Real sutured Heegaard Floer homology

Abstract: We develop a theory of real sutured manifolds and a real Heegaard Floer theory for these manifolds. We develop a notion of real nice diagrams, and prove that our invariant is combinatorially computable. Our theory shares many structural properties with Juhász's sutured Floer homology, as does the topological theory of real sutured manifolds with Gabai's original sutured manifold theory. We also show that our invariant has several new structural properties differentiating it from sutured Floer homology.

• The paper introduces real sutured Heegaard Floer homology, extending conventional sutured Floer theory to manifolds with involutions.
• It establishes combinatorial computability by constructing nice real Heegaard diagrams and explicit real Floer chain complexes.
• It uncovers new phenomena such as failure of Künneth formulas and detection properties, offering novel topological insights.

Real Sutured Heegaard Floer Homology: Construction, Computability, and Novel Phenomena

Introduction and Context

This paper introduces and develops the theory of real sutured Heegaard Floer homology (RSFH), extending the categorical and computational frameworks of Heegaard Floer homology to the class of real sutured manifolds—that is, $3$-manifolds equipped with a sutured structure and an orientation-preserving involution $\tau$ whose fixed point set is of codimension $2$. This work builds fundamentally on the recent advances in the study of real 3-manifolds and their gauge-theoretic invariants, particularly the real versions of Seiberg–Witten and Heegaard Floer theories. Moreover, it parallels and extends Juhász's sutured Floer homology, incorporating the geometric structure induced by involutions and emphasizing combinatorial tractability.

The theory constructed here is not only parallel in its formal properties to classical (unreal) sutured Floer homology but also introduces new structural behaviors resulting from the presence of a real structure. These include the classification of domains in nice diagrams, failure of Künneth-type formulas under certain real operations, and subtle failures in detection properties (such as the real genus).

Definitions and Formalism

The core object, a real sutured manifold $(Y, \gamma, \tau)$, comprises a $3$-manifold $Y$ with a balanced sutured structure $\gamma$ and an orientation-preserving involution $\tau$ with a codimension-$2$ fixed locus. The boundary conditions on $\tau$ ensure compatibility with the sutured decomposition into $\tau$0 and $\tau$1. The definition generalizes standard balanced sutured manifolds to the equivariant category and demands that the set of fixed points of the boundary involution $\tau$2 lies in the set of sutures.

The paper sets up real sutured Heegaard diagrams—Heegaard decompositions endowed with involutive symmetry—and constructs the associated real Floer chain complexes and homology groups, graded over a precisely defined set of real relative $\tau$3 structures. The definition leverages anti-symplectic involutions on symmetric products of the Heegaard surface, and the fixed loci $\tau$4 play a direct role in defining intersection points and the moduli spaces that enter the Floer complex.

A crucial technical development is the extension of the notion of nice diagrams (after Sarkar–Wang) to the real setting, leading to the concept of nice real Heegaard diagrams. This structure yields explicit combinatorial descriptions of moduli spaces—rendering the theory algorithmically computable.

Combinatorial Computability and Nice Diagrams

The paper provides a constructive proof that for any balanced real sutured manifold, there exists a nice real Heegaard diagram for which the associated chain complex, and thus $\tau$5, can be computed purely by combinatorial means. The differential in the real Floer complex can be explicitly expressed via counts of index-$\tau$6 domains of certain prescribed types (bigons, rectangles, certain annuli/tori, etc.) classified under real symmetry constraints.

Strong numerical results include:

• The hat version of real Heegaard Floer homology, $\tau$7, is algorithmically (combinatorially) computable for any based closed real $\tau$8-manifold $\tau$9.
• The classification of real index $2$0 domains in a nice diagram is exhaustive, and mod $2$1 counts of holomorphic representatives for these domains are always $2$2.
• The construction of nice diagrams respects the real structure and is achieved by a finite sequence of real isotopies and handleslides, following a lex ordering of domain badness.

This combinatorial nature facilitates direct calculation in cases of geometric/topological interest, including real link exteriors, strong inversions, and branched covers of knots.

Structural Properties and Decomposition Formulas

Much of the paper is devoted to establishing structural parallels and contrasts between RSFH and its classical counterpart:

• Direct sum decompositions: $2$3 splits as a sum over real relative $2$4 structures, with explicit diagrams realizing the correspondence between diagrammatic generators and these structures.
• Adjunction inequalities: An adjunction inequality analogous to the classic case is proven, but, notably, the bound is often not sharp in the real setting—a phenomenon demonstrated explicitly.
• Surface decomposition formula: A real analog of Juhász's decompositional theorem for sutured Floer homology is established: under suitable hypotheses (notably, open decomposing surfaces with well-behaved intersections with sutures), the RSFH of the decomposed manifold is the direct sum (over outer real $2$5 structures) of the RSFH of the original manifold.
• Arc decompositions: A new kind of decomposition is introduced for real manifolds, corresponding to the removal of equivariant neighborhoods of arc components in the fixed set, leading to an explicit formula for the effect on RSFH and a reduction for computing RSFH for manifolds without such arc components.

Contrasts to the unreal setting are significant:

• Künneth formula failures: It is proved that, unlike in the usual (unreal) theory, RSFH does not satisfy a multiplicative Künneth formula under connected sum operations in general. Constructions are provided where the rank of RSFH is not multiplicative.
• Detection properties fail: The real knot Floer homology in the sense developed here (or as in $2$6) does not detect the real genus or the fiberedness of knots; the most direct analogues to these detection results in the classical theory fail.

A host of further natural operations—puncturing, stabilization, connected sum, guided deletion/arc decomposition—are formulated in the real setting, and their effect on RSFH is explicitly described.

Topological Implications and Examples

The formal theory is applied to several notable scenarios, including:

• Use of real invariants to distinguish exotic smooth structures (e.g., exotic $2$7-knots in $2$8 distinguished by real Seiberg–Witten invariants of their covers).
• Real Floer obstructions to sliceness for cables of the figure-8 knot, paralleling and complementing prior Seiberg–Witten theoretic proofs.
• An infinite family of real structures on $2$9 arising via analysis of covering involutions on branched covers.

The role of the involution is essential: many structural and computational phenomena derive from fixed-point data and the compatibility of Heegaard diagrams and almost complex structures with the involution.

Theoretical and Practical Implications

Theoretical Implications

• Enhanced obstruction sets: RSFH provides new obstructions to topological problems (e.g., sliceness, ribbonness) that are not accessible via the classical theory. Moreover, invariants arising in the presence of a real structure are sensitive to subtleties invisible in the regular Floer context.
• Categorial and TQFT generalizations: The framework provides a natural structure underlying equivariant TQFTs, threading together real monopole and Seiberg–Witten invariants with an algorithmic, diagram-based theory.
• Room for further detection failures: The failure of detection properties in RSFH suggests rich and nuanced algebraic structures in equivariant Floer settings, meriting further study—especially regarding the impact on concordance invariants.

Practical and Computational Implications

• Algorithmic calculation: The combinatorial computability of $(Y, \gamma, \tau)$0 and RSFH opens the door to explicit computations in concrete cases, particularly for real invertible knots, link exteriors with involutions, and branched double covers.
• Extensions to link Floer homology: The techniques and results—especially the handling of multi-based and nice diagrams—inform the computation of real link Floer homology and suggest possible real generalizations of other Floer invariants.
• Adaptation to bordered settings: The independence of the combinatorial proof here from bordered Floer frameworks (though consistent with and complementary to earlier approaches) offers robustness: existing computations in the bordered context can be translated to, or verified by, the explicit diagrammatic framework here.

Discussion and Outlook

The theory developed here substantially extends the algebraic, topological, and computational machinery of Floer theory to the field of real/symmetric 3-manifolds, making substantial progress on the combinatorial and diagrammatic front. Important open questions remain regarding:

• The topological characterization of vanishing RSFH: unlike in the classical theory, tautness does not precisely detect nonvanishing RSFH, opening new directions for the search of topological criteria.
• The refinement and possible tightness of real adjunction inequalities.
• Extensions to other flavors (minus, infinity) of the Heegaard Floer package in the real-equivariant category.
• Deeper algebraic understanding of phenomena (such as the failure of Künneth and detection) unique to the real setting.

Future developments might include:

• Generalization to more general group actions or the study of equivariant Floer theories for non-involutive automorphisms.
• Interplay with the categorification of further quantum/topological invariants using real structures.
• Computation and tabulation of RSFH groups for broad families of real knots and links, yielding new concordance invariants.

Conclusion

This work rigorously develops the theory of real sutured Heegaard Floer homology, proves the algorithmic computability of its combinatorial form, and illuminates the interplay between classical and real topological invariants. Several phenomena are unique to the real category—such as failures of classical detection, adjunction inequality sharpness, and multiplicativity properties—highlighting the necessity and subtlety of the real perspective within Floer theory.

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References:

• Juhász, A. "Holomorphic discs and sutured manifolds." [math.GT/0601443]
• Guth, G., Manolescu, C., "Real Heegaard Floer homology." (Kodigala et al., 2022)
• Sarkar, S., Wang, J. "A combinatorial description of some Heegaard Floer homologies." [math.GT/0607777]
• Lipshitz, R. "A cylindrical reformulation of Heegaard Floer homology." [math.GT/0502404]
• Walt, R. J., "Equivariant jets," [math/0602679]
• Additional references as cited in the original paper, e.g., (2607.11082)