The equivalence of Ekeland-Hofer and equivariant symplectic homology capacities

Jean Gutt; Vinicius G. B. Ramos

REVIEW 4 major objections 3 minor 2 cited by

The equivalence of Ekeland-Hofer and equivariant symplectic homology capacities

T0 review · 4 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper proves that on every star-shaped domain in $\mathbb{R}^{2n}$, the Ekeland–Hofer capacities coincide with the Gutt–Hutchings equivariant symplectic homology capacities at every level $k$.

desk verdict The main theorem is important and the architecture is coherent, but the written proof of the load-bearing chain isomorphism has a real gap: the gluing half of the moduli-space boundary argument is missing. read the letter →

arxiv 2412.09555 v1 pith:KTO7P2ZV submitted 2024-12-12 math.SG

Jean Gutt , Vinicius G. B. Ramos This is my paper · ORCID

classification math.SG MSC 53D4053D35

keywords Ekeland–HofercapacitiesequivariantsymplectichomologyGutt–Hutchingsstar-shapeddomainsHamiltonianactionfunctionalFadell–RabinowitzindexhybridcurvesU-map

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 two seemingly unrelated sequences of symplectic capacities — the classical Ekeland–Hofer capacities, built by minimax arguments over loops in $\mathbb{R}^{2n}$, and the Gutt–Hutchings capacities, built from positive $S^1$-equivariant symplectic homology — are actually the same invariant. The main theorem states that for every star-shaped domain $X \subset \mathbb{R}^{2n}$ and every $k \in \mathbb{N}_0$, $c^{\mathrm{EH}}_k(X) = c^{\mathrm{GH}}_k(X)$. If true, this answers a question posed in 1989 as well as Floer's suggestion from 1988 that Ekeland–Hofer capacities should be understood through $S^1$-equivariant homology. A sympathetic reader would care because it unifies variational and Floer-theoretic approaches to symplectic rigidity and extends the reach of the classical capacities.

What carries the argument

The carrier of the proof is a count of parametrized hybrid curves, denoted $\Phi$. A hybrid curve consists of a negative gradient trajectory of the Hamiltonian action functional in $E = H^{1/2}(S^1, \mathbb{R}^{2n})$ followed by a Floer trajectory, with the base point moving along a gradient flow of the Morse function $\tilde f_N$ on $S^{2N+1}$; the map $\Phi$ counts zero-dimensional such moduli spaces and is shown to be a chain isomorphism between the equivariant Morse and Floer complexes. It must additionally commute with the $U$-map, which is the algebraic shadow of the $S^1$ action. The Fadell–Rabinowitz index is reformulated as the maximal power of $U$ acting nontrivially, and this is what converts the chain isomorphism into an equality of capacities.

What would settle it

A concrete check: compute the filtered equivariant Morse and Floer complexes for a simple admissible Hamiltonian with known Reeb orbits, such as an irrational ellipsoid, and compare the rank of the image of each $U^j$ map; any discrepancy would falsify Theorem 1.8 and hence Theorem 1.2. Alternatively, exhibit a nonconstant hybrid curve from a critical point to itself, which would violate the Crossing Energy Theorem used in Section 5.3.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is a chain-level structural identity that forces the capacity identity. For every admissible Hamiltonian $H$ and every action level $c>0$, there is an isomorphism of chain complexes $CM^{S^1,c}(H) \cong CF^{S^1,c}(H)$ between the equivariant Morse homology of the Hamiltonian action functional and the $S^1$-equivariant Floer homology of $H$, and this isomorphism commutes with the $U$-map. From this, the paper derives that the Fadell–Rabinowitz index of the sublevel sets $\{A_H \le c\}$ equals the algebraic quantity $\kappa_c(H)$ extracted from filtered Floer homology, which in turn is exactly the comparison needed to equate the two capacity sequences on every star-shaped domain.

Load-bearing premise

The proof's load-bearing premise is the compactness and gluing behavior of the hybrid moduli spaces stated in Theorem 5.5: the closure of an index-1 family must be a compact one-manifold whose boundary is exactly the broken trajectories, and the $U$-map must commute with the count; in the paper the gluing direction and the $U$-map commutation are deferred rather than proved.

Editorial extensions

If this is right

The two capacity sequences are interchangeable on star-shaped domains: any lower bound, computation, or existence result proved for one transfers to the other.
Ekeland–Hofer capacities, originally restricted to $\mathbb{R}^{2n}$, inherit the Liouville-domain framework of Gutt–Hutchings capacities, so they can be extended beyond Euclidean domains.
The chain-level isomorphism means that higher structure beyond the capacities — the action filtration and the $U$-module structure — matches between the Morse and Floer sides.
The equality validates Floer's 1988 remark that Ekeland–Hofer capacities must come from $S^1$-equivariant homology, giving that program a precise theorem.

Reading between the lines

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

One implication the paper leaves implicit is that the isomorphism should identify not just the capacities but the filtered morphism groups themselves, so one could compute the full $S^1$-equivariant symplectic homology of a star-shaped domain from Morse data alone.
A testable extension would be to check whether the same equality holds for domains that are not star-shaped but are still Liouville, or for the higher capacities defined by iterating the $U$-map more than once.
The reliance on a Crossing Energy Theorem suggests that a clean proof of that statement would put the isomorphism on firmer ground; constructing the gluing direction in the parametrized setting is the natural next target.

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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

Mike,

Quick read of Gutt–Ramos. The headline is real: they claim c^EH_k = c^GH_k for all star-shaped domains, answering Ekeland–Hofer's 1989 extension question and Floer's S1-equivariant intuition. That's a significant result if it holds. The paper's architecture is clean: reformulate EH capacities in Fadell–Rabinowitz index terms and GH capacities via the U-map, then compare them through a chain isomorphism between equivariant Morse and Floer complexes of the action functional. The structural lemmas (1.6, 1.7, 1.9–1.11) are checkable, and I found no circularity; the two sides are defined independently. Credit where due: the paper is honest about what it needs, and the analytic sections show real work on compactness and transversality.

The soft spot is exactly the one the stress-test flags. Theorem 1.8/5.1 is the load-bearing bridge, and the chain map Φ is defined by counting hybrid curves. For ∂Φ = Φ∂, you need the index-1 moduli space to be a compact 1-manifold whose boundary is precisely the broken trajectories. Theorem 5.5 proves the compactness half—every sequence subconverges to some broken object—but it does not prove gluing: that every broken index-1 object appears as the limit of a one-parameter family, with the right collar structure. Without that, the boundary count is undefined and the claim that Φ is a chain map is asserted, not proved. The U-map compatibility is also deferred mutatis mutandis to GH18, and the 'Crossing Energy Theorem' used for the diagonal is uncited—though that last one is probably standard action-energy. These are genuine gaps, not nits, but they look fillable with standard techniques if the authors supply them.

My bottom line: this deserves a serious referee. The result is important, the framework is coherent, and the gaps are specific. But I wouldn't take the theorem as established from this version. Send it to review; the referee report should ask for the gluing argument and the U-map details. If they come through, this is a strong paper.

Referee Report

4 major / 3 minor

Summary. The paper proves that the Ekeland–Hofer capacities of a star-shaped domain in R^{2n} agree with the Gutt–Hutchings capacities defined from positive S^1-equivariant symplectic homology, for every level k. The proof reduces this to Theorem 1.8, which asserts a chain-complex isomorphism between the S^1-equivariant Morse homology of the Hamiltonian action functional and the S^1-equivariant Floer homology of the same Hamiltonian, compatible with the U-map. The isomorphism is constructed in Section 5 by counting hybrid curves, where the Floer part is coupled to a gradient flow on the auxiliary S^{2N+1} factor.

Significance. If the main theorem is correct, it answers a question raised by Floer in 1988 and gives a rigorous bridge between variational capacities and Floer-theoretic capacities. The paper has real strengths: the reduction in Lemmas 1.9–1.11 is clean; the reformulations in Propositions 1.6 and 1.7 are explicit; the compactness and transversality sections contain substantial and credible arguments, including the estimates (5.2) and (5.3) and the use of Lemma 5.6; and the two sides of the desired equality are defined from independent inputs, so there is no circularity. However, the proof of the load-bearing isomorphism Theorem 1.8 is incomplete as written: the chain map property of the hybrid-curve count is asserted rather than proved, and the U-map compatibility is deferred. These gaps are local in the sense that they concern the proof of Theorem 1.8, but they are central to the paper's main claim.

major comments (4)

[§5.3, Theorem 5.5] The proof that Φ is a chain map is not complete. The text asserts that ∂Φ = Φ∂ follows from the fact that the closure of the index-1 hybrid moduli space is a 1-manifold whose boundary consists of broken trajectories, and it cites Theorem 5.5 for this. Theorem 5.5 proves H^1_loc-precompactness and subconvergence of every sequence to some broken object; it does not prove a gluing theorem showing that every index-1 broken object is the limit of a one-parameter family of smooth hybrid curves, nor that these limits form collar neighborhoods. Without gluing, the algebraic boundary count that defines the chain map equation is not shown to be well-defined. Since Theorem 1.8 is the bridge between the Morse-theoretic Ekeland–Hofer index and the Floer-theoretic Gutt–Hutchings capacities, this gap is load-bearing for Theorems 1.5 and 1.2.
[§5.3, Crossing Energy Theorem] The proof that Φ is an isomorphism uses the assertion that M^hyb((x,p),(x,p)) consists only of the constant solution, attributed to an unstated 'Crossing Energy Theorem'. This theorem is not cited or proved, and the constant-solution claim is what makes the diagonal entries of the matrix for Φ equal to 1. If the assertion fails, Φ need not be invertible, so this is another load-bearing point in the proof of Theorem 1.8.
[§5.3, U-map compatibility] The compatibility of Φ with the U-map, required by Theorem 1.8 and used in Lemma 1.9, is deferred with the phrase 'mutatis mutandis from [GH18]' at the end of §5.3. In the parametrized S^1-equivariant setting, the U-map is not a formal consequence of the non-equivariant statement in [GH18]; the hybrid curve count must be shown to interact correctly with the additional S^1-family or CP^N structure. A proof or a precise citation of a statement covering the present chain model is needed.
[§5.1, Theorem 5.3] The proof of transversality for M^hyb is only sketched. The linearized operator DΘ is declared Fredholm with the stated index by a 'slight modification' of [Hec13, Theorem 4.4], and the implicit function theorem is invoked without proving surjectivity after the generic perturbation X. Since the moduli space has a non-compact domain and an infinite-dimensional unstable-manifold boundary condition, the surjectivity and regularity of the evaluation at zero are not automatic. This affects the well-definedness of the count defining Φ and needs a complete proof.

minor comments (3)

[Theorem 1.2 and throughout] The notation c^CH_k appears in Theorem 1.2 and several later statements; this should be c^GH_k, matching the Gutt–Hutchings capacities and the title.
[Abstract and Section 1] The abstract says 'on all star-shaped domain in R^{2n}'; this should be 'on all star-shaped domains'. There is also a typo 'Inpired' in the first paragraph of Section 1.
[Section 4.2.1] The decomposition H = H+ ⊕ H− is written as a product with S^{2N+1}, but S^{2N+1} is not a vector space. This is a minor abuse of notation and should be clarified, for instance by working with the tangent space at a point or with local coordinates on S^{2N+1}.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the two capacity sequences are defined from independent inputs, and the isomorphism in Theorem 1.8 is constructed rather than assumed.

full rationale

No significant circularity is present. The Ekeland–Hofer capacities are defined in Section 1.2 from the Fadell–Rabinowitz index of sublevel sets of the Hamiltonian action functional, while the Gutt–Hutchings capacities are defined in Section 1.3 from the delta and U maps on positive S1-equivariant symplectic homology; these are independent inputs. The derivation chain is: Proposition 1.6 is a direct reformulation of the Ekeland–Hofer definition, Proposition 1.7 is a direct reformulation of the Gutt–Hutchings definition using the action filtration and U-map axioms, Theorem 1.8 provides the bridge via an explicitly constructed chain map Phi counting hybrid curves, and Lemmas 1.9–1.11 compare the resulting indices. No parameter is fitted and the desired equality is nowhere assumed as an input to the construction of Phi. The only self-citation is the statement in Section 5.3 that 'Phi commutes with the U-map follows mutatis mutandis from [GH18]'; this defers a technical compatibility proof to prior published work by the first author that does not assume the Ekeland–Hofer/Gutt–Hutchings equivalence, so it is independent support rather than circularity. Concerns that Theorem 5.5 proves compactness but not gluing, and that the boundary identification in Section 5.3 is asserted rather than proved, are proof-completeness or correctness risks, not circular reductions: they do not make any displayed equation equal to its own input by construction.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claim rests on standard symplectic-topology machinery (Fadell-Rabinowitz index theory [EH89, EH90], Abbondandolo-Majer Morse theory [AM01, AM05], Bourgeois-Oancea family Floer homology [BO16], and the first author's capacity framework [GH18]) plus technical genericity and compactness assumptions for the hybrid moduli spaces adapted from [AK22] and [Hec13]. No free parameters are fitted to data; the slope L and the small constants ε and δ are construction data. No new entities are postulated.

assumptions (8)

domain assumption Properties of positive S1-equivariant symplectic homology from [GH18]: action filtration, U map, Reeb orbits property, δ map (Proposition 1.3).
c^GH is defined from these structures (Definition 1.4), and Proposition 1.7 reformulates it via κ_c; the present paper does not reprove them.
standard math Fadell-Rabinowitz index theory: monotonicity, subadditivity, and [EH90, Proposition 1], ind_EH(E- ⊕ E0 ⊕ V) = (1/2) dim V for S1-invariant V ⊂ E+.
Used in Lemma 1.10 (subadditivity) and in Lemma 1.11 Step 1 (ellipsoid sublevel computation) to compute the variational side of the equality.
standard math Abbondandolo-Majer Morse homology for strongly indefinite functionals on Hilbert spaces: conditions (M.1) to (M.5) and Theorem 4.7.
The framework of Section 4.1; the action functional A_H on H^{1/2} × S^{2N+1} must satisfy these conditions for the Morse complex CM^{S1}(H) to be defined.
domain assumption Genericity: residual sets H_reg of Hamiltonians making A_H Morse (Proposition 4.15, [AM01]), and residual compact perturbations X making the vector field Morse-Smale and the hybrid moduli spaces transverse (Theorems 4.17 and 5.3).
Transversality for the Morse complex and for M_hyb is established by Baire-category arguments asserted with references; needed for well-defined chain complexes and moduli spaces.
ad hoc to paper Compactness of hybrid moduli spaces: H^1_loc-precompactness from estimates (5.2) and (5.3), Lemma 5.6 (Abouzaid, as in [Rit13]), and the asserted gluing of broken trajectories in Section 5.3.
The proof of Theorem 5.5 and of the chain-map property of Φ; the gluing half is asserted, not demonstrated, in the text.
ad hoc to paper The 'Crossing Energy Theorem', invoked without citation or proof in Section 5.3.
Used to show M_hyb((x,p),(x,p)) is the constant solution, which enters the bijectivity argument for Φ; no reference or statement is given.
domain assumption Known values of the Gutt-Hutchings capacities of ellipsoids: c_κ(B) = a for B = E(a_1, ..., a_n), from [GH18].
Lemma 1.11 Step 2 uses equation (2.1), a = c_κ(B) ≤ c_κ(X) < L, to find the subspace E_κ ⊂ {A_{H_B} ≤ a}; this relies on the independent ellipsoid computation.
domain assumption An arbitrary splitting E0 = E0+ ⊕ E0- identifying the relative Morse index with the parametrized Conley-Zehnder index (Remark 4.9, [Abb01]).
The grading of the Morse complex is matched to the Floer grading via this splitting and the cited identification; the result is asserted to be independent of the choice.

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Pith. "Pith review of The equivalence of Ekeland-Hofer and equivariant symplectic homology capacities." pith.science (2026). https://pith.science/paper/KTO7P2ZV

@misc{pith2026241209555,
  author       = {Pith},
  title        = {Pith review of: The equivalence of Ekeland-Hofer and equivariant symplectic homology capacities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KTO7P2ZV}},
  note         = {Machine review of arXiv:2412.09555}
}

abstract

In this paper, we prove that the Ekeland-Hofer capacities coincide on all star-shaped domain in $\mathbb{R}^{2n}$ with the equivariant symplectic homology capacities defined by the first author and Hutchings, answering a 35 years old question. Along the way, we prove that given a Hamiltonian $H$, the (equivariant) Floer homology of $H$ is chain-complex isomorphic to the Morse homology of the Hamiltonian action functional.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

26 extracted references · 23 canonical work pages · cited by 2 Pith papers

[1]

Morse theory for strongly indefinite functionals and Hamiltonian systems

Alberto Abbondandolo. Morse theory for strongly indefinite functionals and Hamiltonian systems . PhD thesis, Scuola Normale Superiore, 1999

work page 1999
[2]

Morse theory for H amiltonian systems , volume 425 of Chapman & Hall/CRC Research Notes in Mathematics

Alberto Abbondandolo. Morse theory for H amiltonian systems , volume 425 of Chapman & Hall/CRC Research Notes in Mathematics . Chapman & Hall/CRC, Boca Raton, FL, 2001

work page 2001
[3]

Symplectic homology of convex domains and C larke's duality

Alberto Abbondandolo and Jungsoo Kang. Symplectic homology of convex domains and C larke's duality. Duke Math. J. , 171(3):739--830, 2022

work page 2022
[4]

Morse homology on hilbert spaces

Alberto Abbondandolo and Pietro Majer. Morse homology on hilbert spaces. Comm. Pure Appl. Math. , 54:689--760, 2001

work page 2001
[5]

Ordinary differential operators in hilbert spaces and fredholm pairs

Alberto Abbondandolo and Pietro Majer. Ordinary differential operators in hilbert spaces and fredholm pairs. Mathematische Zeitschrift , 243(3):525--562, Mar 2003

work page 2003
[6]

When the morse index is infinite

Alberto Abbondandolo and Pietro Majer. When the morse index is infinite. Internat. Math. Res. Notices , 71:3839--3854, 2004

work page 2004
[7]

A morse complex for infinite dimensional manifolds--part i

Alberto Abbondandolo and Pietro Majer. A morse complex for infinite dimensional manifolds--part i. Advances in Mathematics , 197(2):321--410, 2005

work page 2005
[8]

S^1 -equivariant symplectic homology and linearized contact homology

Fr \'e d \'e ric Bourgeois and Alexandru Oancea. S^1 -equivariant symplectic homology and linearized contact homology. International Mathematics Research Notices , pages 1--89, 2016

work page 2016

Show all 26 references

[9]

Cieliebak, A

K. Cieliebak, A. Floer, and H. Hofer. Symplectic homology. II . A general construction. Math. Z. , 218(1):103--122, 1995

1995
[10]

Symplectic topology and H amiltonian dynamics

Ivar Ekeland and Helmut Hofer. Symplectic topology and H amiltonian dynamics. Math. Z. , 200(3):355--378, 1989

1989
[11]

Symplectic topology and H amiltonian dynamics

Ivar Ekeland and Helmut Hofer. Symplectic topology and H amiltonian dynamics. II . Math. Z. , 203(4):553--567, 1990

1990
[12]

Floer and H

A. Floer and H. Hofer. Coherent orientations for periodic orbit problems in symplectic geometry. Math. Z. , 212(1):13--38, 1993

1993
[13]

Floer and H

A. Floer and H. Hofer. Symplectic homology. I . O pen sets in C ^n . Math. Z. , 215(1):37--88, 1994

1994
[14]

Symplectic capacities from positive S 1--equivariant symplectic homology

Jean Gutt and Michael Hutchings. Symplectic capacities from positive S 1--equivariant symplectic homology. Algebr. Geom. Topol. , 18(6):3537--3600, 2018

2018
[15]

Pseudoholomorphic curves in symplectic manifolds

Mikhail Gromov. Pseudoholomorphic curves in symplectic manifolds. Inventiones Mathematicae , 82:307--347, 1985

1985
[16]

Floer theory and reduced cohomology on open manifolds

Yoel Groman. Floer theory and reduced cohomology on open manifolds. Geom. Topol. , 27(4):1273--1390, 2023

2023
[17]

The positive equivariant symplectic homology as an invariant for some contact manifolds

Jean Gutt. The positive equivariant symplectic homology as an invariant for some contact manifolds. Journal of Symplectic Geometry , 15(4):1019--1069, 2017

2017
[18]

Isomorphic chain complexes of H amiltonian dynamics on tori

Michael Hecht. Isomorphic chain complexes of H amiltonian dynamics on tori. Journal of Fixed Point Theory and Applications , 14(1):165--221, 2013

2013
[19]

Symplectic Invariants and Hamiltonian Dynamics

Helmut Hofer and Eduard Zehnder. Symplectic Invariants and Hamiltonian Dynamics . Springer Basel, 2011

2011
[20]

Positive ( S^1 -equivariant) symplectic homology of convex domains, higher capacities, and C larke's duality

Stefan Matijevic. Positive ( S^1 -equivariant) symplectic homology of convex domains, higher capacities, and C larke's duality. Preprint arXiv:2410.13673 , 2024

2024 arXiv
[21]

Fibered symplectic cohomology and the L eray- S erre spectral sequence

Alexandru Oancea. Fibered symplectic cohomology and the L eray- S erre spectral sequence. J. Symplectic Geom. , 6(3):267--351, 2008

2008
[22]

Alexander F. Ritter. Topological quantum field theory structure on symplectic cohomology. J. Topol. , 6(2):391--489, 2013

2013
[23]

M. Schwarz. Cohomology Operations from S^1 -Cobordisms in Floer Homology . PhD thesis, Swiss Federal Institute of Technology Zurich, 1995

1995
[24]

A biased view of symplectic cohomology

Paul Seidel. A biased view of symplectic cohomology. In Current developments in mathematics, 2006 , pages 211--253. Int. Press, Somerville, MA, 2008

2006
[25]

Functors and computations in F loer homology with applications

Claude Viterbo. Functors and computations in F loer homology with applications. I . Geom. Funct. Anal. , 9(5):985--1033, 1999

1999
[26]

Notes taken during F loer's talk at the MSRI in 1988

Alan Weinstein. Notes taken during F loer's talk at the MSRI in 1988

1988

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Morse theory for strongly indefinite functionals and Hamiltonian systems

Alberto Abbondandolo. Morse theory for strongly indefinite functionals and Hamiltonian systems . PhD thesis, Scuola Normale Superiore, 1999

work page 1999

[2] [2]

Morse theory for H amiltonian systems , volume 425 of Chapman & Hall/CRC Research Notes in Mathematics

Alberto Abbondandolo. Morse theory for H amiltonian systems , volume 425 of Chapman & Hall/CRC Research Notes in Mathematics . Chapman & Hall/CRC, Boca Raton, FL, 2001

work page 2001

[3] [3]

Symplectic homology of convex domains and C larke's duality

Alberto Abbondandolo and Jungsoo Kang. Symplectic homology of convex domains and C larke's duality. Duke Math. J. , 171(3):739--830, 2022

work page 2022

[4] [4]

Morse homology on hilbert spaces

Alberto Abbondandolo and Pietro Majer. Morse homology on hilbert spaces. Comm. Pure Appl. Math. , 54:689--760, 2001

work page 2001

[5] [5]

Ordinary differential operators in hilbert spaces and fredholm pairs

Alberto Abbondandolo and Pietro Majer. Ordinary differential operators in hilbert spaces and fredholm pairs. Mathematische Zeitschrift , 243(3):525--562, Mar 2003

work page 2003

[6] [6]

When the morse index is infinite

Alberto Abbondandolo and Pietro Majer. When the morse index is infinite. Internat. Math. Res. Notices , 71:3839--3854, 2004

work page 2004

[7] [7]

A morse complex for infinite dimensional manifolds--part i

Alberto Abbondandolo and Pietro Majer. A morse complex for infinite dimensional manifolds--part i. Advances in Mathematics , 197(2):321--410, 2005

work page 2005

[8] [8]

S^1 -equivariant symplectic homology and linearized contact homology

Fr \'e d \'e ric Bourgeois and Alexandru Oancea. S^1 -equivariant symplectic homology and linearized contact homology. International Mathematics Research Notices , pages 1--89, 2016

work page 2016