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Floer homotopy theory for monotone Lagrangians

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Pith's one-line read Monotone Lagrangians carry a Steenrod algebra action on Floer cohomology.

desk verdict Ambitious and genuinely novel program for Steenrod actions on monotone Lagrangian Floer cohomology, but the load-bearing smooth-gluing transfer is asserted rather than proved, so the main theorem is conditional until that gap is closed. read the letter →

arxiv 2506.17431 v2 pith:DZU2AL5U submitted 2025-06-20 math.SG math.ATmath.GT

Ciprian Mircea Bonciocat This is my paper · ORCID
classification math.SGmath.ATmath.GT MSC 53D4053D1255S1057R9055N20
keywords LagrangianFloerhomologymonotoneLagrangiansSteenrodalgebraactionPostnikovtruncationcomplexbordismU-braneOh-Pozniakspectralsequencecleanintersections
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 monotone Lagrangians, which previously resisted Floer homotopy theory because disk bubbling produces curvature in high-dimensional moduli spaces, still carry a well-defined Steenrod algebra action on their Lagrangian Floer cohomology. The action is defined over the Postnikov truncation $\tau_{\leq N_\mu-3}MU$ of complex bordism, where $N_\mu$ is the minimal Maslov number, instead of over the sphere spectrum. The advantage is that only moduli spaces below index difference $N_\mu$ are needed as data; the missing higher pieces are filled by obstruction theory, and the truncation is exactly what kills the obstructing homotopy groups. If the construction is correct, Floer cohomology of such Lagrangians acquires operations beyond its ring structure, computable via upgraded Oh-Pozniak spectral sequences and an Albers-PSS isomorphism. The paper applies this to $\mathbb{RP}^n\subset\mathbb{CP}^n$ and extracts new restrictions on clean intersections with a Hamiltonian isotopy, including nonzero characteristic classes $q_i(TC)$ of the intersection component.

What carries the argument

The load-bearing object is an N-truncated R-oriented flow category: the data of compact moduli spaces of Floer trajectories only for index differences $\mu(i)-\mu(j)<N$, each a manifold with corners, together with R-Thom classes for their normal bundles. Out of this data the paper runs the Cohen-Jones-Segal construction over the Postnikov truncation $\tau_{\leq N-3}R$. The truncation is the mechanism: Pontrjagin-Thom collapse maps for pairs with index difference at least $N$ are missing, and extending them from boundary strata has obstructions in degrees at least $N-2$, exactly the homotopy groups that $\tau_{\leq N-3}R$ kills. For Lagrangians, the U-brane is the input that supplies the U-structure on the flow category; an index theorem proves the reduced index map $\Omega(U/O)\to\Omega^\infty KO$ is an equivalence of $E_\infty$-groups, so the U-brane criterion is purely topological. The upgraded Oh-Pozniak spectral sequence then computes the resulting Steenrod action from clean intersections, with $E_1$-page built from MU-Thom spectra over the intersection components.

What would settle it

Find a monotone pair satisfying the paper's hypotheses in which the compactified moduli space of index difference $N_\mu-1$ is not a smooth manifold with corners, for example if the gluing profile transferred from the Liouville setting produces a non-smooth transition function at the Maslov-index $N_\mu$ disk-bubble stratum in a projective example; then the truncated flow category $F(M,L)$ is not well-defined and the Steenrod action cannot exist.

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Extended reading notes

Core claim

The central claim, Theorem B (Theorem 2.26), is that for a closed embedded monotone Lagrangian $L\subset M$ with $\pi_1(M,L)=0$, $\pi_2(M,L)=\mathbb{Z}$, minimal Maslov number $N_\mu$, and a U-brane, there is a well-defined action $A^*_A(\tau_{\leq N_\mu-3}MU) \circlearrowleft HF^*(M,L;A)$, independent of the auxiliary Hamiltonian and almost-complex structure. The action comes from an $N_\mu$-truncated flow category $F(M,L)$, whose compactified moduli spaces are required only for index differences below $N_\mu$, fed into the Cohen-Jones-Segal construction over the truncated ring spectrum; the missing Pontrjagin-Thom data for larger index differences are supplied by obstruction theory, and the Postnikov truncation is chosen so the obstructions vanish. A U-brane, a null-homotopy of the map $L\times^h_ML\to U/O\to BO$ that induces a U-structure, is supplied by topological hypotheses, via an index theorem identifying the reduced index map $\Omega(U/O)\to\Omega^\infty KO$ with an equivalence. As a sample application, for $\mathbb{RP}^n\subset\mathbb{CP}^n$ with $n$ odd the induced Milnor primitive operations $Q_i$ force any clean intersection $\{p\}\sqcup C$ to satisfy explicit $Q_i$-formulas on $H^*(C;\mathbb{F}_2)$ and imply the nonzero characteristic classes $q_i(TC)=y_{2^{i+1}-1}$.

Load-bearing premise

The construction rests on the premise that a smooth-gluing theorem originally proved in the Liouville setting also holds for the monotone Lagrangian moduli spaces used here, so that every compactified moduli space below index difference $N_\mu$ is a genuine smooth manifold with corners; the paper does not prove that transfer, it only notes that the proof avoids the Liouville structure.

Editorial extensions

If this is right

  • Any monotone Lagrangian satisfying the hypotheses and admitting a U-brane has Floer cohomology that is a graded module over $A^*_A(\tau_{\leq N_\mu-3}MU)$, so its Steenrod operations are invariants of the Lagrangian, not of the chosen Hamiltonian.
  • The Albers-PSS isomorphism and the Oh-Pozniak spectral sequence respect the Steenrod action, so in the PSS range the operations on $HF^*$ are read off from $H^*_{\mathrm{sing}}(L;\mathbb{F}_2)$.
  • For $\tau_{\leq N_\mu-3}MU$-modules with finite homotopy support there are generalized Floer cohomology theories, meaning truncated Morava K-theories or similar spectra give additional obstructions beyond ordinary cohomology.
  • In the projective-space case, any clean intersection $\{p\}\sqcup C$ of $\mathbb{RP}^n$ with a Hamiltonian pushoff has the $\mathbb{F}_2$-cohomology of $\mathbb{RP}^{n-1}$, satisfies the $Q_i$-formulas (2.6)-(2.7), and has $q_i(TC)\neq 0$, ruling out many homologically plausible components.

Reading between the lines

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

  • The obstruction-theoretic fill-in suggests a dictionary: Steenrod operations over $\tau_{\leq N_\mu-3}MU$ that fail to lift to the next truncation should encode the same disk-bubble curvature that appears as the classical $d^2$ differential, giving a cohomological window into real Gromov-Witten invariants of the Lagrangian.
  • With a model structure on pro-spectra in which homology isomorphisms are the weak equivalences, the entire pro-spectrum, not just its Steenrod action, should become a Hamiltonian-isotopy invariant, upgrading Theorem 2.26 to a stable homotopy type.
  • The same truncation trick should apply to pairs of distinct Lagrangians, with $N_\mu$ replaced by the gcd of their minimal Maslov numbers; the periodicity range, and therefore the usable Albers-PSS range, would shrink accordingly.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper introduces N-truncated flow categories, which retain only moduli spaces of dimension at most N−2, and associates to an N-truncated R-oriented flow category a module prospectrum over the Postnikov truncation τ≤N−3R (Theorem A, §1). This is used to define an action of a Steenrod algebra on the Floer cohomology of the flow category, and generalized cohomology theories under a finite-homotopy-support condition. For a closed embedded monotone Lagrangian L⊂M satisfying π1(M,L)=0, π2(M,L)=Z and Nμ≥3, and equipped with a U-brane, the paper constructs an Nμ-truncated flow category F(M,L) from compactified Floer moduli spaces and obtains an action of A^*_A(τ≤Nμ−3MU) on HF^*(M,L;A), independent of auxiliary choices (Theorem B, Thm. 2.26). It also upgrades the Oh–Pozniak spectral sequence and the Albers–PSS isomorphism to Steenrod-module statements, and applies these to clean intersections of RP^n⊂CP^n (n odd), deriving restrictions on the topology and characteristic classes of the intersection component C, including non-vanishing of qi(TC) (Thm. 2.52).

Significance. If the construction can be completed, the paper would give one of the first systematic Floer homotopy-type invariants for monotone Lagrangians with minimal Maslov number at least 5, where disk-bubble curvature is not an issue below the truncation level. The obstruction-theoretic filling of the missing Pontrjagin–Thom maps (Prop. 1.35) is a clean and potentially useful idea, and the lift of Milnor primitives to τ≤rMU (Prop. 1.23) is concrete. The RP^n application yields explicit, falsifiable restrictions on clean intersections that are not visible to ordinary Floer cohomology; for instance, the non-vanishing of qi(TC) is a concrete new prediction. The main value of the paper is therefore both structural and computational, and the announced results would be a substantial advance if the analytic and categorical gaps are closed.

major comments (4)
  1. [§2.2, Thm. 2.5] The smooth manifold-with-corners structure on the partial compactifications M_{x̃,ỹ} is the foundation for Thm. 2.6, and hence for the flow category F(M,L) and every subsequent invariant, but the proof is an assertion that Large's Liouville-setting proof [23, §6] transfers verbatim. The manuscript gives no argument that Large's gluing and exponential-decay estimates are independent of the Liouville form, and it does not address the fact that in the monotone setting the exclusion of disk and sphere bubbles is a separate Maslov-index argument (Thm. 2.6) that is not part of the cited theorem. Since the continuation moduli spaces in Prop. 2.25 inherit the same dependence, this transfer should be proved in detail or stated as an explicit standing assumption.
  2. [§2.6, proof of Thm. 2.30] The comparison of the Floer and Morse Cohen–Jones–Segal constructions is moved to the topological category, where the argument depends on the spaces W_i of broken flow lines with open end from [9, Def. 3.2, Thm. 3.7]; the paper says these 'should be contractible' but gives no proof. This contractibility is needed for the Pontrjagin–Thom maps that identify the E_1-page with Thom spectra over MU-local systems, so the Oh–Pozniak spectral sequence upgrade is not established unless this assertion is proved or replaced by a precise reference.
  3. [§2.5, Prop. 2.25 and Thm. 2.26] The claimed independence of the Steenrod action from the Hamiltonian and almost-complex structure is proved only by a sketch. The construction of the Nμ-truncated continuation flow category again invokes the unproved smooth-gluing transfer of Thm. 2.5 for the mixed moduli spaces M_{x̃_I,ỹ_II}, and the proof does not explain how the U-brane is used to obtain compatible MU-orientations on these mixed moduli spaces, nor how the finite-range approximation of Prop. 1.38 is compatible with the continuation maps. These details are load-bearing for the 'well-defined' statement in Theorem B.
  4. [§1.5, Prop. 1.38] The finite-range approximation is the bridge between the possibly infinite indexing set and the finite CJS construction, but its proof is only a sketch. The statement that adding one index 'attaches a cell' and hence leaves cohomology, as a module over the Steenrod algebra, unchanged in a fixed finite range requires a precise cofiber-sequence argument, especially because the Pontrjagin–Thom maps for the new cells are constructed by the obstruction-theoretic extension of Prop. 1.35. Since Theorem 1.25 and the pro-spectrum description depend on this proposition, the proof needs to be expanded.
minor comments (6)
  1. [§1.5, Prop. 1.38] The first sentence of the proposition uses τ≤N−2R where the surrounding text and the Steenrod-action statement use τ≤N−3R; this is presumably a typo.
  2. [§2.2, Thm. 2.6] The proof that the action filtration embeds the indexing set into (Z,<) is compressed; it would help to state explicitly that the set of capped chords with a fixed action value is finite, since the intersection Φ1(L)∩L is compact after a nondegenerate perturbation.
  3. [§2.3, Thm. 2.15] The group-theoretic step in the proof ('they can only be a summand of themselves via the identity') is stated without proof and is not immediate for groups with Z/2 summands; a short argument or a reference is needed.
  4. [§2.4, Rmk. 2.19] The non-canonical choice of homotopy in Proposition 2.18 produces a U-structure, but the paper does not explain why the resulting Steenrod action in Theorem 2.26 is independent of this choice; this point should be clarified.
  5. [§1, Defs. 1.2 and 1.15] The definition of an R-orientation of a truncated flow category is given for the existing moduli spaces only, while the obstruction-theoretic extension later fills in the missing pairs; the domain of the 'higher PT' maps and the compatibility with Def. 1.15 should be spelled out.
  6. [Throughout §1 and §2] The paper cites [9] for many foundational results (e.g., Def. 2.9, Prop. 5.7, §6); listing the exact statements imported from [9] would make the dependence transparent and simplify verification.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Steenrod action is constructed from flow-category moduli data and the RP^n restrictions are derived, not assumed; self-citations to [9] supply prior machinery rather than re-importing the conclusion.

full rationale

No circular derivation found. The central object, the Steenrod action of Theorem B, is constructed from the moduli spaces of an N_mu-truncated flow category via Pontrjagin-Thom maps and a Cohen-Jones-Segal pro-spectrum; it is an output, not an input that is later read back. The obstruction-theoretic extension of the missing top PT maps (Prop. 1.35-Cor. 1.37) uses only the vanishing of pi_{>=N-2} tau_{<=N-3}R and is contractible, so no fitted parameter is being renamed as a prediction. The RP^n results (Thm. 2.47 and Thm. 2.52) are derived consequences of the constructed action together with Oh's computation and the Oh-Pozniak spectral sequence; the local-system shifts and grading parameter r are constrained by the known periodicity, not fitted to the claimed non-vanishing q_i(TC). The paper does rely heavily on the author's earlier [9] for the CJS construction, continuation maps, and the Morse flow-category Thom-spectrum identification, but these are prior independent mathematical results with their own hypotheses and proofs, not circular imports of the present theorem. The manuscript itself flags genuine gaps: Thm. 2.5 transfers Large's smoothness result from Liouville manifolds to the monotone setting by assertion ('the proof of the smoothness result does not use the Liouville structure'); Rmk. 1.40 admits that no model structure on pro-spectra is available for the desired invariance; the proof of Thm. 2.30 relies on spaces W_i from [9] 'which should be contractible'; and Thm. 2.27 says details are omitted. These are rigor-support concerns, not circular reductions: none of them assumes the conclusion it is used to prove. Because no step in the derivation reduces to its own inputs by definition or by a fitted parameter, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The central claim rests on several unproved inputs: the author's previous preprint [9] for core CJS machinery, Large's smooth gluing theorem transferred from Liouville to monotone settings, the contractibility of W_i spaces, and the existence of Postnikov truncations of E1 ring spectra. No numerical parameters are fitted anywhere in the paper.

assumptions (4)
  • domain assumption Correctness of the main technical results in the author's previous paper [9], including the CJS construction, existence and homotopy-uniqueness of embeddings, and the Morse flow category Thom spectrum theorem.
    Used as black boxes throughout: embeddings in Prop 1.6, recovering Floer homology in Prop 1.34, Morse flow category in the proof of Thm 2.30, and continuation maps in §2.5. [9] is a same-author preprint with no independent verification.
  • domain assumption Large's smooth gluing theorem [23] gives smooth manifold-with-corners structures on compactified moduli spaces, and the proof transfers from Liouville manifolds to monotone symplectic manifolds.
    Invoked in Thm 2.5 and Thm 2.6 to construct the N_μ-truncated flow category. The paper notes Large's setup is Liouville manifolds and asserts the smoothness proof does not use the Liouville structure.
  • ad hoc to paper The spaces W_i of broken flow lines with open end from [9] are contractible in the Lagrangian setting.
    Assumed in the proof of Thm 2.30 to carry out the Cohen-Jones-Segal construction for Morse flow categories in the topological category; the paper says 'which should be contractible'.
  • standard math Postnikov truncations of connective E1 ring spectra exist as E1 ring spectra, and the forgetful functor to spectra commutes with τ_{≤r}.
    Invoked in Def 1.22 and used in Cor 1.37 and Thm 1.25. This is standard in the literature (e.g., Lurie's Higher Algebra, Dugger-Shipley).
invented entities (1)
  • U-brane
    purpose: A null-homotopy of the map L ×^h_M L → U/O → BO, used to induce a U-structure on the Lagrangian flow category and thereby a Steenrod algebra action over τ_{≤N_μ-3}MU.
    Introduced in Def 2.21. Existence is shown for RP^n ⊂ CP^n with odd n, but the entity itself is a mathematical definition with no falsifiable empirical handle outside the theory.

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Pith. "Pith review of Floer homotopy theory for monotone Lagrangians." pith.science (2026). https://pith.science/paper/DZU2AL5U

@misc{pith2026250617431,
  author       = {Pith},
  title        = {Pith review of: Floer homotopy theory for monotone Lagrangians},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DZU2AL5U}},
  note         = {Machine review of arXiv:2506.17431}
}
abstract

We circumvent one of the roadblocks in associating Floer homotopy types to monotone Lagrangians, namely the curvature phenomena occurring in high dimensions. Given $N \ge 3$ and $R$ a connective $\mathbb E_1$-ring spectrum, there is a notion of an $N$-truncated, $R$-oriented flow category, to which we associate a module prospectrum over the Postnikov truncation $\tau_{\le N - 3}R$. This endows ordinary Floer cohomology with an action of the Steenrod algebra over $\tau_{\le N-3}R$, and also induces certain generalized cohomology theories. We give sufficient conditions for a closed embedded monotone Lagrangian to admit such well-defined invariants for $N = N_\mu$ the minimal Maslov number, and $R = MU$ complex bordism. Finally, we formulate Oh-Pozniak type spectral sequences for these invariants, and show that in the case of $\mathbb{RP}^n \subset \mathbb{CP}^n$ they provide further restrictions on the topology of clean intersections with a Hamiltonian isotopy, not detected by ordinary Floer (co)homology.

Figures

Figures reproduced from arXiv: 2506.17431 by the authors.

Figure 1
Figure 1. Depiction of the spectral sequence for clean intersection {p} ⊔ C (left) and for the whole of RPn (right), in the sample case n = 7 and r = 1, following notation conventions of part II of the theorem. Theorem 2.47 − Point + Connected, part II. Assume n ≥ 3 odd. Further, denote by r ∈ {0, . . . , n} the unique residue such that µ(p) ≡ r − 1 (mod n + 1), and by yd the generator of Hd (C; F2) for 0 ≤ d ≤ n − 1. Then, f… view at source ↗
Figure 2
Figure 2. Depiction of the two ranges in which d + 2i+1 must lie, for the identities (2.6), (2.7) to hold, as a function of r, in the sample case of n = 9 [PITH_FULL_IMAGE:figures/full_fig_p030_2.png] view at source ↗

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Reference graph

Works this paper leans on

42 extracted references · 33 canonical work pages · cited by 2 Pith papers

  1. [9]

    C. M. Bonciocat. Revisiting the Cohen-Jones-Segal construction in Morse-Bott theory, 2025. URL: https://arxiv.org/abs/2409.11278, arXiv:2409.11278

    arXiv 2025

  2. [1]

    Abouzaid and A

    M. Abouzaid and A. J. Blumberg. Arnold Conjecture and Morava K-theory, 2021. URL:https:// arxiv.org/abs/2103.01507, arXiv:2103.01507

    arXiv 2021

  3. [2]

    Abouzaid and A

    M. Abouzaid and A. J. Blumberg. Foundation of Floer homotopy theory I: Flow categories, 2024. URL: https://arxiv.org/abs/2404.03193, arXiv:2404.03193

    arXiv 2024

  4. [3]

    P. Albers. A Lagrangian Piunikhin-Salamon-Schwarz morphism and two comparison homomorphisms in Floer homology.Int. Math. Res. Not. IMRN, (4):Art. ID rnm134, 56, 2008.doi:10.1093/imrn/rnm134

    work page doi:10.1093/imrn/rnm134 2008

  5. [4]

    M. Ando, A. J. Blumberg, D. Gepner, M. J. Hopkins, and C. Rezk. An infinity-categorical approach to R-line bundles, R-module thom spectra, and twisted R-homology.Journal of Topology, 7(3):869–893, 2014

    work page 2014

  6. [5]

    M. F. Atiyah.K-theory. Advanced Book Classics. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, second edition, 1989. Notes by D. W. Anderson

    work page 1989

  7. [6]

    S. Auyeung. Local Lagrangian Floer homology of quasi-minimally degenerate intersections.Journal of Topology and Analysis, pages 1–38, 2023

    work page 2023

  8. [7]

    Basterra

    M. Basterra. André–Quillen cohomology of commutative S-algebras.Journal of pure and applied algebra, 144(2):111–143, 1999

    work page 1999

Show all 42 references
  1. [8]

    K. Blakey. Floer homotopy theory and degenerate Lagrangian intersections, 2024. URL: https:// arxiv.org/abs/2410.11478, arXiv:2410.11478

    2024

  2. [10]

    J. D. Christensen and D. C. Isaksen. Duality and pro-spectra.Algebr. Geom. Topol., 4:781–812, 2004. doi:10.2140/agt.2004.4.781

    2004 doi

  3. [11]

    R. L. Cohen. Floer homotopy theory, realizing chain complexes by module spectra, and manifolds with corners. In Algebraic topology, volume 4 of Abel Symp., pages 39–59. Springer, Berlin, 2009. URL: https://doi.org/10.1007/978-3-642-01200-6_3, doi:10.1007/978-3-642-01200-6\_3

    2009 doi

  4. [12]

    R. L. Cohen. The Floer homotopy type of the cotangent bundle.Pure Appl. Math. Q., 6(2):391–438,

  5. [13]

    R. L. Cohen, J. D. S. Jones, and G. B. Segal. Floer’s infinite-dimensional Morse theory and homotopy theory. InThe Floer memorial volume, volume 133 ofProgr. Math., pages 297–325. Birkhäuser, Basel, 1995

    1995

  6. [14]

    Côté and Y

    L. Côté and Y. B. Kartal. Equivariant Floer homotopy via Morse-Bott theory, 2024. URL:https: //arxiv.org/abs/2309.15089, arXiv:2309.15089

    2024 arXiv

  7. [15]

    Daemi and K

    A. Daemi and K. Fukaya. Atiyah-Floer conjecture: a formulation, a strategy to prove and generaliza- tions. arXiv preprint arXiv:1707.03924, 2017

    2017 arXiv

  8. [16]

    Daemi, K

    A. Daemi, K. Fukaya, and M. Lipyanskiy. Lagrangians, SO(3)-instantons and the Atiyah-Floer conjec- ture. arXiv preprint arXiv:2109.07032, 2021

    2021 arXiv

  9. [17]

    Dugger and B

    D. Dugger and B. Shipley. Postnikov extensions of ring spectra. Algebraic & Geometric Topology, 6(4):1785–1829, 2006

    2006

  10. [18]

    Fukaya, Y.-G

    K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono.Exponential decay estimates and smoothness of the moduli space of pseudoholomorphic curves, volume 299. American Mathematical Society, 2024

    2024

  11. [19]

    Gaunce Jr, J

    L. Gaunce Jr, J. P. May, M. Steinberger, et al. Equivariant stable homotopy theory, volume 1213. Springer, 2006

    2006

  12. [20]

    K. Jänich. Vektorraumbündel und der Raum der Fredholm-Operatoren. Mathematische Annalen, 161:129–142, 1965. URL:http://eudml.org/doc/161303

    1965

  13. [21]

    P. B. Kronheimer and T. S. Mrowka. Knot homology groups from instantons.Journal of Topology, 4(4):835–918, 2011

    2011

  14. [22]

    N. J. Kuhn and C. J. Lloyd. Computing the Morava K–theory of real Grassmannians using chromatic fixed point theory.Algebraic & Geometric Topology, 24(2):919–950, 2024

    2024

  15. [23]

    T. Large. Spectral Fukaya Categories for Liouville Manifolds. PhD thesis, Massachusetts Institute of Technology, 2021. URL:https://dspace.mit.edu/handle/1721.1/139233

    2021

  16. [24]

    A. Lazarev. Homotopy theory of A-infinity ring spectra and applications to MU-modules.K-theory, 24(3):243–281, 2001. 36 CIPRIAN M. BONCIOCAT

    2001

  17. [25]

    Lipshitz and S

    R. Lipshitz and S. Sarkar. A Khovanov stable homotopy type.Journal of the American Mathematical Society, 27(4):983–1042, Apr. 2014. URL:http://dx.doi.org/10.1090/S0894-0347-2014-00785-2, doi:10.1090/s0894-0347-2014-00785-2

    2014 doi

  18. [26]

    Lurie.Higher Algebra

    J. Lurie.Higher Algebra. Princeton University, Princeton, NJ, 2017

    2017

  19. [27]

    Mahowald

    M. Mahowald. Ring spectra which are Thom complexes.Duke Math. J., 46(3):549–559, 1979. URL: http://projecteuclid.org/euclid.dmj/1077313574

    1979

  20. [28]

    Manolescu

    C. Manolescu. Seiberg-Witten-Floer stable homotopy type of three-manifolds withb1 = 0.Geom. Topol., 7:889–932, 2003. doi:10.2140/gt.2003.7.889

    2003 doi

  21. [29]

    J. May, F. Quinn, N. Ray, and I. Tornehave.E-infinity Ring Spaces and E-infinity Ring Spectra. Lecture Notes in Mathematics. Springer Berlin Heidelberg, 2006. URL:https://books.google.com/books? id=bpV8CwAAQBAJ

    2006

  22. [30]

    J. P. May. E-infty ring spaces and E-infty ring spectra.Lecture Notes in Mathematics, 1977

    1977

  23. [31]

    J. Milnor. The Steenrod algebra and its dual, pages 357–382. URL: https://www.worldscientific. com/doi/abs/10.1142/9789814401319_0006, arXiv:https://www.worldscientific.com/doi/pdf/ 10.1142/9789814401319_0006, doi:10.1142/9789814401319_0006

    doi

  24. [32]

    Y.-G. Oh. Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks I.Communica- tions on pure and applied mathematics, 46(7):949–993, 1993

    1993

  25. [33]

    Y.-G. Oh. Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings.In- ternational Mathematics Research Notices, 1996(7):305–346, 01 1996.arXiv:https://academic.oup. com/imrn/article-pdf/1996/7/305/2247012/1996-7-305.pdf, doi:10.1155/S1073792896000219

    1996 doi

  26. [34]

    Porcelli and I

    N. Porcelli and I. Smith. Bordism of flow modules and exact Lagrangians, 2024. URL:https://arxiv. org/abs/2401.11766, arXiv:2401.11766

    2024 arXiv

  27. [35]

    Porcelli and I

    N. Porcelli and I. Smith. Spectral Floer theory and tangential structures, 2024. URL:https://arxiv. org/abs/2411.03257, arXiv:2411.03257

    2024 arXiv

  28. [36]

    N. W. Porcelli. Families of relatively exact Lagrangians, free loop spaces and generalised homology. Selecta Mathematica, 30(2):21, 2024

    2024

  29. [37]

    M. Poźniak. Floer homology, Novikov rings and clean intersections. InNorthern California Symplectic Geometry Seminar, pages 119–181. American Mathematical Society, 1999

    1999

  30. [38]

    Rezchikov

    S. Rezchikov. Cyclotomic Structures in Symplectic Topology, 2024. URL:https://arxiv.org/abs/ 2405.18370, arXiv:2405.18370

    2024 arXiv

  31. [39]

    S. Schwede. Orthogonal spectra and stable homotopy theory, 2022

    2022

  32. [40]

    P. Seidel. Lagrangian two-spheres can be symplectically knotted. Journal of Differential Geometry, 52(1):145–171, 1999

    1999

  33. [41]

    N. Yagita. On the Steenrod algebra of Morava K-theory.Journal of the London Mathematical Society, 2(3):423–438, 1980. Department of Mathematics, Stanford University, 450 Jane Stanford W ay, Building 380, Stanford, CA 94305-2125, USA. Email address: ciprianb@stanford.edu

    1980

  34. [2010]

    doi:10.4310/PAMQ.2010.v6.n2.a6

    2010 doi

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.