Naturality in real Heegaard Floer theory

Ciprian Manolescu; Gary Guth

REVIEW 2 major objections 4 minor 198 references

Naturality in real Heegaard Floer theory

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper proves real Heegaard Floer homology is natural: diagram-move paths induce canonical maps, making HFR^o a functor with an equivariant mapping class group action.

desk verdict Naturality in real Heegaard Floer theory is a serious, substantial paper, but Definition 2.25 has a genus typo that makes the trade pentagon empty as printed; the intended fix is clear and the paper deserves a careful referee. read the letter →

arxiv 2608.00256 v1 pith:JWUWAYAD submitted 2026-07-31 math.GT

Gary Guth , Ciprian Manolescu This is my paper · ORCID

classification math.GT MSC 57K18

keywords realHeegaardFloerhomologynaturalityequivariantmappingclassgroupstronginvariantA4singularityinvolutivesuturedtransitivesystems

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

The paper upgrades real Heegaard Floer homology from an invariant defined only up to isomorphism to a natural one. For each flavor, HFR^o is shown to be a functor from based real 3-manifolds to F[U]-modules, with isotopic equivariant diffeomorphisms acting identically. The same argument makes real link Floer and real sutured Floer natural, and yields an involutive version of the invariant. A sympathetic reader should care because naturality is exactly the ingredient needed to move from diagram-by-diagram computations to functorial cobordism maps, and because it gives well-defined equivariant mapping class group actions.

What carries the argument

The carrying object is the graph of real isotopy diagrams with edges given by real equivalences, {1}- and Z/2-stabilizations, and equivariant diffeomorphisms. A strong real Heegaard invariant is a weak invariant obeying five axioms; the new axiom is trade invariance for the simple trade loop (Definition 2.25), a pentagon canceling two {1}-stabilizations against one Z/2-destabilization. The paper derives this loop from the codimension-2 A4({1}) singularity—a fixed-point singularity with normal form x1^2 - x2^2 + x3^5 in a 2-parameter family—classified via an equivariant splitting lemma, and verifies the axiom by counting rigid real-invariant holomorphic rectangles in a cylindrical reformulati

What would settle it

Compute the mod-2 count of the holomorphic curves contributing to the simple trade on a small explicit diagram, such as the pentagon of Figure 2.4 on genus-one summands. If the resulting map on CFR is not the identity, Axiom (5) fails and Theorem 8 and Theorem 1 collapse. A complementary check: trace the same A4({1}) loop around the two resolutions of Section 6.6 and compare the induced maps.

Watch

Extended reading notes

Core claim

The central result, Theorem 8, states that real sutured Floer homology, real Heegaard Floer homology, and real link Floer homology are strong real Heegaard invariants: they satisfy functoriality, commutativity, continuity, handleswap invariance, and the new trade invariance. Theorem 1 then states that HFR^o is a functor RMan_* to F[U]-Mod whose value is isomorphic to the groups of [GM], with isotopic diffeomorphisms inducing identical maps. In effect, any sequence of real Heegaard moves connecting two diagrams of the same manifold gives the same canonical isomorphism; the paper proves this by showing that the associated graph has no monodromy and assembling the groups into a transitive syste

Load-bearing premise

The load-bearing premise is that the simple trade loop—the pentagon of moves in Definition 2.25, whose proof is checked in Section 8 by counting holomorphic curves—induces the identity map on real Heegaard Floer homology; if that count were wrong, the transitive system of Section 2.5 would fail and Theorem 1 would collapse.

Editorial extensions

If this is right

HFR^o carries a well-defined action of the based equivariant mapping class group.
The conjugation involution is defined up to chain homotopy, yielding involutive real Heegaard Floer homology HFRI^o as an invariant.
Real link and real sutured Floer homology inherit naturality, giving mapping class group actions on those invariants.
The canonical isomorphisms allow different Heegaard diagram computations to be glued into one transitive system, so the invariant is path-independent.
This is the first step toward functoriality of real Heegaard Floer homology under equivariant cobordisms.

Reading between the lines

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

One consequence the authors leave implicit: if the trade count holds, naturality transfers to any construction built functorially from the real Floer complex, so future equivariant cobordism maps will automatically be independent of decompositions.
The mapping class group action may let real link Floer homology distinguish symmetries of strongly invertible links that ordinary link Floer ignores; this is testable on simple examples by comparing U-action and grading shifts.
A concrete independent check: on a small genus-one model diagram with three handles, compute the Floer map around the simple trade pentagon; the identity result is a finite mod-2 count.
The involutive real invariant HFRI^o could provide an equivariant homology-cobordism obstruction, an extension the paper frames but does not develop.

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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

First thing you should know: this paper is the real naturality result for real Heegaard Floer homology, not a routine translation of JTZ21. The new ingredients are real: the A4({1}) fixed-set singularity, the crossover move, and the simple trade loop. The authors also come clean about AI-drafted lemmas and about what is imported from their own prior work. That's honest and useful.

The soft spot, and it's a real one, is Definition 2.25. The simple trade pentagon as printed has H5 = (Σ#Σ0), while H4 = (Σ#Σ0#Σ1#Σ2). Since h is declared to be an equivariant diffeomorphism H4→H5, no such map exists: the genera differ by 2. So the pentagon is empty as stated, and Axiom 2.26(5) is vacuous for it. The intended fix is obvious from the geometry—H5 should also have genus g(Σ)+3, i.e. Σ#Σ0#Σ1#Σ2—and Section 6.6's resolution of the A4 bifurcation into exactly this loop confirms that reading. But this is not a cosmetic issue: the trade axiom is the one axiom with no unreal counterpart, and the paper's main theorem rests on it. The authors need to correct the definition explicitly and make sure the proof in Section 8 is checking the corrected pentagon, not a different object.

The other caveats are milder. Section 8, where the trade computation lives, is cut off in the material I saw, so I cannot vouch for it; that computation needs independent checking. Theorem 9's proof is explicitly a sketch deferring to [JTZ21]; that's reasonable for a paper following such a close template, but a referee should ask for enough detail to confirm the real-setting modifications. The AI-drafted classification lemmas deserve a look, but the disclosure itself is a point in the authors' favor.

Overall verdict: the architecture is sound, the new geometric content is genuinely new, and the flaws are local. Send it to peer review. This is exactly the kind of paper that needs a serious referee who will check the trade computation and the corrected definition, not a desk reject.

Referee Report

2 major / 4 minor

Summary. The paper develops naturality for real Heegaard Floer homology. It sets up a graph of real Heegaard diagrams with moves including real handleslides, {1}- and Z/2-stabilizations, a new crossover move, and a new "simple trade" loop. It defines weak and strong real Heegaard invariants and proves (Theorem 8) that real sutured Floer homology, real Heegaard Floer homology, and real link Floer homology are strong invariants. Theorems 9–10 convert strong invariance into transitive systems and canonical maps, yielding Theorem 1: functors HFR^◦ : RMan_* → F[U]-Mod, independent of the choice of Heegaard diagram, with isotopic diffeomorphisms inducing identical maps, hence an equivariant mapping class group action. The strategy follows JTZ21: singularity classification in Sections 3–4, diagram construction and loop decomposition in Sections 5–6, the metatheorem in Section 7, and verification in Section 8. The paper also defines involutive real Heegaard Floer homology using naturality of the conjugation involution.

Significance. If correct, the paper is a significant step: it upgrades real Heegaard Floer homology from an isomorphism-class invariant to a natural functor with equivariant mapping class group actions, a prerequisite for equivariant cobordism maps. The paper is well organized and careful about isolating what is new: the only axiom with no unreal counterpart is Trade invariance, Axiom (5), and the authors identify the A4({1}) singularity as its source. The singularity classification and the diagram calculus are presented in detail, and the proof strategy is transparent and falsifiable. The main weakness is that the new simple trade loop is currently not well-defined as printed because the genus data are inconsistent; since Theorem 9 and Section 6.6 rely on this loop being a nonempty, concrete object, the definitional issue must be fixed before the central claim can be accepted.

major comments (2)

[§2.4, Definition 2.25 and Axiom 2.26(5)] The genera assigned to the pentagon vertices are incompatible with the declared edge h. Definition 2.25(1) gives H1 = Σ#Σ0 (genus g+1), H2 = Σ#Σ0#Σ1 (genus g+2), H3 = H4 = Σ#Σ0#Σ1#Σ2 (genus g+3), and H5 = Σ#Σ0 (genus g+1). Item (2) then declares h: H4 → H5 to be an equivariant diffeomorphism, but no equivariant diffeomorphism exists between surfaces of different genera. Moreover, the stated combination of two {1}-stabilizations and one Z/2-destabilization does not close the genus count as printed. If H5 is meant to be Σ#Σ0#Σ1#Σ2, the definition should say so explicitly; if the edge assignments or the genus effect of Z/2-stabilization are different, that convention must be stated. As written, Axiom 2.26(5) is either vacuous or refers to a pentagon that is not the one analyzed in Section 6.6. This is load-bearing: Theorem 9 and the resolution of D2 in Section 6.6 depend on the simple trade
[§6.6 and §8] The proof of Trade invariance — the only axiom without an unreal counterpart — is described as "requiring analyzing specific holomorphic curves" analogous to JTZ21's handleswap verification, but the actual moduli/index computation is not carried out in the text provided for review. Since Axiom 2.26(5) is the single new ingredient that upgrades the weak real invariant to a strong one, the referee needs to see either the full holomorphic-curve count or a precise citation to a place where it is performed. In addition, Section 6.6 states that the six remaining strata around the A4({1}) singularity are glued "at a point, a loop around which is exactly a simple trade loop," but the edge-by-edge identification with Definition 2.25 is not shown. Once the genus mismatch is corrected, this identification should be written out explicitly.

minor comments (4)

[§2.3, Definition 2.14] The definition of a Z/2-stabilization is ambiguous. It says there is a disk D ⊂ Σ1 and a punctured torus T ⊂ Σ2 with Σ1 \ D = Σ2 \ T, while also requiring D ∩ τ(D) = ∅ and T ∩ τ(T) = ∅. Since τ(T) is then contained in Σ2 \ T, the equation Σ1 \ D = Σ2 \ T seems inconsistent unless T denotes an orbit of two punctured tori. Please clarify the notation and state explicitly how the genus changes.
[§2.4, Remark 2.24] Remark 2.24 refers to "Lemma 2.23," but the object is Definition 2.23. The cross-reference should be corrected.
[§2.6, Proof of Theorem 11] The proof contains hard-to-parse expressions such as "Fδ(H0)→H0" and "Fδ(H0) ◦ F(δH0)". The arrows and compositions should be rewritten with explicit homotopy classes and source/target objects so the reader can follow the reduction to a single diagram H0.
[§8.2] The assertion that "the space of conformal structures on the rectangle which respect the real structure is zero-dimensional, as the symmetry forces the cross-ratio to be one" deserves a proof or a reference. The real locus of the moduli space of four-pointed spheres is not automatically a single point, and this dimension statement is used in the claim that real rectangle counting maps are chain maps.

Circularity Check

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No significant circularity: naturality is an independent upgrade of the authors' prior weak invariant, not a reduction to it.

full rationale

The paper's central claim is an upgrade: [GM] established only that real Heegaard Floer homology is a weak real Heegaard invariant, i.e. that Heegaard moves induce isomorphisms between isomorphism classes. The present paper defines a graph of diagrams and axioms for a strong invariant, then proves that every sufficiently generic 2-parameter family of real gradients has no monodromy by resolving each codimension-2 bifurcation into the elementary loops listed in Definition 2.26. The genuinely new and load-bearing input is the simple trade axiom, which is not assumed from [GM] or [JTZ21]; the paper states it must be checked by a new holomorphic curve computation (Section 1, Section 8). This is structurally distinct from a fitted parameter or a self-definitional reduction. The citation of [JTZ21] supplies a proof strategy and meta-framework, not the real-specific result; the real-specific content (real singularities, real gradient bifurcations, real moves, crossover, simple trade) is derived in the body of the paper. The apparent genus mismatch in Definition 2.25 is a plausibly typographical correctness issue that would make a stated arrow ill-posed, but it does not make any derivation equivalent to its own inputs; it is not a circularity. No parameter is fitted to data, no prediction is statistically forced, and no central result is obtained simply by renaming an existing invariant.

Assumptions & free parameters 0 free parameters · 5 assumptions · 4 invented entities

The central claim rests on four classes of imported background: (1) the JTZ21 naturality meta-framework; (2) the invariance-up-to-isomorphism results of [GM], [Xiab], and [BGX] (Theorems 5–7); (3) standard analytic foundations of Lagrangian/Heegaard Floer theory; (4) Nagase connectivity of real Heegaard diagrams. In-paper but load-bearing are the real singularity classification to codimension 2 and the completeness of the bifurcation list, which generate exactly the moves and loops treated. There are no free parameters in the fitting sense; the genuinely new objects (crossover, simple trade, A4, HFRI) are well-defined mathematical constructs with internal verification.

assumptions (5)

domain assumption JTZ21 naturality meta-framework: a strong Heegaard invariant (five axioms) yields a transitive system and hence a natural functor (Theorem 9 here is the real analogue of [JTZ21, Theorem 2.38]).
Sections 2.4–2.6 restate the machinery; the meta-theorem converting strong invariants into natural ones is imported from [JTZ21], extended by one new axiom (Trade invariance).
domain assumption Weak invariance of real Heegaard Floer/link/sutured homology: invariance up to isomorphism under real moves, from [GM] (Theorem 5), [Xiaa] (Theorem 6), and [BGX] (Theorem 7).
The naturality upgrade is built on top of these quoted results; a gap in any of them propagates to Theorems 1–4.
ad hoc to paper Completeness of the real singularity/bifurcation classification: generic 1- and 2-parameter families of real functions/gradients exhibit only A2/A3(Z/2) and A2/A4({1}) singularities and bifurcation types (A)–(E).
Proved in Sections 3–4 (with AI-drafted lemmas per the disclosure) but load-bearing: any missing codimension-2 phenomenon would produce an unaccounted loop in the graph of real Heegaard diagrams and break the path-independence theorem.
standard math Standard Floer-theoretic analytic foundations: transversality, Gromov compactness, continuation maps, and the correspondence between J-holomorphic strips/rectangles and real-invariant polygons (Section 8.1–8.2).
Section 8 assumes the usual analytical package, plus the fact that the real conformal structure on a rectangle is unique (cross-ratio forced to 1).
standard math Nagase connectivity: any two real Heegaard diagrams of the same real sutured manifold are connected by real moves (Proposition 2.19; strengthened as Proposition 2.29).
Imported from [Nag79]/[GM]; needed so that path-independence (Theorem 9) defines a single transitive system over all diagrams.

invented entities (4)

Crossover move independent evidence
purpose: New real Heegaard diagram move capturing a {1}-orbit quasi-transversal tangency (Definition 5.18); occurs in codimension-1 families and is decomposed into a {1}-stabilization, handleslides, and a {1}-destabilization (Section 6.1).
Precisely defined combinatorially; its monodromy is computed in type (A3)/(A4)/(B4)/(E) links, so an independent reader can check the decomposition directly.
Simple trade loop independent evidence
purpose: Pentagon loop (Definition 2.25) sourced from the A4({1}) singularity, canceling two {1}-stabilizations against one Z/2-destabilization; adds the new Trade invariance axiom (Definition 2.26(5)) with no unreal counterpart.
The paper verifies by explicit holomorphic curve count that the loop acts trivially (Section 8); this is the single most checkable-and-checkable-again claim in the paper.
A4({1}) singularity independent evidence
purpose: New codimension-2 fixed-set singularity in real functions: normal form x1^2 − x2^2 + x3^5 + λ1 x3 + λ2 x3^3; its bifurcation diagram is computed in Lemma 3.12.
The normal form and bifurcation set (λ2 = 9λ1^2/20) are explicit and directly checkable.
Involutive real Heegaard Floer homology HFRI^◦
purpose: Invariant of (Y, τ, s) defined as the homology of the mapping cone of 1 + ι on the real Heegaard Floer complex (Section 9, Theorem 4); conjugates the real structure.
Well-defined as an isomorphism-class invariant assuming the paper's naturality, but no computations or applications are exhibited in the visible text; its utility is prospective.

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Cite this review

Pith. "Pith review of Naturality in real Heegaard Floer theory." pith.science (2026). https://pith.science/paper/JWUWAYAD

@misc{pith2026260800256,
  author       = {Pith},
  title        = {Pith review of: Naturality in real Heegaard Floer theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JWUWAYAD}},
  note         = {Machine review of arXiv:2608.00256}
}

read the original abstract

In a previous paper we defined real Heegaard Floer homology, an invariant of three-manifolds equipped with involutions. Here we prove that real Heegaard Floer homology is natural, and that it admits an action of the equivariant mapping class group. We also establish naturality of real link Floer homology and real sutured Floer homology, and give a definition of involutive real Heegaard Floer homology.

Figures

Figures reproduced from arXiv: 2608.00256 by the authors.

**Figure 2.1.** A (simple) Z/2-stabilization [PITH_FULL_IMAGE:figures/full_fig_p006_2_1.png] view at source ↗

**Figure 2.2.** Just as before, if H1 and H2 are isotopy diagrams, then H2 is obtained from H1 by a {1}- (de)stabilization if they have representatives (Σ2, α2, β2, τ2) and (Σ1, α1, β1, τ1), respectively, such that (Σ2, α2, β2, τ2) is obtained from (Σ1, α1, β1, τ1) by a {1}-(de)stabilization. D T τ ′ τ [PITH_FULL_IMAGE:figures/full_fig_p007_2_2.png] view at source ↗

**Figure 2.3.** A simple real handleswap loop. As usual in Heegaard Floer theory, the alpha curves are in red and the beta curves in blue. Also, the fixed point set is in green. The purple lines represent real equivalences, and the dashed line represents a diffeomorphism. See Remark 4.14 [PITH_FULL_IMAGE:figures/full_fig_p009_2_3.png] view at source ↗

Figures from the paper (49 more)

**Figure 2.4.** Figure 2.4: A simple trade loop. The black lines indicate (de)stabilizations. The letters in the circles indicate which feet of the handles are identified, and how. (We emphasize that for a {1}-stabilization, while the attaching circles are symmetric along a vertical axis, the g…

**Figure 2.5.** Figure 2.5: Here is the equivariant basepoint twist displayed in the coordinates described in Lemma 2.39; the vertical direction displays slices of Σ0 along the x-axis of the image of δ(Σ0). The involution is given by rotation about the axis normal to the page. Horizontal slices…

**Figure 3.1.** Figure 3.1: A ± 3 (Z/2) and A4({1}) bifurcation diagrams for n = 1. 4. Families of real gradients We now turn to families of real gradients, following [JTZ21, Section 5]. See also PalisTakens [PT83], Carneiro-Palis [CP89], Vegter [Veg85], and Bao-Lawson [BL]. Definition 4.1. An…

**Figure 4.1.** Figure 4.1: The directional manifold of a hyperbolic critical point p µ, as well as the various vectors. Let Xg (Y ) be the set of all gradient vector fields on Y . It is well known that the set of Morse-Smale vector fields on Y is open and dense in Xg (Y ). Let Xg (Y, τ ) be th…

**Figure 4.2.** Figure 4.2: Two 1-parameter families of equivariant isotopies. The fixed set of the involution, C, is drawn here in green; the involution is given by reflection. The red and blue curves A and τ (A) represent slices of the stable and unstable manifolds which are interchanged by t…

**Figure 4.3.** Figure 4.3: Some possible bifurcations appearing in 1-parameter families of real gradient vector fields. The gradient flow always goes upwards. (a) an index 1-2 birth-death in a {1}-orbit; (b) an index 1-2 birth-death in a Z/2-orbit; (c) an index 0-1 birth-death and a symmetric …

**Figure 4.4.** Figure 4.4: Codimension 2 bifurcations of types (A1) and (A2), which involve simultaneous orbits of tangency between a one-dimensional and a twodimensional submanifold. (B3) The vector field Xµ has a saddle-node orbit (Z/2)·p µ and a hyperbolic orbit (Z/2)·p µ with a single or…

**Figure 4.5.** Figure 4.5: Codimension 2 bifurcations of types (A3) and (A4) that consist of simultaneous orbits of tangency involving at least one 1-dimensional submanifold. Type (C) bifurcations consist of two simultaneous saddle-node orbits. These, of course, can appear in several configur…

**Figure 4.6.** Figure 4.6: Codimension-2 bifurcations of type (B) involving pairs of quasitransversal orbits of tangency. {1}-(B4) p1 p Z/2-(B4) p1 p p (B5) [PITH_FULL_IMAGE:figures/full_fig_p035_4_6.png]

**Figure 4.7.** Figure 4.7: Codimension-2 bifurcations of type (B) involving a single real quasi-transversal orbit of tangency. p µ 2 are critical points of index 1. There are, of course, two possible configurations, depending on whether or not p1 = p2. Secondary bifurcations appear in the pres…

**Figure 4.8.** Figure 4.8: Codimension-2 bifurcations of type (C): these come in three varieties, depending on the orbit types of the saddle nodes. In the first frame, we have varied the shading to emphasize the different kinds of transverse orbits that may appear. (D1) (D2) p1 p2 p3 p2 p1 [P…

**Figure 4.9.** Figure 4.9: Codimension-2 bifurcations of type (D) involving a real singularity of codimension 2. intersecting (Z/2)·Wu (p µ 1 ) or between (Z/2)·Wu (p µ 1 ) and a stable manifold of dimension 2 intersecting (Z/2) · Wu (τ (p µ 2 )). See [PITH_FULL_IMAGE:figures/full_fig_p036_4_9.png]

**Figure 4.10.** Figure 4.10: Type (E) codimension-2 bifurcations. In the third frame, we have drawn the directional manifold in pink. Definition 4.19. A real sutured function on (Y, γ, τ ) is a smooth function f : Y → [−1, 1] which satisfies: (1) f −1 (±1) = R±(γ) and f −1 (0) ⊃ s(γ); (2) f has…

**Figure 5.1.** Figure 5.1: A generalized real handleslide of type (m, n) = (2, 3). A A B B (3, 11, 3) D τ (D) T τ (T) [PITH_FULL_IMAGE:figures/full_fig_p043_5_1.png]

**Figure 5.2.** Figure 5.2: A Z/2-stabilization of type (k, ℓ, c) = (3, 11, 3). Definition 5.15. A real Heegaard diagram (Σ′ , α′ , β ′ , τ ′ ) is obtained from (Σ, α, β, τ ) by a generalized Z/2-stabilization of type (k, ℓ, c) if there is a disk D ⊂ Σ and a punctured torus T ⊂ Σ ′ as well as c…

**Figure 5.3.** Figure 5.3: A generalized {1}-stabilization of type (4). Definition 5.18. A real Heegaard diagram (Σ′ , α′ , β ′ , τ ′ ) is obtained from (Σ, α, β, τ ) by a crossover of type (k, ℓ) if there are twice-punctured disks P ⊂ Σ and P ′ ⊂ Σ ′ as well as distinguished curves α ∈ α, α ′…

**Figure 5.4.** Figure 5.4: A crossover of type (3, 2). Proposition 5.19. Let {(ft, vt)}t∈[−1,1] be a generic 1-parameter family of real functions and real gradient-like vector fields on (Y, γ, τ ) that has a bifurcation at t = 0. If the bifurcation at t = 0 is not an index 1-2 birth-death nor …

**Figure 5.5.** Figure 5.5: The construction of the two invariant splitting surfaces on either side of a bifurcation associated to a real quasi-transversal orbit of tangency. Definition 5.22. Let Hi = (Σi , [αi ], [βi ], τi) be real isotopy diagrams for 1 ≤ i ≤ 4. A generalized distinguished re…

**Figure 5.6.** Figure 5.6: Commuting crossovers, i.e. the link of a bifurcation of type (A3) with (k1, ℓ1) = (2, 1) and (k2, ℓ2) = (3, 1). We see two commuting crossovers. (A4a): Here, p µ 1 is a hyperbolic singularity of index 1 with a {1}-orbit of tangency from τ (p 0 1 ) to p 0 1 ; addition…

**Figure 5.7.** Figure 5.7: A commutation of a real handleslide and a crossover, or the link of a bifurcation of type (A4a). (A4b): In this case, p 0 1 is a hyperbolic singularity of index 1 with a {1}-orbit of tangency from τ (p 0 1 ) to p 0 1 ; additionally, p µ 2 and p µ 3 are hyperbolic sin…

**Figure 5.8.** Figure 5.8: A commutation of a real handleslide and a crossover corresponding to the link of a bifurcation of type (A4b). of type (k1, ℓ2 + (ℓ1 − k2)). Pick a point x ∈ C1, and define H1 using a separating surface Σ1 ∈ Σ R(fx, vx). Since S1 is a handleslide stratum, we can defin…

**Figure 5.9.** Figure 5.9: The link of a bifurcation of type (A4c). In this example k1 = k2 = 1, and ℓ1 = ℓ2 = 2. Say there are m flows to p from index 1 critical points and ℓ flows from p to index 2 critical points (if p is a {1}-stabilization, i.e. τ (p) = p, then m = ℓ). If p is a Z/2-stabi…

**Figure 5.10.** Figure 5.10: A commutation of a real handleslide and a Z/2-stabilization corresponding to the link of a bifurcation of type (B1) appearing in a Z/2- orbit. Here, the two green arcs on either side of the diagrams are identified and represent a portion of the fixed point set. This…

**Figure 5.11.** Figure 5.11: A commutation of a real handleslide and a {1}-stabilization, which corresponds to the link of a bifurcation of type (B1) appearing in a trivial orbit. This example has m = ℓ = 3 and k1 = k2 = 1. take Σ4 = Σ1, as on the unstabilized side of the diagram, they are sepa…

**Figure 5.12.** Figure 5.12: The link of a bifurcation of type (B2) appearing in a Z/2-orbit. In this example, k = 3, ℓ = 2, c = 1, m1 = 2, and m2 = 1. Again, flows out of p µ 1 are partitioned into two subsets. Suppose there are n flows from p to critical points of index 2, r1 +r2 flows from p…

**Figure 5.13.** Figure 5.13: The link of a bifurcation of type (B2) appearing in a trivial orbit. In this example, k = 1, m1 = 2 and m2 = 1. In this example, there are no additional handleslides after the crossover occurs. There are two diffeomorphism strata relating Σ2 and Σ′ 2 as well as Σ3 a…

**Figure 5.14.** Figure 5.14: The link of a bifurcation of type (B3) appearing in a trivial orbit. In this example, k = 3. H2 → H3 is of type (n, m + s + c1, c1), H1 → H4 is of type (n, m + (t + ℓ + c 2 2 )s, c1 + sc2 2 ), and H4 → H2 is of type (ℓ + c2 + t, k + s, c2). The underlying Heegaard s…

**Figure 5.15.** Figure 5.15: The link of a bifurcation of type (B3) appearing in a Z/2-orbit. Here, the two green arcs on either side of the diagrams are identified and represent a portion of the fixed point set. In this example, k = 4, ℓ = 3, and c = 3. construct Σ′ 3 as a Z/2-stabilization. T…

**Figure 5.16.** Figure 5.16: A commutation of a crossover and a Z/2-stabilization corresponding to a bifurcation of type (B4) involving a Z/2-stabilization. In this example, n = 2, r1 = 1, r2 = 0, m = 2, c = 0, ℓ1 = 1, ℓ2 = 2. We highlight some of the more subtle aspects of the figure. The sta…

**Figure 5.17.** Figure 5.17: A commutation of a crossover and a {1}-stabilization corresponding to a bifurcation of type (B4) involving a {1}-stabilization. (E1): In this case, we have index 1 critical points p µ 1 and p µ 2 and a pair of orbits of tangency from p 0 1 to τ (p 0 2 ) and p 0 2 t…

**Figure 5.18.** Figure 5.18: A bifurcation of type (B5) involving a {1}-stabilization. The two green arcs on either side of the diagrams are identified and represent a portion of the fixed point set. In this example, m = 1, n = 2, c = 1. The case that p µ 1 and p µ 2 are distinct is similar. We…

**Figure 5.19.** Figure 5.19: The link of a bifurcation of type (C) involving two generalized Z/2-stabilizations. In each quadrant, there are four pairs of arcs which are identified, e.g. the black arcs decorated with a single arrow on the right side of the diagram are glued together. Here, we h…

**Figure 5.20.** Figure 5.20: The link of a bifurcation of type (C) involving a generalized Z/2-stabilization and a generalized {1}-stabilization. Again, there is a pair of green arcs which are identified and represent portions of the fixed point set. In this example, m = 3, n = 2, ℓ = 1, t = 1,…

**Figure 5.21.** Figure 5.21: The link of a bifurcation of type (C) involving two generalized {1}-stabilizations. In this example, k = 1 and ℓ = 2. p1 p2 p3 B A B A (1, 2 + 2, 2) (1, 3, 0) [PITH_FULL_IMAGE:figures/full_fig_p066_5_21.png]

**Figure 5.22.** Figure 5.22: The link of a bifurcation of type (D1). In this example, c2 = 2, c3 = 0, k = 2, and ℓ = 3. (2) V1 is a properly embedded 1-dimensional submanifold-with-boundary of D2 \ V0, and [PITH_FULL_IMAGE:figures/full_fig_p066_5_22.png]

**Figure 5.23.** Figure 5.23: The link of a bifurcation of type (D2). In this example k = 3 and ℓ = 1. In the first region, we have highlighted the pair of symmetric tori which are used in the Z/2-destabilization. (3) each point x ∈ V0 has a neighborhood Nx such that the pair (Nx, V ∩ Nx) is hom…

**Figure 5.24.** Figure 5.24: The link of a bifurcation of type (E1). Here, we have drawn the handleswap around the annulus A. Each time the foot crosses C, the annuli A and B intersect, though for clarity we have only drawn A. In this example, k = 2, ℓ = 3, and r = 2. triangulation of S 1 such …

**Figure 5.25.** Figure 5.25: The link of a bifurcation of type (E2). In this example, k = ℓ = 2. 6.1. Codimension-1 resolutions. First, we break down generalized stabilizations (of both types) and crossovers. The case of generalized {1}-stabilizations is the most straightforward, and follows ju…

**Figure 6.1.** Figure 6.1: The two kinds of stabilization slides. Now, suppose H′ is obtained from H by a crossover of type (k, ℓ). Suppose this crossover involves a neighborhood A of a curve α and a neighborhood B of the beta curve β = τ (α). Let α1, . . . , αk and αk+1, . . . , αk+ℓ be the a…

**Figure 6.2.** Figure 6.2: Switching the orientation involved in a {1}-stabilization of type (k). The top and bottom of the figure represent the two ways of resolving the stabilization. handle. See the transition between the second and third chambers on the top row of [PITH_FULL_IMAGE:figures…

**Figure 6.3.** Figure 6.3: Two ways of resolving a crossover of type (k, ℓ) and an interpolation between these resolutions. In both cases, there are k + 1 + ℓ real handleslides. Here, and later, the dashed purple curve represents the handleslide involving the curves created in the {1}-stabili…

**Figure 6.4.** Figure 6.4: Switching the orientation involved in a Z/2-stabilization. The top and bottom of the figure represent two ways of resolving the stabilization. Bifurcations of type (A4b) are the simplest. We simply resolve the (k, ℓ)-crossover into a simple {1}-stabilization, followe…

**Figure 8.1.** Figure 8.1: A class of rectangles admitting a unique (real) holomorphic representative. we have ind(ψ˜) = 2 indR(ψ) + σ(L0, x) − σ(L1, y) 2 (8) as well. 8.3. Real Heegaard Floer homology. Given a real sutured manifold (Y, γ, τ ), the associated real Heegaard Floer complex is def…

**Figure 8.2.** Figure 8.2: The diagrams EZ/2 and E{1} appearing in Lemma 8.15 [PITH_FULL_IMAGE:figures/full_fig_p086_8_2.png]

**Figure 8.3.** Figure 8.3: The four-manifold Σ × ♢ used to define the holomorphic quadrilateral maps. (J ′3 ′ ) Near the cylindrical ends of ♢, the almost complex structure J agrees with cylindrical almost complex structures on Σ × [0, 1] × R satisfying condition (J5 ′ ) above. (J ′4 ′ ) The …

**Figure 8.4.** Figure 8.4: A quadruple diagram used for computing the real handleslides in a trade loop. Consider the real quadruple diagram Qa,b = (Σ0, αb, αa, βa, βb, τ0) shown in the first frame of [PITH_FULL_IMAGE:figures/full_fig_p096_8_4.png]

**Figure 8.5.** Figure 8.5: A triple stabilization of a real Heegaard diagram for (Σ2(U∪U), τ ). The two domains contributing to the differential are shaded. and Tαc ∩ Tβc = c. For Qc,d, we write Tαc ∩ Tβc = c and Tαd ∩ Tβd = d. For these rectangle counting maps, we have the analogue of Lemma 8…

**Figure 8.6.** Figure 8.6: The quadruple diagrams used for computing the real handleslides in a trade loop. Therefore, the map Φg can be computed as the composition of the three handleslide maps described in ?? 8.36?? 8.37. To prove invariance under simple trade loops, we must prove the follow…

**Figure 8.7.** Figure 8.7: The quadruple diagrams used for computing the real handleslides in a real handleswap loop. Proof. The proof follows by an equivariant neck-stretching argument. Since QL and QR are disjoint, the arguments in [JTZ21, Proposition 9.31] apply here. Indeed, pairs of real-…

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

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