Hat-Homology Lattice Overview

• Hat-homology lattice is a reduced construction in lattice homology theories that refines Floer-type invariants across topological, string, and singularity frameworks.
• It is obtained by setting U=0 or modding out U-torsion in lattice complexes, using combinatorial or cubical structures to capture relative homological data.
• The approach underpins spectral sequence collapses and equivalences with Heegaard Floer homology, offering refined invariants for plumbed 3-manifolds and singularities.

The “hat-homology lattice” denotes a family of hat or reduced constructions in lattice-type homology theories, rather than a single universally fixed invariant. In the literature on plumbed $3$-manifolds, the hat flavor is obtained by setting $U=0$ in a lattice complex generated by characteristic vectors and subsets of vertices; in string-theoretic models for sutured Floer homology it is a homology generated by string diagrams without contractible closed curves; and in analytic and filtered lattice homologies of curve and surface singularities it is realized by relative groups such as $H_*(S_n,S_{n-1},\mathbb{Z})$. Across these settings, the common feature is a combinatorial or cubical “lattice” supporting a reduced theory that models or refines Floer-type invariants (Ozsváth et al., 2012, Mathews et al., 2012, Némethi, 2023, Némethi, 2023).

1. Terminological scope and recurrent structures

In the topological lattice homology of plumbing trees, the lattice is built from pairs $[K,E]$, where $K$ is a characteristic cohomology class and $E$ is a subset of the vertex set. In the string-homological setting of $(D^2,F)$, the relevant combinatorial lattice is the set of noncrossing string diagrams up to isotopy, modulo bypass relations, and it is described as a lattice in the vector space sense. In the singularity-theoretic setting, the lattice is $\mathbb{Z}^r$ together with a weight function, and the homology is extracted from a tower of cubical subcomplexes $S_n$ (Mathews et al., 2012, Némethi, 2023, Némethi, 2023).

A useful summary is as follows.

Setting	Hat construction	Output
Plumbing trees	Set $U=0$ in the lattice complex	Hat flavor of lattice homology
$U=0$0	$U=0$1 with no contractible closed curves	$U=0$2
Curve and surface singularities	$U=0$3	Hat or reduced lattice homology

The collected literature also shows that the adjective “hat” is not implemented in a single way. In plumbing-tree lattice homology it is the $U=0$4 specialization; in filtered singularity theories it is a relative construction with trivial $U=0$5-action; in string homology it is enforced by the exclusion of contractible closed curves and by a differential given by crossing resolutions. This indicates that “hat-homology lattice” is best understood as a reduced combinatorial layer within a broader lattice-homological framework (Ozsváth et al., 2012, Mathews et al., 2012, Némethi, 2023).

2. Hat flavor in lattice homology of plumbing trees

For a plumbing tree $U=0$6, the lattice complex is defined by

$U=0$7

with differential

$U=0$8

Here $U=0$9 is the set of characteristic vectors in $H_*(S_n,S_{n-1},\mathbb{Z})$0, and the exponents $H_*(S_n,S_{n-1},\mathbb{Z})$1 and $H_*(S_n,S_{n-1},\mathbb{Z})$2 are non-negative integers defined combinatorially (Zemke, 2021).

In the completed theory of Ozsváth, Stipsicz, and Szabó, the chain complex is

$H_*(S_n,S_{n-1},\mathbb{Z})$3

with the same differential pattern. The complex has a delta-grading by $H_*(S_n,S_{n-1},\mathbb{Z})$4, and for torsion Spin$H_*(S_n,S_{n-1},\mathbb{Z})$5 structures it carries an explicit Maslov grading. The homology splits according to Spin$H_*(S_n,S_{n-1},\mathbb{Z})$6 structures on $H_*(S_n,S_{n-1},\mathbb{Z})$7 (Ozsváth et al., 2012).

The hat flavor is obtained by setting $H_*(S_n,S_{n-1},\mathbb{Z})$8. In the plumbing-tree setting this produces the reduced or hat version of lattice homology. For rational homology spheres, the completed and hat versions determine each other, and analogous statements hold for the hat theory in the spectral-sequence framework (Ozsváth et al., 2012). In Zemke’s formulation, the reduced version is also described by setting $H_*(S_n,S_{n-1},\mathbb{Z})$9 or by modding out by $[K,E]$0-torsion; for rational homology spheres, the completion or reduction does not lose information and matches the reduced versions in both lattice homology and Heegaard Floer homology (Zemke, 2021).

3. Spectral sequences, equivalence with Heegaard Floer homology, and $[K,E]$1-actions

A central structural result is the spectral sequence from lattice homology to Heegaard Floer homology of the plumbed $[K,E]$2-manifold. For a plumbing tree $[K,E]$3, there is a spectral sequence whose $[K,E]$4 page is isomorphic to lattice homology and which converges to $[K,E]$5; the construction is based on the link surgery formula and respects the Spin$[K,E]$6 splitting. For torsion Spin$[K,E]$7 structures the identification preserves absolute Maslov grading, and for non-torsion structures it preserves relative grading (Ozsváth et al., 2012).

Ozsváth, Stipsicz, and Szabó showed that if $[K,E]$8 is type-$[K,E]$9, equivalently in particular if it has at most two bad vertices, then the spectral sequence collapses at $K$0, giving an isomorphism between lattice homology and Heegaard Floer homology. Zemke later proved Némethi’s conjecture in full generality: if $K$1 is the boundary of a plumbing of a tree of disk bundles over $K$2, then the lattice homology of $K$3 coincides with the Heegaard Floer homology of $K$4, and if $K$5, the isomorphism is relatively graded (Ozsváth et al., 2012, Zemke, 2021).

For $K$6, Zemke also gave a conjectural description of the $K$7-action on lattice homology. If $K$8 denotes the meridian corresponding to a vertex $K$9, then the endomorphism

$E$0

is extended $E$1-equivariantly, and for $E$2 one sets

$E$3

The paper proves that $E$4 depends only on the class of $E$5 in $E$6, up to chain homotopy, and conjectures that the isomorphism

$E$7

respects the structure of modules over $E$8 (Zemke, 2021).

4. String-homological realization and the bypass lattice

A distinct use of the hat construction occurs in dimensionally reduced sutured Floer homology for $E$9. Here $(D^2,F)$0 is a marked surface with $(D^2,F)$1 alternating points on $(D^2,F)$2, and string diagrams are immersed oriented $(D^2,F)$3-manifolds in $(D^2,F)$4 with boundary on $(D^2,F)$5, up to homotopy with endpoints fixed. The hat string complex is

$(D^2,F)$6

and its differential is

$(D^2,F)$7

where $(D^2,F)$8 is obtained by resolving the crossing $(D^2,F)$9, while resolutions that introduce contractible closed curves are set to zero. The complex satisfies $\mathbb{Z}^r$0, and it is filtered by minimal intersection number among representatives (Mathews et al., 2012).

Its homology,

$\mathbb{Z}^r$1

has a particularly rigid form. All cycles are represented by non-crossing string diagrams, that is, by sets of sutures, and the only relation among these generators is the bypass relation: $\mathbb{Z}^r$2 for every bypass triple $\mathbb{Z}^r$3. Consequently,

$\mathbb{Z}^r$4

and this homology is canonically isomorphic to the sutured Floer homology $\mathbb{Z}^r$5. For $\mathbb{Z}^r$6, the space is isomorphic to $\mathbb{Z}^r$7, hence is $\mathbb{Z}^r$8-dimensional over $\mathbb{Z}^r$9. The paper also states that for non-alternating markings the complex is acyclic, so $S_n$0 (Mathews et al., 2012).

In this setting, the phrase “hat-homology lattice” refers very explicitly to the quotient of noncrossing string diagrams by bypass relations. The paper further notes that this quotient has a standard basis given by basis chord diagrams, linking the hat theory to contact-topological bypass exactness and to combinatorial models inspired by string topology (Mathews et al., 2012).

5. Filtered hat lattices for curve and surface singularities

For an isolated curve singularity $S_n$1 with $S_n$2 irreducible components, filtered lattice homology is built on the lattice $S_n$3 with weight

$S_n$4

where $S_n$5 is the multivariable Hilbert function and $S_n$6. For each $S_n$7, one defines

$S_n$8

and then

$S_n$9

The $U=0$0-action is induced by inclusions $U=0$1. The hat version is

$U=0$2

and it is presented as the analogue of the hat version in Heegaard Floer theory, with trivial $U=0$3-action. For each fixed $U=0$4, the level filtration $U=0$5 induces a homological spectral sequence with

$U=0$6

converging to the graded pieces of $U=0$7. The collection of first pages is equivalent to the motivic Poincaré series of $U=0$8, and for plane curve singularities the first page can be identified with the Heegaard Floer link homology of the link of the singularity (Némethi, 2023).

For normal surface singularities with rational homology sphere link $U=0$9, the homological version is again built from a tower $U=0$00, now using a Riemann–Roch type weight function

$U=0$01

and cube weights

$U=0$02

The homology is

$U=0$03

with $U=0$04 acting by inclusion $U=0$05. If an embedded curve singularity $U=0$06 is fixed, the embedded link $U=0$07 induces a filtration $U=0$08, and hence a spectral sequence

$U=0$09

Each page is a triple graded $U=0$10-module, and all entries on all pages are new invariants of the decorated pair $U=0$11. The hat version is

$U=0$12

which is bigraded by homological degree and weight, has trivial $U=0$13-action, and has graded Euler characteristic $U=0$14 (Némethi, 2023).

These singularity-theoretic constructions show that hat-homology lattices can be substantially finer than an unfiltered lattice homology. In the curve case, the first page already encodes motivic data and, for plane curves, Heegaard Floer link homology; in the surface case, the filtration by embedded curve data produces new triple-graded invariants and multivariable Poincaré series, with further connections to Jacobi theta series (Némethi, 2023, Némethi, 2023).

6. Link lattice complexes and plumbed links

A link-level extension of the lattice-homological framework is provided by the link lattice complex for plumbed links. If $U=0$15 is a plumbing graph with non-arrow vertices $U=0$16 and arrow vertices $U=0$17, the link lattice complex

$U=0$18

is defined as a quotient of the classical lattice complex $U=0$19. It is generated by those $U=0$20 with $U=0$21, while the differential sums only over $U=0$22. The resulting complex is a module over

$U=0$23

where $U=0$24, and the actions satisfy $U=0$25 for all $U=0$26, with $U=0$27 and $U=0$28 commuting for different $U=0$29 and $U=0$30 (Borodzik et al., 2022).

This complex carries Maslov and Alexander gradings and an $U=0$31-module structure over the coefficient ring. The main theorem states that if $U=0$32 is a rational homology sphere, then for each Spin$U=0$33 structure $U=0$34,

$U=0$35

as $U=0$36-modules over the relevant ring, where $U=0$37 is the completed link Floer complex. For plumbed L-space links, the link Floer complex is formal and is a free resolution of its homology, yielding an algorithm to compute link Floer complexes, in particular for algebraic links, from the multivariable Alexander polynomial (Borodzik et al., 2022).

Although this is not itself a hat construction in the narrow sense, it clarifies how the lattice viewpoint extends from $U=0$38-manifold invariants to link Floer complexes. A plausible implication is that many “hat-homology lattice” constructions are best viewed as reduced faces of a larger module-theoretic framework in which lattice data controls Floer complexes, gradings, and resolutions (Borodzik et al., 2022).

7. General lattice-homological categorification

A further generalization replaces the geometric origin of the lattice by an arbitrary weight function on $U=0$39. The general construction assigns to a lattice $U=0$40 and a weight function $U=0$41 a bigraded $U=0$42-module

$U=0$43

where $U=0$44 is the union of cubes all of whose vertices have weight at most $U=0$45, and $U=0$46 is induced by inclusion $U=0$47. This abstraction includes the analytic lattice homologies of singularities and shows that lattice homology is not confined to plumbing graphs or Floer-theoretic models (Némethi et al., 27 Mar 2026).

When the weight function arises from Hilbert functions of valuative multifiltrations on a Noetherian $U=0$48-algebra $U=0$49 and a finitely generated $U=0$50-module $U=0$51, the theory applies to realizable submodules $U=0$52, with integrally closed ideals as the basic case when $U=0$53. The independence theorem states that if two collections of extended discrete valuations realize the same submodule $U=0$54, then the resulting lattice homology modules are isomorphic as bigraded $U=0$55-modules, even if the lattices and weight functions differ. Their Euler characteristic is $U=0$56. In this way one associates a well-defined lattice homology to any quotient $U=0$57 with $U=0$58 realizable (Némethi et al., 27 Mar 2026).

The geometric applications include categorifications of the delta invariant $U=0$59 of a reduced curve singularity, the geometric genus $U=0$60, the irregularity $U=0$61, and various plurigenera of isolated normal singularities. This suggests that hat-type reductions belong to a broader categorification program in which reduced lattice layers record significantly more information than the numerical invariants recovered by Euler characteristic alone (Némethi et al., 27 Mar 2026).