] Remark 3.2 claims that K is a quadratic-linear-scalar algebra with relations including ZW = U, and uses this to derive σW = U T^{-1}σ. This is inconsistent with the definition of K in Section 1.1, where I0·K·I0 = F[W,Z] contains no element U and I0·K·I1 = 0. In particular, the product σU is zero in K, so the manipulation σW = T^{-1}σ ZW = T^{-1}σU is not valid in K. The remark should be corrected or removed; as written it undermines the claim that K! is obtained by a standard Koszul-duality recipe.

minor comments (4)

[§2.3] The text uses 'Mauer-Cartan' where 'Maurer-Cartan' is standard.
[§3.1, proof of Lemma 3.1] In the proof of Lemma 3.1, 'I1 · K1 · I1' should be 'I1 · K! · I1'.
[§5.1] The displayed definition of the symmetry E on K! reads 'E(w)=z, E(w)=z'; the second instance should be 'E(z)=w'.
[§4.4, proof of Proposition 4.11] The notation 'K Y' for a module is confusing; the text should consistently use a module name such as M.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the Koszul duality theorem is proved from explicit definitions, with unproved curved-dg associativity as a correctness gap only.

full rationale

The paper's central derivation chain is not circular. The algebra K and the surgery modules are taken from the author's prior work [Zem21, Zem23], but K is also explicitly recalled in Section 1.1, and the categories KMod(U),b and K!Mod(U),b are redefined in this paper (Sections 2.4, 4.3) rather than imported as black boxes. The Koszul dual algebra K! is defined by explicit generators, relations, differential, and curvature (Section 3.1), and the dualizing bimodules K[Tr]K! and K![Co]K are constructed explicitly via the homological perturbation lemma (Lemma 3.9). Theorem 1.3 is proved by direct computations in Propositions 3.18-3.22; Proposition 4.11 then derives the category equivalence from Theorem 1.3 plus standard box-tensor associativity. No parameter is fitted, no prediction reduces to an input, and no uniqueness theorem from the author's prior work is invoked to force the construction. The main caveat is in Proposition 4.11, where the paper extends [LOT15, Lemma 2.3.14(3)] to curved dg-algebras with the sentence 'the same proof works for curved dg-algebras'; this is an unproved assertion and a possible correctness gap, but it is a reliance on an external lemma, not a circular reduction to the theorem being proved. Likewise the separatedness claim for K!([Co]⊠L⊠[Co])K is asserted without proof; again this is proof incompleteness, not circularity. Self-citations to [Zem21, Zem23, CZZ24a] provide the surrounding surgery theory and examples but do not carry the logical weight of the new equivalence.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

The central claims rest on the definitions of the surgery algebra K and the bordered Floer formalism from prior work, on the standard homological perturbation lemma, and on a stated-but-unproved extension of a lemma to curved algebras. No free parameters are fitted to data. The new algebra K! is an invented structure, but it is explicitly defined and its duality equations are verified, so it does not have the 'graviton' problem.

assumptions (3)

standard math The homological perturbation lemma as stated in Lemma 2.6 and Lemma 2.7.
Used to construct the trace bimodule and the A-infinity algebra K!∞; the proofs are cited as standard.
domain assumption The definition and properties of the surgery algebra K and its module categories from Zemke's prior works [Zem21, Zem23].
The paper recalls the definitions but the geometric content, that these modules compute Heegaard Floer invariants, is taken as input from previous papers.
domain assumption The extension of [LOT15, Lemma 2.3.14(3)] from uncurved to curved dg-algebras.
Invoked in the proof of Proposition 4.11 to obtain strict associativity of box tensor products; the extension is asserted without proof.

invented entities (1)

K!, the Koszul dual curved dg-algebra independent evidence
purpose: Dual algebra encoding the same surgery formulas as K with smaller rank
Explicitly defined in Section 3.1 with relations and differential; the duality equations are verified in Theorem 1.3.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Koszul duality and the link surgery formula." pith.science (2026). https://pith.science/paper/DBWEDDLT

@misc{pith2026250709964,
  author       = {Pith},
  title        = {Pith review of: Koszul duality and the link surgery formula},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DBWEDDLT}},
  note         = {Machine review of arXiv:2507.09964}
}

abstract

In previous works, the author described an associative algebra whose $A_\infty$-module categories encode the Heegaard Floer Dehn surgery formulas. In this article, we describe the Koszul dual of this algebra. We construct dualizing bimodules, and prove several equivalences of categories. The constructions of this paper have applications to computational problems involving the link surgery formula.

Figures

Figures reproduced from arXiv: 2507.09964 by the authors.

**Figure 3.1.** The six configurations which contribute to the zsθ coefficient of the differential of K! [Co] K ⊠ K[Tr]K!. Definition 4.1. We say that a type-D module K!M is cobonsai if there is an n0 > 0 so that each summand b1 ⊗ · · · ⊗ bn ⊗ y of δ n (x), where bi ∈ K! is a monomial, has the [PITH_FULL_IMAGE:figures/full_fig_p033_3_1.png] view at source ↗

**Figure 4.1.** Two diagrams which encode summands of δ n of K! [Co] K ⊠ KM. The left tree is irrelevant to the cobonsai-ness of the tensor product, since it vanishes when we compose with (I + ε): K! → K! +. The right tree is relevant to the cobonsai-ness of the tensor product. the module input to the root. We consider the action mT on the tensor product K[Tr]K! ⊠ K! N. For each n ≥ 0, we take the iterated structure map δ n of K! … view at source ↗

**Figure 4.2.** A left module input tree T (left), and a right expansion T ′ of T (right) Since K[Tr]K! is strictly unital, summands of δ 1 of K! N which are weighted by elements of I ⊗ N make no contribution to mT (such differentials only contribute to m1, which is never a factor of mT ). Since the structure maps of K[Tr]K! are algebraically [PITH_FULL_IMAGE:figures/full_fig_p035_4_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

5 extracted references · 1 canonical work pages

[2010]

[MOT25] Ciprian Manolescu, Peter S

e-print, arXiv:1011.1317 [math.GT]. [MOT25] Ciprian Manolescu, Peter S. Ozsv´ ath, and Dylan P. Thurston,Grid diagrams and Heegaard Floer invariants, Ann. of Math. (2)201(2025), no. 1, 1–78. [OS04] Peter Ozsv´ ath and Zolt´ an Szab´ o,Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2)159(2004), no. 3, 1027–1158. [OS...

arXiv 2025
[2020]

[MO10] Ciprian Manolescu and Peter S

e-print, arXiv:2009.05222 [math.GT]. [MO10] Ciprian Manolescu and Peter S. Ozsv´ ath,Heegaard Floer homology and integer surgeries on links,

work page arXiv 2009
[2021]

[Zem23] ,A general Heegaard Floer surgery formula,

e-print, arXiv 2109.11520 [math.GT]. [Zem23] ,A general Heegaard Floer surgery formula,

arXiv
[2023]

Department of Mathematics, University of Oregon, Eugene, OR, USA Email address:izemke@uoregen.edu

e-print, arXiv 2308.15658 [math.GT]. Department of Mathematics, University of Oregon, Eugene, OR, USA Email address:izemke@uoregen.edu

arXiv
[2024]

[CZZ24b] ,satellites, GitHub, 2024.https://github.com/ian-zemke/satellites, commit 0aac7b6

e-print, arXiv 2412.05755 [math.GT]. [CZZ24b] ,satellites, GitHub, 2024.https://github.com/ian-zemke/satellites, commit 0aac7b6. [GN16] Eugene Gorsky and Andr´ as N´ emethi,Links of plane curve singularities areL-space links, Algebr. Geom. Topol.16(2016), no. 4, 1905–1912. [LOT11] Robert Lipshitz, Peter S. Ozsv´ ath, and Dylan P. Thurston,Heegaard Floer h...

arXiv 2016

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [2010]

[MOT25] Ciprian Manolescu, Peter S

e-print, arXiv:1011.1317 [math.GT]. [MOT25] Ciprian Manolescu, Peter S. Ozsv´ ath, and Dylan P. Thurston,Grid diagrams and Heegaard Floer invariants, Ann. of Math. (2)201(2025), no. 1, 1–78. [OS04] Peter Ozsv´ ath and Zolt´ an Szab´ o,Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2)159(2004), no. 3, 1027–1158. [OS...

arXiv 2025

[2] [2020]

[MO10] Ciprian Manolescu and Peter S

e-print, arXiv:2009.05222 [math.GT]. [MO10] Ciprian Manolescu and Peter S. Ozsv´ ath,Heegaard Floer homology and integer surgeries on links,

work page arXiv 2009

[3] [2021]

[Zem23] ,A general Heegaard Floer surgery formula,

e-print, arXiv 2109.11520 [math.GT]. [Zem23] ,A general Heegaard Floer surgery formula,

arXiv

[4] [2023]

Department of Mathematics, University of Oregon, Eugene, OR, USA Email address:izemke@uoregen.edu

e-print, arXiv 2308.15658 [math.GT]. Department of Mathematics, University of Oregon, Eugene, OR, USA Email address:izemke@uoregen.edu

arXiv

[5] [2024]

[CZZ24b] ,satellites, GitHub, 2024.https://github.com/ian-zemke/satellites, commit 0aac7b6

e-print, arXiv 2412.05755 [math.GT]. [CZZ24b] ,satellites, GitHub, 2024.https://github.com/ian-zemke/satellites, commit 0aac7b6. [GN16] Eugene Gorsky and Andr´ as N´ emethi,Links of plane curve singularities areL-space links, Algebr. Geom. Topol.16(2016), no. 4, 1905–1912. [LOT11] Robert Lipshitz, Peter S. Ozsv´ ath, and Dylan P. Thurston,Heegaard Floer h...

arXiv 2016