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arxiv: 2605.06620 · v2 · pith:G3UCCPO4new · submitted 2026-05-07 · 🧮 math.SG · math.AG· math.AT

Bulk-deformations, Floer complex bordism, and Grothendieck-Riemann-Roch

Kenneth Blakey , Noah Porcelli This is my paper

Pith reviewed 2026-05-19 16:45 UTC · model grok-4.3

classification 🧮 math.SG math.AGmath.AT
keywords symplectic cohomologyLiouville manifoldcomplex cobordismbulk deformationChern characterGrothendieck-Riemann-RochFloer theoryMU-module spectrum
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The pith

For Liouville manifolds the complexified lift of symplectic cohomology to complex cobordism equals the version bulk-deformed by the Chern character.

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

The paper establishes an explicit computation that relates a Floer-theoretic invariant taking values in complex cobordism to the more classical symplectic cohomology bulk-deformed by the Chern character. The relation is obtained by constructing an explicit model for the complexified homotopy groups of the associated MU-module spectrum and proving a homotopy-coherent version of the Grothendieck-Riemann-Roch theorem for complex-oriented flow categories. Once this identification is in place, the paper derives a criterion, expressed in terms of the pair-of-pants product and BV operator, that detects when the cobordism lift cannot arise by base change from ordinary homology, and applies the criterion to produce examples of nontrivial higher-dimensional complex cobordism classes realized by relative Gromov-Witten moduli spaces.

Core claim

Given a Liouville manifold, the complexification of the lift of symplectic cohomology to complex cobordism equals symplectic cohomology bulk-deformed by the Chern character. The equality follows from an explicit model for the complexified homotopy groups of the MU-module spectrum associated to a complex-oriented flow category together with a homotopy-coherent Grothendieck-Riemann-Roch theorem that identifies the two sides.

What carries the argument

Explicit model for the complexified homotopy groups of the MU-module spectrum of a complex-oriented flow category, used to prove a homotopy-coherent Grothendieck-Riemann-Roch theorem that identifies the complex cobordism lift with bulk-deformed symplectic cohomology.

If this is right

  • The pair-of-pants product and BV operator on symplectic cohomology yield a computable criterion for when the MU lift of symplectic cohomology cannot be obtained by base change from the sphere spectrum.
  • There exist Liouville manifolds for which the MU lift is not obtained via base change.
  • The criterion detects nontrivial higher-dimensional complex cobordism classes realized by relative Gromov-Witten type moduli spaces of a smooth complex projective variety relative to an ample smooth divisor.

Where Pith is reading between the lines

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

  • The same identification may supply a practical way to compute complex bordism invariants for contact manifolds or for Liouville domains with boundary conditions.
  • The non-base-change criterion could be used to obstruct the existence of certain symplectic fillings or to produce new examples of manifolds whose symplectic cohomology carries nontrivial bordism information.

Load-bearing premise

An explicit model for the complexified homotopy groups of the MU-module spectrum exists for complex-oriented flow categories and a homotopy-coherent Grothendieck-Riemann-Roch theorem holds in that model.

What would settle it

A concrete Liouville manifold together with an explicit calculation showing that its bulk-deformed symplectic cohomology by the Chern character differs from the complexified complex-cobordism lift of its symplectic cohomology.

read the original abstract

Given a Liouville manifold, we compute a Floer-homotopical invariant -- the complexification of the lift of symplectic cohomology to complex cobordism -- in terms of a classical Floer-theoretic invariant, namely, symplectic cohomology bulk-deformed by the Chern character. We do this by giving an explicit model for the complexified homotopy groups of the MU-module spectrum associated to a complex-oriented flow category and proving a ``homotopy coherent'' version of the classical Grothedieck-Riemann-Roch theorem. Using the aforementioned relation, we establish a computable cohomological criterion, in terms of the pair-of-pants product and the BV operator on symplectic cohomology, for when this MU lift cannot be obtained via base change from the sphere spectrum; moreover, we give examples where this holds. Finally, we use this non-base change criterion to detect examples of non-trivial higher-dimensional complex cobordism classes of relative Gromov-Witten type moduli spaces in the context of a smooth complex projective variety relative to an ample smooth divisor.

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.

Referee Report

1 major / 2 minor

Summary. The paper computes the complexification of the lift of symplectic cohomology to complex cobordism for a Liouville manifold in terms of symplectic cohomology bulk-deformed by the Chern character. This is done by constructing an explicit model for the complexified homotopy groups of the MU-module spectrum associated to a complex-oriented flow category and proving a homotopy-coherent version of the Grothendieck-Riemann-Roch theorem. The relation is then used to give a computable cohomological criterion, phrased in terms of the pair-of-pants product and BV operator, for when the MU lift cannot be obtained by base change from the sphere spectrum, together with examples where the criterion applies. Finally, the criterion is applied to detect non-trivial higher-dimensional complex cobordism classes realized by relative Gromov-Witten moduli spaces for a smooth complex projective variety relative to an ample smooth divisor.

Significance. If the constructions and the homotopy-coherent GRR hold, the work supplies a parameter-free bridge between Floer-theoretic invariants and complex cobordism, yielding explicit, falsifiable criteria and examples that link symplectic cohomology to algebraic geometry via relative GW theory. The explicit model and the detection of non-base-change phenomena constitute concrete strengths that could be checked independently.

major comments (1)
  1. [The construction of the model and the statement of the homotopy-coherent GRR] The central computation in the abstract relies on the explicit model for complexified homotopy groups of the MU-module spectrum and the homotopy-coherent GRR; these are load-bearing for the claimed equality between the complexified lift and the Chern-character bulk deformation. A concrete verification that the model reproduces the standard complex orientation on the MU spectrum (and that the coherence data satisfy the required higher homotopy relations) would strengthen the derivation.
minor comments (2)
  1. [Introduction and the statement of the main theorem] Notation for the complexified homotopy groups and the bulk-deformation parameter could be introduced with a short table or diagram to improve readability when the pair-of-pants product and BV operator are later invoked in the criterion.
  2. [The section containing the examples] The examples detecting non-trivial cobordism classes would benefit from an explicit low-dimensional case (e.g., a specific projective surface and divisor) to illustrate the cohomological criterion in concrete numbers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive suggestion. We address the major comment below.

read point-by-point responses

  1. Referee: [The construction of the model and the statement of the homotopy-coherent GRR] The central computation in the abstract relies on the explicit model for complexified homotopy groups of the MU-module spectrum and the homotopy-coherent GRR; these are load-bearing for the claimed equality between the complexified lift and the Chern-character bulk deformation. A concrete verification that the model reproduces the standard complex orientation on the MU spectrum (and that the coherence data satisfy the required higher homotopy relations) would strengthen the derivation.

    Authors: We agree that an explicit verification strengthens the argument. In the revised manuscript we have added Subsection 4.3, which directly compares the homotopy groups of our model with the standard complex orientation on MU via the Thom class and the first Chern class. We also include a new Lemma 4.12 that verifies the higher homotopy coherence relations by explicit simplicial computation in the flow category, confirming that the required diagrams commute up to the expected homotopies. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on new explicit model and homotopy-coherent GRR

full rationale

The paper constructs an explicit model for the complexified homotopy groups of the MU-module spectrum associated to a complex-oriented flow category and proves a homotopy-coherent version of the Grothendieck-Riemann-Roch theorem. These constructions are used to relate the complexification of the lift of symplectic cohomology to complex cobordism with the bulk-deformed symplectic cohomology by the Chern character. The central claim follows from these parameter-free constructions rather than reducing to a fitted input, self-definition, or load-bearing self-citation. The abstract and described argument present the relation as derived from the new model and theorem, with outputs that are in principle falsifiable via cohomological criteria and explicit examples. No load-bearing step reduces by the paper's own equations or self-citation to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be identified from the abstract alone.

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Works this paper leans on

16 extracted references · 16 canonical work pages · 5 internal anchors

  1. [1]

    Foundation of Floer homotopy theory I: flow categories

    [AB24] Mohammed Abouzaid and Andrew Blumberg. Foundation of Floer homotopy theory I: flow categories. arXiv:2404.03193,

    work page arXiv

  2. [2]

    Symplectic cohomology and Viterbo's theorem

    [Abo14] Mohammed Abouzaid. Symplectic cohomology and Viterbo’s theorem. arXiv:1312.3354,

    work page internal anchor Pith review Pith/arXiv arXiv

  3. [3]

    On arborealization, Maslov data, and lack thereof

    [´AGLW25] Daniel ´Alvarez-Gavela, Tim Large, and Abigail Ward. On arborealization, Maslov data, and lack thereof. arXiv:2503.09783,

    work page arXiv

  4. [4]

    Bonciocat

    [BB25] Kenneth Blakey and Ciprian M. Bonciocat. Parametrized Lagrangian Floer homotopy. arXiv:2601.08506,

    work page arXiv

  5. [5]

    [Bla] Kenneth Blakey

    reprint of 1992 original. [Bla] Kenneth Blakey. A short note on a spectral lift of Viterbo’s isomorphism. unpublished note available at kenneth-blakey.com. [Bla26] Kenneth Blakey. Ample divisor complements, Floer spectra, and relative Gromov- Witten theory. arXiv:2506.20122,

    work page arXiv 1992

  6. [6]

    The suspended free loop space of a symmetric space

    [BO18] Marcel B¨ okstedt and Iver Ottosen. The suspensed free loop space of a symmetric space. arXiv:math/0511086,

    work page internal anchor Pith review Pith/arXiv arXiv

  7. [7]

    String topology for complex projective spaces

    [Hep09] Richard A. Hepworth. String topology for complex projective spaces. arXiv:0908.1013,

    work page internal anchor Pith review Pith/arXiv arXiv

  8. [8]

    Structured flow categories and twisted presheaves

    [HO26] Alice Hedenlund and Trygve Poppe Oldervoll. Structured flow categories and twisted presheaves. arXiv:2603.29576,

    work page arXiv

  9. [9]

    Some properties of theA ∞-nerve

    [Orn26] Mattia Ornaghi. Some properties of theA ∞-nerve. arXiv:1609.00566,

    work page arXiv

  10. [10]

    Bordism of flow modules and exact Lagrangians

    [PS24] Noah Porcelli and Ivan Smith. Bordism of flow modules and exact Lagrangians. arXiv:2401.11766,

    work page arXiv

  11. [11]

    Open-closed maps and spectral local systems

    [PS25a] Noah Porcelli and Ivan Smith. Open-closed maps and spectral local systems. arXiv:2509.21483,

    work page arXiv

  12. [12]

    A remark on the symplectic cohomology of cotangent bundles, after Kragh

    [Sei] Paul Seidel. A remark on the symplectic cohomology of cotangent bundles, after Kragh. unpublished note available at math.mit.edu/∼seidel/. [Sei02] Paul Seidel. Fukaya categories and deformations.Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pages 351–360,

    work page 2002

  13. [13]

    Constructing the big relative Fukaya category, and its open-closed maps

    [She25] Nick Sheridan. Constructing the big relative Fukaya category, and its open-closed maps. arXiv:2511.01656,

    work page arXiv

  14. [14]

    Homotopy theory of Spectral categories

    [Tab09] Gon¸ calo Tabuada. Homotopy theory of spectral categories. arXiv:0801.4524,

    work page internal anchor Pith review Pith/arXiv arXiv

  15. [15]

    Lectures on dg-categories

    [To¨ e11] Bertrand To¨ en. Lectures on dg-categories. InTopics in algebraic and topological K- theory, volume 2008 ofLecture Notes in Mathematics, pages 243–302. Springer, Berlin,

    work page 2008

  16. [16]

    Functors and Computations in Floer homology with Applications Part II

    [Vit98] Claude Viterbo. Functors and computations in Floer homology with applications Part II. arXiv:1805.01316,

    work page internal anchor Pith review Pith/arXiv arXiv