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arxiv: 2602.21152 · v2 · pith:K7Y6UTBVnew · submitted 2026-02-24 · 🧮 math.SG

Equivariant Floer cohomology for contactomorphisms of quotient spaces

Dylan Cant , Eric Kilgore , Jun Zhang This is my paper

Pith reviewed 2026-05-21 12:49 UTC · model grok-4.3

classification 🧮 math.SG
keywords equivariant Floer cohomologycontact orderabilityquotient contact manifoldsnonlinear Maslov indexcontactomorphismssymplectic fillingsmapping conesdiscriminant crossings
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The pith

Equivariant contact Floer cohomology shows that quotients of fillable contact manifolds are orderable.

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

This paper sets out to prove that contact manifolds obtained as quotients of fillable contact manifolds by finite group actions compatible with the filling are orderable, with real projective spaces serving as the main example. A sympathetic reader cares because orderability equips the contactomorphism group with a partial order that constrains the existence of positive loops and connects to questions in contact dynamics. The authors build an equivariant version of contact Floer cohomology carrying a k[[x]]-module structure and use it to construct an analogue of Givental's nonlinear Maslov index. Mapping cones of continuation maps in this theory detect crossings of the discriminant, and the whole construction is lifted to chain level by viewing contact Floer cohomology as an infinity-functor on an infinity-categorification of the Eliashberg-Polterovich orderability relation.

Core claim

The central claim is that an equivariant formulation of contact Floer cohomology, equipped with its k[[x]]-module structure, yields a nonlinear Maslov index analogue in which mapping cones of continuation maps detect crossings with the discriminant; this detection, once lifted to an infinity-functor on a suitable infinity-categorification of the Eliashberg-Polterovich relation, establishes orderability for the contactomorphism groups of quotients of fillable contact manifolds such as RP^{2n-1}.

What carries the argument

The equivariant contact Floer cohomology as a k[[x]]-module whose mapping cones of continuation maps detect crossings of the discriminant.

If this is right

  • The contactomorphism group of RP^{2n-1} is orderable for every n.
  • Any contact manifold obtained as a quotient of a fillable manifold by a compatible finite group action is orderable.
  • The nonlinear Maslov index analogue supplies a new algebraic invariant for paths of contactomorphisms on these quotients.
  • Orderability follows whenever the module structure on equivariant Floer cohomology changes across the discriminant.

Where Pith is reading between the lines

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

  • The same module-structure technique might apply to quotients by groups that are not finite if suitable compatibility conditions can be formulated.
  • Explicit low-dimensional calculations of the index on RP^3 could provide an independent check on whether positive loops exist.
  • The infinity-functor lift may connect to other chain-level constructions in symplectic geometry that handle non-canonical choices.

Load-bearing premise

The finite group actions on the contact manifold must be compatible with a symplectic filling in a way that permits a well-defined equivariant contact Floer cohomology theory and the associated k[[x]]-module structure.

What would settle it

A concrete computation for a loop in the contactomorphism group of RP^3 that crosses the discriminant yet produces a mapping cone whose k[[x]]-module structure fails to register the crossing would falsify the detection mechanism.

Figures

Figures reproduced from arXiv: 2602.21152 by Dylan Cant, Eric Kilgore, Jun Zhang.

Figure 1
Figure 1. Figure 1: Elements of M(∆2 ). It is a convenient fact that: Lemma 2.2. If a straight line path ℓ passes through vq, then ℓ −1 (vq) = {q}. Proof. Suppose that ℓ(τ ) = vq. The total integral of the speed one-form over [0, τ ] is equal to q, since the speed one-form is exact with primitive: Xjθj . 24This definition is closely related to other spaces of paths in the simplex, e.g., [Ada56], [GPS20, Remark 4.1], and [Par1… view at source ↗
Figure 2
Figure 2. Figure 2: Cubical coordinates on the moduli space of straight line paths are ultimately derived from the points p1 and p2. Proof. The identification is as follows: each straight line path ℓ in M(∆n ) intersects the face opposite v0 at some time τ1(ℓ), at a point p1 = ℓ(τ1(ℓ)). There is a unique straight line path F(ℓ) passing through v1 such that: • F(ℓ)(τ ) lies on the line segment connecting v1 and p1 for τ ∈ [1, … view at source ↗
Figure 3
Figure 3. Figure 3: Graph of auxiliary functions f : R → R and ρ : R → R. We require the derivative ρ ′ (t) to be supported in the interior of the interval where f = 1. For similar use of such auxiliary functions f, ρ in setting up the continuation equation, we refer the reader to [Can23, CHK23, Can24b].29 The role of f will be the following: if ψη,t is a Hamiltonian isotopy, extended to all times t ∈ R by the rule ψη,t+1 = ψ… view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of a solution to (16). With our coho￾mological conventions, γ+ will be considered as the “input.” Let us call Borel data non-degenerate provided the time-1 map ψη,1 has only non-degenerate fixed points when η is a pole. In this case, it follows from (B3) that the solutions u are asymptotic to stationary solutions over non-degenerate orbits of Xvk,±,t and Xv0,+,t at the two ends. At this stage,… view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of the domain of a solution in M(Φ, k, ±), when Φ is a n-simplex. The equation a solution (ℓ, π, u) must solve is expressed in terms of n sub-intervals [si , si + wi ] determined by ℓ, and the values of Φ(ℓ). This leads to the second regularity requirement imposed on Φ: (∗2) the parametric moduli space of continuation cylinders M(Φ, k, ±) is cut transversally (in the parametric sense). See, e.… view at source ↗
Figure 6
Figure 6. Figure 6: The case when xi(ℓn) converges to 0 but the other coordinates converge in (0, 1], si remains bounded but si+1 converges to −∞; all of the differences si+1 − sj remain bounded if j ≥ i + 1. Moreover, wi+1 converges to 1. retranslations un(s + si+1(ℓn), t). Moreover, ℓn converges to a straight-line path which passes through vi , and we consider the limit ℓ∞ as split into the concatenation ℓ+, ℓ− of its restr… view at source ↗
Figure 7
Figure 7. Figure 7: Restriction of F to the critical circle has 2p crit￾ical points (shown for p = 3). Here v1,e is required to be a positive pole. This V then has exactly p zeros of index n for each n ≥ 0, which are related to each other by the G action, and τ acts by shifting the index by 2. Choosing distinguished critical points v0,e, v1,e, of index 0 and 1 respectively, we have a natural labeling of the rest vn,g by the d… view at source ↗
Figure 8
Figure 8. Figure 8: A 2-simplex in Q(W) with j = 0. The [01] and [02] faces both have j = 0, while the [12] face has j = −1. This simplex in Q(W) also has the extra data of a 0 simplex in G(W) (one should imagine the id vertex is decorated with a vector field on W). Proposition 3.10. The simplicial set Q(W) is an ∞-category. Proof. The argument is only mildly more complicated than that of Propo￾sition 3.8, and we leave the pr… view at source ↗
Figure 9
Figure 9. Figure 9: The input asymptotic of the continuation cylin￾der is degenerate, if j ≥ 0, and u solves the holomorphic curve equation on the region s ≥ 1. If j = −1 then the in￾put asymptotic is non-degenerate Borel data. If any of the cubical coordinates xi(ℓ), with i ≤ j, are sufficiently close to zero, then the curve will be genuinely holomorphic in more regions, by axiom (N4), some of the levels we stabilize to be t… view at source ↗
Figure 10
Figure 10. Figure 10: Shown with j = 2 and n = 3 and cubical coor￾dinates (x1, x2) = (1/4, 1/4). The time s∗ is always to the right of the interval Ij+1. A straight-line path ℓ ∈ Mint(∆n ) has cubical coordinates x1, . . . , xn−1. There is a unique straight line path ℓ (j) ∈ Mint(∆j ) whose cubical coor￾dinates are x1, . . . , xj−1. The times τ1, . . . , τj , as in §2.1.1, of ℓ (j) determine line segments ℓ (j) |[τi−1,τi] on t… view at source ↗
Figure 11
Figure 11. Figure 11: The set-up for the moduli space M1,2(Σ) and co￾ordinate x1 = 1/8 shown. The symbols X0, X1 represent the Borel–Morse data (on the interval I1, one has a continuation from X0 to X1). The contact isotopy φx1,t is the endpoint of the first level, and eventually equals id as x1 → 0. Having constructed the infinity functor Q ∗ (W) → NdgCh(k[[x]]), the con￾struction of the functor with domain P(Y ) satisfying T… view at source ↗
Figure 12
Figure 12. Figure 12: Cartoon of the composition of PSS followed by (39) The glued Hamiltonian connections are similarly determined by vector fields XR,η,s,t, YR,η,s,t, where R is a gluing parameter. These glued connections still satisfy (iv) and (v). In addition, they satisfy the property that: (vi) XR,η,s,t = f ′ (t)XH for s ≤ −sR, and XR,η,s,t = 0 for s ≥ sR, for some sR. 41see [BC25, §3.2] for an introduction to Hamiltonia… view at source ↗
Figure 13
Figure 13. Figure 13: Domain of an extender is the symplectization SY decomposed into a negative end, a positive end, and a compact part in between. There is an ideal restriction E → C1(Y ) given by the flow generated by Hs,t on the positive end. Given an extender ψs,t ∈ E, and a subset κ ∈ π0(ΛY ) of free homotopy classes of loops, we will define in §4.4 the local Floer chain homotopy type 43 CFloc(ψs,t; κ) for H1,t following… view at source ↗
Figure 14
Figure 14. Figure 14: Convex cut-off function γ(x) used in the exten￾der ansatz equals 0 in a neighborhood of x ≤ 0 and equals x − ϵ for x ≥ 1. Assume that γ ′′(x) > 0 if γ ′ (x) ∈ (0, 1). Lemma 4.10. The isotopy Ψs,t generated by (41) satisfies (E1) through (E5), but maybe not axiom (E6) (in other words, we do not show all the orbits of the extender ansatz have the same action). Proof. Recall that Ks,t is determined from Hs,t… view at source ↗
Figure 15
Figure 15. Figure 15: Chopping up a 1-simplex into a concatenation so that each piece has either 0 or 1 intersection with the discriminant; the shaded parts have 1 intersection with the discriminant, located at the midpoint. The idea now is to pick the partition s0 < · · · < sn so that sj −sj−1 becomes very small. Using this trick, we first determine what happens if the case σs,1 has no intersections with the discriminant: Lem… view at source ↗
Figure 16
Figure 16. Figure 16: Regions in the G-filling W of Y . The length parameter L should be considered as quite large. 4.4.3. Proof of Theorem 4.1. In this section we prove that the cone of the morphism CFeq(φ0,t) → CFeq(φ1,t) associated to the 1-simplex φs,t arising as the ideal restriction of the extender ψs,t lies in the local Floer cohomology chain homotopy class CFloc(ψs,t, κ) from Definition 4.20, where κ is the collection … view at source ↗
Figure 17
Figure 17. Figure 17: Pair of pants surface and the flow tree T. The moduli space used to define the product consists of rigid pairs (¯τ, u), where τ ∈ T, and u : Σ → W is a solution to the Floer equation associated to the Hamiltonian connection Hη obtained by setting η = ¯τ (p(z)) where p : Σ → T is an appropriate map sending the pair of pants surface onto [PITH_FULL_IMAGE:figures/full_fig_p079_17.png] view at source ↗
read the original abstract

This paper establishes the orderability of contact manifolds which are quotients of fillable contact manifolds under finite group actions compatible with the filling, the prototypical example being $\mathbb{R}P^{2n-1}$ as the quotient of $S^{2n-1}$. Our approach employs an equivariant formulation of the so-called contact Floer cohomology theory. This leads us to develop an analogue of Givental's nonlinear Maslov index using the $\mathbf{k}[[x]]$-module structure on an equivariant version of contact Floer cohomology. A key idea is that mapping cones of continuation maps detect crossings with the discriminant (recall that Givental's index is a continuous integer valued function on the complement of the discriminant). To properly handle the inherent non-canonicity in defining such mapping cones, we lift the structure of contact Floer cohomology to chain level by defining it as an $\infty$-functor on a suitable $\infty$-categorification of the Eliashberg-Polterovich orderability relation on the universal cover of the contactomorphism group.

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

2 major / 3 minor

Summary. This paper establishes the orderability of contact manifolds which are quotients of fillable contact manifolds under finite group actions compatible with the filling, the prototypical example being RP^{2n-1} as the quotient of S^{2n-1}. It employs an equivariant formulation of contact Floer cohomology to develop an analogue of Givental's nonlinear Maslov index via the k[[x]]-module structure. Mapping cones of continuation maps detect crossings with the discriminant, with non-canonicity handled by lifting the structure to an ∞-functor on an ∞-categorification of the Eliashberg-Polterovich orderability relation on the universal cover of the contactomorphism group.

Significance. If the constructions are carried through rigorously, the result would be a meaningful advance in contact geometry by extending orderability to a new class of quotient manifolds and supplying a Floer-theoretic detection mechanism for the discriminant. The ∞-categorical handling of continuation-map cones addresses a recurring technical obstacle in the field and may prove reusable in other equivariant or non-canonical settings.

major comments (2)
  1. [§3] §3: The construction of the equivariant contact Floer cohomology on the quotient relies on the finite group action being compatible with a symplectic filling to ensure the theory descends; however, the argument that this compatibility produces well-defined, transverse equivariant moduli spaces is only outlined and requires a complete Fredholm and compactness analysis to support the subsequent claims.
  2. [§5.2] §5.2, around the definition of the mapping cone: the assertion that the cone of the lifted continuation map is non-trivial precisely when the path crosses the discriminant is central to the nonlinear Maslov index analogue, yet the precise chain-level or ∞-categorical mechanism that guarantees this detection property for the quotient case is not fully derived from the Eliashberg-Polterovich relation.
minor comments (3)
  1. [§4] The notation for the ∞-categorification of the Eliashberg-Polterovich relation would benefit from an explicit description of the objects, 1-morphisms, and higher cells, perhaps accompanied by a diagram.
  2. [Introduction] Several references to prior work on equivariant Floer cohomology and ∞-categorical symplectic geometry appear to be missing or outdated; a brief comparison paragraph situating the present lift against existing approaches would improve readability.
  3. [Theorem 1.1] In the statement of the main theorem, the precise range of dimensions n for which the RP^{2n-1} example holds should be stated explicitly rather than left implicit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment and constructive feedback on our manuscript. The comments identify areas where additional detail will strengthen the exposition, and we address each point below with plans for revision.

read point-by-point responses

  1. Referee: [§3] §3: The construction of the equivariant contact Floer cohomology on the quotient relies on the finite group action being compatible with a symplectic filling to ensure the theory descends; however, the argument that this compatibility produces well-defined, transverse equivariant moduli spaces is only outlined and requires a complete Fredholm and compactness analysis to support the subsequent claims.

    Authors: We agree that the current outline in §3 would benefit from a more self-contained Fredholm and compactness argument. In the revised version we will expand this section to include a complete analysis: we will verify that the finite group action, being compatible with the symplectic filling, induces equivariant almost complex structures for which the linearized operators are Fredholm with the expected index, and that the standard SFT-style compactness arguments carry over to the equivariant setting by controlling the energy and using the group-equivariance to rule out bubbling. This will explicitly confirm that the resulting moduli spaces are transverse and that the descended theory on the quotient is well-defined. revision: yes

  2. Referee: [§5.2] §5.2, around the definition of the mapping cone: the assertion that the cone of the lifted continuation map is non-trivial precisely when the path crosses the discriminant is central to the nonlinear Maslov index analogue, yet the precise chain-level or ∞-categorical mechanism that guarantees this detection property for the quotient case is not fully derived from the Eliashberg-Polterovich relation.

    Authors: We appreciate this observation. The manuscript already lifts the continuation maps to an ∞-functor on the ∞-categorification of the Eliashberg-Polterovich relation and uses the k[[x]]-module structure to detect non-triviality of cones. However, the explicit chain-level identification of when the cone becomes non-trivial in the quotient setting can be made more transparent. In the revision we will insert a short subsection deriving this detection property directly from the ∞-categorical data, showing that a crossing of the discriminant corresponds to a non-trivial homotopy class in the cone via the equivariant action and the module structure, thereby completing the derivation from the underlying orderability relation. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper constructs an equivariant contact Floer cohomology theory for quotients under compatible finite group actions, then develops an analogue of Givental's nonlinear Maslov index via the k[[x]]-module structure and mapping cones of continuation maps. These are lifted to an ∞-functor on an ∞-categorification of the Eliashberg-Polterovich orderability relation to address non-canonicity. No quoted equations or steps in the abstract reduce the target orderability result to a self-definition, fitted parameter, or load-bearing self-citation by construction. The approach builds on established external frameworks in symplectic geometry (Floer theory, ∞-categories, prior orderability relations) without the central claim collapsing into its inputs. This is a standard non-circular technical development.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on standard domain assumptions from symplectic geometry plus the new equivariant and infinity-categorical structures introduced in the paper.

axioms (2)
  • domain assumption Existence of symplectic fillings and transversality for holomorphic curves in the equivariant setting.
    Invoked to define the contact Floer cohomology and its equivariant version.
  • domain assumption The Eliashberg-Polterovich orderability relation on the universal cover of the contactomorphism group admits an infinity-categorification.
    Used to lift the structure to chain level and handle non-canonicity of mapping cones.

pith-pipeline@v0.9.0 · 5713 in / 1383 out tokens · 53329 ms · 2026-05-21T12:49:13.018969+00:00 · methodology

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