arxiv: 2605.09082 · v1 · submitted 2026-05-09 · 🧮 math.SG

Recognition: no theorem link

Generation of immersed Lagrangians by cocores

Wonbo Jeong , Dogancan Karabas , Sangjin Lee

Authors on Pith no claims yet

Pith reviewed 2026-05-12 01:49 UTC · model grok-4.3

classification 🧮 math.SG

keywords Lagrangian immersionsWeinstein manifoldsgeneration theoremsChekanov-Eliashberg algebrabounding cochainsLegendrian liftsexact Lagrangianssymplectic geometry

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The pith

Exact Lagrangian immersions in Weinstein manifolds are generated by their Lagrangian cocores when equipped with an augmentation of the Legendrian lift's Chekanov-Eliashberg algebra.

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

The paper extends an existing generation theorem from the embedded case to exact Lagrangian immersions. It establishes that any such immersion in a Weinstein manifold, when equipped with an augmentation of the Chekanov-Eliashberg algebra of its Legendrian lift or an equivalent bounding cochain, is generated by the Lagrangian cocores. A reader would care because this decomposition reduces the study of immersed objects to simpler core pieces and their algebraic data. The result supplies a concrete way to build and analyze these immersions using the given augmentation condition. It applies directly to objects that arise naturally in symplectic geometry.

Core claim

We extend the generation theorem of Chantraine--Dimitroglou Rizell--Ghiggini--Golovko to exact Lagrangian immersions in Weinstein manifolds. We prove that an exact Lagrangian immersion equipped with an augmentation of the Chekanov--Eliashberg algebra of its Legendrian lift, or equivalently, equipped with a corresponding bounding cochain, is generated by the Lagrangian cocores.

What carries the argument

Lagrangian cocores, which act as the generating set for the immersed Lagrangian once the augmentation or bounding cochain is fixed.

If this is right

The wrapped Floer homology of the immersion is determined by the homologies of its cocores.
Immersed Lagrangians satisfying the augmentation condition admit a structural decomposition into basic pieces.
Invariants of such immersions become computable from the data of the cocores alone.
The theorem supplies a criterion for when an immersion can be assembled from cocores.
Previous generation results for embedded Lagrangians now apply to this larger class of immersed objects.

Where Pith is reading between the lines

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

The algebraic condition may produce new relations among immersed objects in the Fukaya category.
Low-dimensional Weinstein manifolds provide concrete test cases where the generation can be checked by direct computation.
The result suggests that similar generation statements might hold after relaxing exactness under additional hypotheses.
It connects the study of immersed Lagrangians to existing work on bounding cochains without requiring new machinery.

Load-bearing premise

The immersion must be exact, sit inside a Weinstein manifold, and carry an augmentation of the Chekanov-Eliashberg algebra of its Legendrian lift or an equivalent bounding cochain.

What would settle it

An explicit exact Lagrangian immersion in a Weinstein manifold that admits an augmentation yet fails to be generated by its cocores would disprove the statement.

read the original abstract

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 extends the CDRG generation theorem to exact immersed Lagrangians in Weinstein manifolds under the augmentation or bounding cochain condition.

read the letter

The main takeaway is that this paper extends the Chantraine-Dimitroglou Rizell-Ghiggini-Golovko generation theorem to exact Lagrangian immersions in Weinstein manifolds. An immersion equipped with an augmentation of the Chekanov-Eliashberg algebra of its Legendrian lift, or equivalently a bounding cochain, is generated by the Lagrangian cocores. The result stays within the standard exact and Weinstein setting without adding new machinery. It builds directly on the cited prior theorem, which keeps the argument grounded and avoids circularity. The abstract states the claim cleanly, and the conditions line up with existing work on wrapped Fukaya categories and Legendrian contact homology. That consistency is a strength. The soft spots are limited. The abstract gives no proof details, so the adaptation steps for the immersed case are not visible here, though the stress-test found no internal inconsistencies. The augmentation requirement is standard but narrows the result to those immersions that satisfy it, which is fine as long as it is stated explicitly. No free parameters or invented entities show up. This paper is for symplectic geometers already working on generation results and Legendrian invariants. A reader familiar with the original CDRG theorem will see it as a natural incremental step. It deserves a serious referee to verify the full proof. I would send it to peer review.

Referee Report

0 major / 1 minor

Summary. The paper extends the generation theorem of Chantraine--Dimitroglou Rizell--Ghiggini--Golovko to exact Lagrangian immersions in Weinstein manifolds. It proves that an exact Lagrangian immersion equipped with an augmentation of the Chekanov--Eliashberg algebra of its Legendrian lift, or equivalently equipped with a corresponding bounding cochain, is generated by the Lagrangian cocores.

Significance. If the result holds, the generalization is significant for symplectic geometry. It broadens the scope of generation results from embedded exact Lagrangians to immersed ones in Weinstein manifolds under the standard augmentation or bounding-cochain hypothesis. This strengthens connections between the wrapped Fukaya category and Legendrian contact homology, and could enable new computations for immersed objects. The paper builds directly on prior work without introducing free parameters or ad-hoc axioms.

minor comments (1)

The abstract states the main theorem clearly but does not indicate the dimension of the Weinstein manifold or whether the immersion is required to be in a specific codimension; adding this would improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the accurate summary of our main result, and the recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; extends external prior theorem

full rationale

The derivation extends the generation theorem of Chantraine--Dimitroglou Rizell--Ghiggini--Golovko to exact immersed Lagrangians in Weinstein manifolds under the augmentation/bounding cochain hypothesis. The abstract and stated conditions rely on established results from wrapped Fukaya categories and Legendrian contact homology without any self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the central claim to its own inputs by construction. The proof chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard domain assumptions in symplectic geometry plus the existence of the augmentation condition; no free parameters or new entities are introduced.

axioms (2)

domain assumption The ambient space is a Weinstein manifold
Invoked to set the context for the generation theorem.
domain assumption The immersion admits an augmentation of the Chekanov-Eliashberg algebra or equivalent bounding cochain
This is the key hypothesis under which generation holds.

pith-pipeline@v0.9.0 · 5351 in / 1035 out tokens · 48673 ms · 2026-05-12T01:49:32.063647+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

158 extracted references · 158 canonical work pages

[1]

Generation of immersed Lagrangians by cocores , author=

work page
[2]

2024 , eprint=

Mirror Construction for Nakajima Quiver Varieties , author=. 2024 , eprint=

work page 2024
[3]

and Weyman, J

Derksen, Harm and Weyman, Jerzy and Zelevinsky, Andrei , TITLE =. Selecta Math. (N.S.) , FJOURNAL =. 2008 , NUMBER =. doi:10.1007/s00029-008-0057-9 , URL =

work page doi:10.1007/s00029-008-0057-9 2008
[4]

Hermes, Stephen , TITLE =. J. Algebra , FJOURNAL =. 2016 , PAGES =. doi:10.1016/j.jalgebra.2016.03.041 , URL =

work page doi:10.1016/j.jalgebra.2016.03.041 2016
[5]

Nadler, David and Zaslow, Eric , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2009 , NUMBER =. doi:10.1090/S0894-0347-08-00612-7 , URL =

work page doi:10.1090/s0894-0347-08-00612-7 2009
[6]

Keating, Ailsa , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s00039-015-0353-4 , URL =

work page doi:10.1007/s00039-015-0353-4 2015
[7]

arXiv preprint arXiv:2212.12564 , year=

T-structures on dg-categories and derived deformations , author=. arXiv preprint arXiv:2212.12564 , year=

work page arXiv
[8]

Be ilinson, A. A. and Bernstein, J. and Deligne, P. , TITLE =. Analysis and topology on singular spaces,. 1982 , MRCLASS =

work page 1982
[9]

2022 , eprint=

Rabinowitz Fukaya categories and the categorical formal punctured neighborhood of infinity , author=. 2022 , eprint=

work page 2022
[10]

King, A. D. , TITLE =. Quart. J. Math. Oxford Ser. (2) , FJOURNAL =. 1994 , NUMBER =. doi:10.1093/qmath/45.4.515 , URL =

work page doi:10.1093/qmath/45.4.515 1994
[11]

and Soibelman, Y

Kontsevich, Maxim and Soibelman, Yan , TITLE =. Mirror symmetry and tropical geometry , SERIES =. 2010 , ISBN =. doi:10.1090/conm/527/10400 , URL =

work page doi:10.1090/conm/527/10400 2010
[12]

2008 , eprint=

Stability structures, motivic Donaldson-Thomas invariants and cluster transformations , author=. 2008 , eprint=

work page 2008
[13]

Bridgeland, Tom and Smith, Ivan , TITLE =. Publ. Math. Inst. Hautes \'Etudes Sci. , FJOURNAL =. 2015 , PAGES =. doi:10.1007/s10240-014-0066-5 , URL =

work page doi:10.1007/s10240-014-0066-5 2015
[14]

, TITLE =

Keller, Bernhard , TITLE =. Representations of algebras and related topics , SERIES =. 2011 , ISBN =. doi:10.4171/101-1/3 , URL =

work page doi:10.4171/101-1/3 2011
[15]

Amiot, Claire , TITLE =. Ann. Inst. Fourier (Grenoble) , FJOURNAL =. 2009 , NUMBER =. doi:10.5802/aif.2499 , URL =

work page doi:10.5802/aif.2499 2009
[16]

Akaho, Manabu and Joyce, Dominic , TITLE =. J. Differential Geom. , FJOURNAL =. 2010 , NUMBER =

work page 2010
[17]

, TITLE =

Nadler, David , TITLE =. SIGMA Symmetry Integrability Geom. Methods Appl. , FJOURNAL =. 2014 , PAGES =. doi:10.3842/SIGMA.2014.018 , URL =

work page doi:10.3842/sigma.2014.018 2014
[18]

Functors of wrapped

Gao, Yuan , journal=. Functors of wrapped

work page
[19]

Exact, graded, immersed

Alston, Garrett and Bao, Erkao , journal=. Exact, graded, immersed. 2018 , publisher=

work page 2018
[20]

2025 , eprint=

Reflexive dg categories in algebra and topology , author=. 2025 , eprint=

work page 2025
[21]

2024 , eprint=

Finite dimensional 2-cyclic Jacobian algebras , author=. 2024 , eprint=

work page 2024
[22]

To\"en, Bertrand and Vaqui\'e, Michel , TITLE =. Ann. Sci. \'Ecole Norm. Sup. (4) , FJOURNAL =. 2007 , NUMBER =. doi:10.1016/j.ansens.2007.05.001 , URL =

work page doi:10.1016/j.ansens.2007.05.001 2007
[23]

International

Keller, Bernhard , TITLE =. International. 2006 , ISBN =

work page 2006
[24]

Keller, Bernhard , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 1998 , NUMBER =. doi:10.1016/S0022-4049(96)00085-0 , URL =

work page doi:10.1016/s0022-4049(96)00085-0 1998
[25]

Brav, Christopher and Dyckerhoff, Tobias , TITLE =. Compos. Math. , FJOURNAL =. 2019 , NUMBER =. doi:10.1112/s0010437x19007024 , URL =

work page doi:10.1112/s0010437x19007024 2019
[26]

2016 , eprint=

Relative Calabi-Yau completions , author=. 2016 , eprint=

work page 2016
[27]

Representations of algebras and related structures , SERIES =

Keller, Bernhard and Wang, Yu , TITLE =. Representations of algebras and related structures , SERIES =. [2023] 2023 , ISBN =

work page 2023
[28]

and Jeong, W

Bae, Hanwool and Jeong, Wonbo and Kim, Jongmyeong , TITLE =. Internat. J. Math. , FJOURNAL =. 2025 , NUMBER =. doi:10.1142/S0129167X25500193 , URL =

work page doi:10.1142/s0129167x25500193 2025
[29]

, TITLE =

Christ, Merlin , TITLE =. Forum Math. Sigma , FJOURNAL =. 2022 , PAGES =. doi:10.1017/fms.2022.1 , URL =

work page doi:10.1017/fms.2022.1 2022
[30]

arXiv preprint arXiv:1604.00114 , year=

Wrapped microlocal sheaves on pairs of pants , author=. arXiv preprint arXiv:1604.00114 , year=

work page arXiv
[31]

On partially wrapped

Sylvan, Zachary , journal=. On partially wrapped. 2019 , publisher=

work page 2019
[32]

Canonaco, Alberto and Ornaghi, Mattia and Stellari, Paolo , TITLE =. Doc. Math. , FJOURNAL =. 2019 , PAGES =

work page 2019
[33]

and Schapira, P

Kashiwara, M. and Schapira, P. , isbn=. Sheaves on Manifolds: With a Short History ``. 2002 , publisher=

work page 2002
[34]

Floer homology of cotangent bundles and the loop product , author=. Geom. Topol. , volume=

work page
[35]

, journal=

Abouzaid, M. , journal=. A geometric criterion for generating the

work page
[36]

, journal=

Abouzaid, M. , journal=. A cotangent fibre generates the

work page
[37]

, journal=

Abouzaid, M. , journal=. On the wrapped

work page
[38]

and Seidel, P

Abouzaid, M. and Seidel, P. , journal=. An open string analogue of

work page
[39]

and Smith, I

Abouzaid, M. and Smith, I. , journal=. Exact

work page
[40]

Discreteness of silting objects and t-structures in triangulated categories , author=. Proc. Lond. Math. Soc. , volume=

work page
[41]

Cluster categories for algebras of global dimension 2 and quivers with potential , author=. Ann. Inst. Fourier , volume=

work page
[42]

Orthogonally spherical objects and spherical fibrations , author=. Adv. Math. , volume=

work page
[43]

and Logvinenko, T

Anno, R. and Logvinenko, T. , journal=. Spherical

work page
[44]

Asplund, Johan , TITLE =. J. Topol. , FJOURNAL =. 2023 , NUMBER =. doi:10.1112/topo.12289 , URL =

work page doi:10.1112/topo.12289 2023
[45]

and Kwon, M

Bae, H. and Kwon, M. , journal=. A computation of the ring structure in wrapped

work page
[46]

arXiv:2006.06016 , year=

On the composition of two spherical twists , author=. arXiv:2006.06016 , year=

work page arXiv 2006
[47]

arXiv:2107.06079 , year=

Entropy of the composition of two spherical twists , author=. arXiv:2107.06079 , year=

work page arXiv
[48]

and Kim, J

Barbacovi, F. and Kim, J. , journal=. On

work page
[49]

Stability conditions in families , author=. Publ. Math. Inst. Hautes

work page
[50]

Faisceaux pervers , author=. Ast

work page
[51]

I , author=

Lagrangian cobordism. I , author=. J. Amer. Math. Soc. , volume=

work page
[52]

and Cornea, O

Biran, P. and Cornea, O. , journal=. Lagrangian cobordism and

work page
[53]

and Cornea, O

Biran, P. and Cornea, O. , journal=. Cone-decompositions of

work page
[54]

Lagrangian shadows and triangulated categories , author=. Ast

work page
[55]

and Cornea, O

Biran, P. and Cornea, O. and Zhang, J. , journal=. Triangulation and persistence:

work page
[56]

and Kapranov, M

Bondal, A. and Kapranov, M. , journal=. Representable functors,

work page
[57]

Compositio Math

Reconstruction of a variety from the derived category and groups of autoequivalences , author=. Compositio Math. , volume=

work page
[58]

t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general) , author=

Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general) , author=. J. K-theory , volume=

work page
[59]

Stability conditions on triangulated categories , author=. Ann. Math. , volume=

work page
[60]

Algebraic geometry---Seattle

Spaces of stability conditions , author=. Algebraic geometry---Seattle. 2005 Part 1 , series=

work page 2005
[61]

Tilting theory and cluster combinatorics , author=. Adv. Math. , volume=

work page
[62]

and Paris-Romaskevich, O

Cantat, S. and Paris-Romaskevich, O. , journal=. Automorphisms of compact

work page
[63]

and Darp

Chan, A. and Darp. Periodic trivial extension algebras and fractionally. arXiv:2012.11927 , year=

work page arXiv 2012
[64]

Chantraine, Baptiste and Dimitroglou Rizell, Georgios and Ghiggini, Paolo and Golovko, Roman , TITLE =. Ann. Sci. \'Ec. Norm. Sup\'er. (4) , FJOURNAL =. 2024 , NUMBER =. doi:10.24033/asens.2570 , URL =

work page doi:10.24033/asens.2570 2024
[65]

and Frauenfelder, U

Cieliebak, K. and Frauenfelder, U. and Oancea, A. , journal=. Rabinowitz

work page
[66]

and Oancea, A

Cieliebak, K. and Oancea, A. , journal=. Symplectic homology and the

work page
[67]

arXiv:2008.13165 , year=

Multiplicative structures on cones and duality , author=. arXiv:2008.13165 , year=

work page arXiv 2008
[68]

Dynamics of bimeromorphic maps of surfaces , author=. Amer. J. Math. , volume=

work page
[69]

The influence of Solomon Lefschetz in geometry and topology , series=

Dynamical systems and categories , author=. The influence of Solomon Lefschetz in geometry and topology , series=

work page
[70]

, journal=

Drinfeld, V. , journal=. D

work page
[71]

and Lekili, Y

Ekholm, T. and Lekili, Y. , journal=. Duality between

work page
[72]

, journal=

Elagin, A. , journal=. Calculating dimension of triangulated categories:

work page
[73]

Three notions of dimension for triangulated categories , author=. J. Algebra , volume=

work page
[74]

Koszul duality patterns in

Etg. Koszul duality patterns in. Geom. Topol. , volume=

work page
[75]

, journal=

Fan, Y.-W. , journal=. Entropy of an autoequivalence on

work page
[76]

, journal=

Fan, Y.-W. , journal=. On entropy of

work page
[77]

, journal=

Fan, Y.-W. , journal=. Systolic inequalities for

work page
[78]

arXiv:2008.06159 , year=

Asymptotic shifting numbers in triangulated categories , author=. arXiv:2008.06159 , year=

work page arXiv 2008
[79]

and Filip, S

Fan, Y.-W. and Filip, S. and Haiden, F. and Katzarkov, L. and Liu, Y. , journal=. On pseudo-

work page
[80]

Categorical polynomial entropy , author=. Adv. Math. , volume=

work page

Showing first 80 references.