Contemporary Mathematics

The lattice cohomology of S_{−d}^{3}(K)

András Némethi and Fernando Román

1. Introduction

Let {K_{ℓ}}_{ℓ=1}^{ν} be a collection of algebraic knots determined by certain irreducible plane curve singularities, and define their connected sum K := K_{1}# ⋯ #K_{ν} ⊂ S^{3}. The aim of the present note is the computation of the lattice cohomology of the 3–manifold M := S_{−d}^{3}(K) obtained by (−d)– surgery of the 3–sphere S^{3} along K. The final version describes the lattice cohomology ℍ^{∗}(M) in terms of the subtle distribution properties of the semigroups {Γ_{ℓ}}_{1≤ℓ≤ν}.

Lattice cohomology of negative definite plumbed 3–manifolds was introduced in [13] and [16]. Since surgery manifolds constitute a key family of 3–manifolds, several articles target their study focusing on the case when K is an algebraic knot: [14] treats the negative integer surgery case, [15] the negative rational surgery case (this second preprint was published as part of [18]).

Any such surgery 3–manifold M = S_{−d}^{3}(K) is the link of a certain normal surface singularity. In this way, the lattice cohomology makes a bridge between the analytic invariants of this singularity and topological–combinatorial invariants of M. For the analytic aspects and connections the reader is invited to inspect [13] and [16]. The present note emphasizes mostly the combinatorial aspects of ℍ^{∗}(M). This includes the connection with the Seiberg–Witten and Heegaard–Floer theories.

The Heegaard–Floer theory was developed by Oszváth and Szabó, see [23] and their long list of publication. The theory associates to any oriented 3–manifold several invariants, the most important one is HF^{+}(M). Conjecturally, the lattice cohomology contains all the information about the Heegaard–Floer homology of M, provided that M is a negative definite plumbed manifold. The starting point of the correspondence appeared in [25], later it was developed in [13, 16]. For the corresponding connections, conjectures and partial results see also section 8. They include a precise correspondence for ν small. If ν = 1 (in this case the corresponding graph is ‘almost rational’, hence the result follows already from [13]) then ℍ^{0}(M) = HF_{even}^{+}(M), and HF_{odd}^{+}(M) = ℍ^{q}(M) = 0 for q ≥ 1. Theorem (8.2.1) of the present work treats the case ν = 2 as well: ℍ^{0}(M) = HF_{even}^{+}(−M), ℍ^{1}(M) = F H_{odd}^{+}(−M), and ℍ^{q}(M) = 0 for q ≥ 2. (For the Heegaard–Floer invariants of surgery manifolds in terms of the filtered chain homotopy type of the Heegaard–Floer complex associated with the pair (S^{3}, K) see also [26].)

More generally, for arbitrary ν, in (4.2.11) we prove that ℍ^{q}(M) = 0 whenever q ≥ ν. Moreover, in section 9 we show that this vanishing result is sharp. Indeed, we determine explicitly the lattice cohomology when each {K_{ℓ}}_{ℓ=1}^{ν} is a torus knots of type (2, 3), and we show that ℍ^{ν−1}(M) = ℤ.

The connection with the Seiberg–Witten theory is described in section 7.1. We prove that the normalized Euler–characteristic of the lattice cohomology coincides with the normalized Seiberg– Witten invariants of M. In particular, this provides a purely combinatorial formula for the Seiberg– Witten invariants. (This was generalized recently by the first author in [20].)

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2000 Mathematics Subject Classification. Primary. 14E15, 57M27, 32Sxx, Secondary. 14Bxx, 57R57, 58Kxx.

Key words and phrases. 3–manifolds, ℚ–homology spheres, lattice cohomology, Ozsváth–Szabó invariant, Heegaard Floer homology, Seiberg–Witten invariant, knot surgeries, plane curve singularities, algebraic semigroups.

The first author is partially supported by OTKA Grants. The second author is partially supported by FPI Grant.

Although in this article we do not emphasize the special case provided by the hypersurface superisolated singularities, this constitutes a main motivation for us. For the corresponding connections with the classification of the rational cuspidal projective plane curves, see [14, 15].

Usually, the direct computation of the lattice cohomology is very hard. Our computation here is based on a crucial ‘reduction theorem’, the main result of [8], which reduces the rank of the plumbing lattice to ν, the number of ‘bad’ vertices in the plumbing graph of M.

A few words about the organization of the sections. Section 2 contains the necessary material about the plumbing graph of M, its connection with the classical invariants of the algebraic knots (Alexander polynomial, semigroups). It also presents some generalities regarding the spin^{c}– structures and Seiberg–Witten invariants of M. The next section is a review of the lattice cohomology. It also contains a ‘new’ Mayer–Vietoris type formula for the normalized Euler–characteristic (which will be used later). Section 4 contains the lattice reduction theorem and its consequences applied for this situation. This is continued in the next two section, where all the invariants are described in terms of the distribution properties of the semigroups. Section 7.1 determines the normalized Euler–characteristic in terms of the Seiberg–Witten invariants, while section 8 contains the connection with Heegaard–Floer theory. In the last two sections we list some examples.

2. The 3–manifold S_{−d}^{3}(K)

2.1. Let us fix a positive integer d and a collection of algebraic knots {K_{ℓ}}_{ℓ=1}^{ν}; these are knots determined by isolated irreducible plane curve singularities. In this section we describe the oriented 3manifold M = S_{−d}^{3}(K), obtained by (−d)-surgery along the connected sum K = K_{1}# ⋯ #K_{ν} ⊂ S^{3} of the knots K_{ℓ}. In (2.2) we review some of the complete invariants of algebraic knots which codify the isotopy types K_{ℓ} ⊂ S^{3}. Then we provide the negative definite plumbing representation of S_{−d}^{3}(K), and we determine some invariants of this manifold, including its Seiberg–Witten invariants.

2.2. Review of algebraic knots. Let K ⊂ S^{3} be an algebraic knot, i.e. the link of an irreducible plane curve singularity f; see [4] and [5] as general references for their invariants. Then K is an iterated torus knot. We will assume that f is not smooth, i.e. K is not an unknot. The isotopy type of K ⊂ S^{3} is completely characterized by any of the following invariants listed below: Newton pairs, linking pairs, Alexander polynomial, semigroup, embedded resolution graph (or, equivalently, plumbing graph of the knot).

2.2.1. The Newton pairs of f consist of r ≥ 1 pairs of integers {(p_{i}, q_{i})}_{i=1}^{r}, where p_{i} ≥ 2, q_{i} ≥ 1, q_{1} > p_{1} and gcd(p_{i}, q_{i}) = 1. They appear naturally in the ‘normal form’ of the equation of f. On the other hand, in topological discussions one usually uses the ‘linking pairs’ (or, the decorations of the splice diagram, cf. [5]) (p_{i}, a_{i})_{i=1}^{r}, where

a_{1} = q_{1} and a_{i+1} = q_{i+1} + p_{i+1}p_{i}a_{i} for i ≥ 1.

2.2.2. The Alexander polynomial Δ(t) of K_{f} ⊂ S^{3}, normalized by Δ(1) = 1, in terms of the pairs (p_{i}, a_{i})_{i} is given by

Δ(t) = \frac{(t^{a_{1}p_{1}p_{2}···p_{r}} − 1)(t^{a_{2}p_{2}···p_{r}} − 1) ⋯ (t^{a_{r}p_{r}} − 1)(t − 1)}{(t^{a_{1}p_{2}···p_{r}} − 1)(t^{a_{2}p_{3}···p_{r}} − 1) ⋯ (t^{a_{r}} − 1)(t^{p_{1}···p_{r}} − 1)}.

The degree of Δ is the Milnor number μ of f.

2.2.3. The original definition of the semigroup Γ is analytic: it is the semigroup of the intersection multiplicities of f with all possible analytic germs. This is equivalent with the following combinatorial descriptions. Γ is the sub-semigroup of ℤ_{≥0} (with 0 ∈ Γ) generated by the integers β̄_{0} = p_{1}p_{2} ⋯ p_{r}, β̄_{k} = a_{k}p_{k+1} ⋯ p_{r} for 1 ≤ k ≤ r − 1, and β̄_{r} = a_{r}. This is also equivalent with the next identity (cf. [7]), which will be used in the sequel:

(2.2.4)  \frac{Δ(t)}{1 − t} = ∑_{k∈Γ} t^{k}.

In fact, #(ℤ_{≥0} \ Γ) = μ/2, cf. [11]. The largest element of ℤ_{≥0} \ Γ is μ − 1, and for 0 ≤ k ≤ μ − 1 one has the symmetry: k ∈ Γ if and only if μ − 1 − k ∉ Γ.

The lattice cohomology of S_{−d}^{3}(K) 3

The integer δ := #(ℤ_{≥0} \ Γ) = μ/2 is called the delta-invariant of f, which also equals the minimal Seifert genus of K ⊂ S^{3}.

2.2.5. The embedded minimal good dual resolution graph G(f) of the pair (ℂ^{2}, f^{−1}(0)), or equivalently, the minimal (negative definite) plumbing graph of the pair (S^{3}, K) will be denoted schematically by

[figure: K; −1; v_{0}; G^{∗}]

In fact, this is a tree of the following shape

[figure: K; ⋯; −1; v_{0}; G(f) :]

where the dash-lines represent strings. Here we emphasize only the r nodes (vertices of valency 3) and end–vertices. The graph has some additional decorations: each vertex has a self–intersection (Euler number) and a multiplicity decoration (the vanishing order of the pull–back of f along the corresponding exceptional divisor), while the arrowhead has the multiplicity decoration 1. For example, the multiplicity of the unique (−1)–vertex v_{0} is m = a_{r}p_{r}, cf. [5]. One can prove that m > μ. For more details, see [4, 5] or [12], section 4.I.

If we disregard the arrowhead, the graph G^{∗} transforms into a plumbing graph of S^{3}, hence can be blown down completely.

2.3. The plumbing graph of S_{−d}^{3}(K). In the sequel we consider ν algebraic knots {K_{ℓ}}_{ℓ=1}^{ν}. Associated with K_{ℓ}, let r_{ℓ} be the number of linking pairs, (p_{i}^{(ℓ)}, a_{i}^{(ℓ)})_{i=1}^{r_{ℓ}} the corresponding linking pairs, Δ_{ℓ}, Γ_{ℓ}, μ_{ℓ} and δ_{ℓ} the Alexander polynomial, semigroup, Milnor number and delta– invariant respectively, while G_{ℓ} the graph of (S^{3}, K_{ℓ}), i.e. the embedded resolution graph of the pair (ℂ^{2}, f_{ℓ}^{−1}(0)). Let Z_{ℓ} be the divisor of (the pullback of) f on G_{ℓ}, and let its coefficient on the (−1)–vertex be denoted by m_{ℓ}, which equals m_{ℓ} = a_{r_{ℓ}}^{(ℓ)}p_{r_{ℓ}}^{(ℓ)} (see above). Then, similarly as in [14, 15], the plumbing graph G(M) of the oriented 3–manifold M = S_{−d}^{3}(K) is obtained from the graphs {G_{ℓ}}_{ℓ} (by deleting their multiplicities but unmodified otherwise) by adding a new vertex v_{+} with self–intersection −d − ∑_{ℓ} m_{ℓ} connected to the vertices v_{0,ℓ} of G_{ℓ}. (This statement can be checked by Kirby calculus).

[figure: v_{+}; −d − ∑_{ℓ} m_{ℓ}; −1; v_{0,1}; G_{1}^{∗}; G(M) :; −1; v_{0,ν}; G_{ν}^{∗}; ⋮; ⋮]

In the sequel we will fix this plumbing representation of M, and we assume that d > 0. In particular, the above plumbing graph has a negative definite intersection form.

Our goal is to compute the graded roots, lattice cohomologies and the Seiberg–Witten invariants associated with M.

2.4. Invariants of plumbing graphs/plumbed manifolds. First, we recall some notations and generalities regarding plumbing graphs.

2.4.1. Generalities. Let us assume that G is a connected negative definite plumbing graph. G can be realized as the resolution graph of some normal surface singularity (X, 0), and the link M of (X, 0) can be considered as the plumbed 3–manifold associated with G. In the sequel we assume that M is a rational homology sphere, or, equivalently, that G is a tree and all the genera decorations are zero. For more details regarding this section, see e.g. [13, 14, 15, 16].

Let X̃ be the smooth 4-manifold with boundary M obtained either by plumbing disc bundles along G, or via the resolution π : X̃ → X of (X, 0) with resolution graph G. Then L = H_{2}(X̃, ℤ) is generated by {E_{j}}_{j∈J}, the cores of the plumbing construction (or the irreducible components of the exceptional divisor E := π^{−1}(0) of π). L is a lattice via the (negative definite) intersection form I := {(E_{j}, E_{i})}_{j,i}. Let L′ be the dual lattice {l′ ∈ L ⊗ ℚ : (l′, L) ⊂ ℤ}. L′ is generated by the (anti)dual elements E_{j}^{∗} defined via (E_{j}^{∗}, E_{k}) = −δ_{jk} (the negative of the Kronecker symbol).

Set H := L′/L. Then H_{1}(M, ℤ) = H. The set of characteristic elements are defined by

Char := {k ∈ L′ : (k, x) + (x, x) ∈ 2ℤ for any x ∈ L}.

The unique rational cycle k_{can} ∈ L′ which satisfies the system of adjunction relations (k_{can}, E_{j}) = −(E_{j}, E_{j}) − 2 for all j is called the canonical cycle. Then Char = k_{can} + 2L′. There is a natural action of L on Char by l ∗ k := k + 2l whose orbits are of type k + 2L. Obviously, H acts freely and transitively on the set of orbits by [l′] ∗ (k + 2L) := k + 2l′ + 2L.

The first Chern class realizes an identification between the spin^{c}–structures Spin^{c}(X̃) on X̃ and Char ⊂ L′. Spin^{c}(X̃) is an L′ torsor compatible with the above action of L′ on Char.

All the spin^{c}–structures on M are obtained by restriction Spin^{c}(X̃) → Spin^{c}(M), Spin^{c}(M) is an H torsor, and the actions are compatible with the factorization L′ → H. Hence, one has an identification of Spin^{c}(M) with the set of L–orbits of Char, and this identification is compatible with the action of H on both sets. In this way, any spin^{c}-structure of M will be represented by an orbit [k] := k + 2L ⊂ Char. The canonical spin^{c}–structure corresponds to [−k_{can}].

Recall that the Seiberg–Witten invariant associates with any [k] ∈ Spin^{c}(M) is a rational number sw(M, [k]) ∈ ℚ. Moreover,

(2.4.2)  sw(M, [−k]) = sw(M, [k]) = −sw(−M, [k]),

where −M denotes M with opposite orientation.

2.4.3. The Seiberg–Witten invariants of S_{−d}^{3}(K). Assume that M = S_{−d}^{3}(K) with plumbing graph G as in (2.3). The base elements E_{+}, respectively {E_{0,ℓ}}_{ℓ=1}^{ν}, correspond to the vertices v_{+} and {v_{0,ℓ}}_{ℓ=0}^{ν}.

We define Δ(t) := ∏_{ℓ} Δ_{ℓ}(t). Its degree is μ := ∑_{ℓ} μ_{ℓ}; and we also set δ := ∑_{ℓ} δ_{ℓ} = μ/2.

Since Δ(1) = 1 and Δ′(1) = δ (use e.g. the formula of (2.2.2)), one gets

Δ(t) = 1 + δ(t − 1) + (t − 1)^{2} · Q(t)

for some polynomial Q(t) = ∑_{i=0}^{μ−2} α_{i}t^{i} of degree μ − 2 with integral coefficients.

If ν = 1, then all the coefficients {α_{i}}_{i=0}^{μ−2} are strict positive. In fact, α_{i} = #{k ∉ Γ_{1} : k > i} [14], hence

δ = α_{0} ≥ α_{1} ≥ ⋯ ≥ α_{μ−2} = 1 (for ν = 1).

If ν = 2 then still α_{i} ≥ 0 for all i, but the monoteneity is lost; while for ν ≥ 3 some of the coefficients might be even negative. Nevertheless, in any situation (see e.g. the proof of (7.1.3) part (b)):

(2.4.4)  α_{0} = δ and α_{μ−2} = 1  (for any ν).

Similarly as in [14, 15], or [3, (8.1)] (where the case ν = 1 is treated), one obtains that L′/L is the cyclic group with d elements generated by [E_{+}^{∗}]. (This can be verified as follows: first one checks that det(−I) = d, hence |H| = d. Since all G_{ℓ}^{∗} are unimodular, the coefficient of E_{+} in E_{+}^{∗} is 1/d, hence [E_{+}^{∗}] has order d in L′/L = H.)

Theorem 2.4.5. For any a ∈ {0, 1, …, d − 1} one has

sw(S_{−d}^{3}(K), [k_{can} + 2aE_{+}^{∗}]) = − ∑_{l∈ℤ_{≥0}} α_{a+ld} + \frac{(μ − 2 + d − 2a)^{2}}{8d} − \frac{1}{8}.

(k_{can} + 2aE_{+}^{∗})^{2} + |J | = −\frac{(μ − 2 + d − 2a)^{2}}{d} + 1.

Proof. The first statement is proved for ν = 1 in [3, (8.1)] (where E_{j}^{∗} is used with opposite sign). The proof of the general case runs in the same way based on [3, (5.0.3)], and on the fact that any component of G \ v_{+} represent S^{3}. Similarly, the second statement follows from [3, (5.0.2)]. □

3. Review of lattice cohomologies

3.1. Preliminaries, ℤ[U]–modules. One can associate lattice cohomologies to different objects at different levels (see below). All the time, the output is a graded ℤ[U]–module. Some special ℤ[U]–modules will stay as building blocks of the cohomology, we review their definitions first.

Consider the graded ℤ[U]–module ℤ[U, U^{−1}], and (following [25]) denote by T_{0}^{+} its quotient by the submodule U · ℤ[U]. This has a grading in such a way that deg(U^{−d}) = 2d (d ≥ 0). Similarly, for any n ≥ 1, the quotient of ℤ⟨U^{−(n−1)}, U^{−(n−2)}, …, 1, U, …⟩ by U · ℤ[U] (with the same grading) defines the graded module T_{0}(n). Hence, T_{0}(n), as a ℤ–module, is freely generated by 1, U^{−1}, …, U^{−(n−1)}, and has finite ℤ-rank n.

More generally, for any graded ℤ[U]–module P with d–homogeneous elements P_{d}, and for any r ∈ ℚ, we denote by P [r] the same module graded (by ℚ) in such a way that P [r]_{d+r} = P_{d}. Then set T_{r}^{+} := T_{0}^{+}[r] and T_{r}(n) := T_{0}(n)[r]. (Hence, for m ∈ ℤ, T_{2m}^{+} = ℤ⟨U^{−m}, U^{−m−1}, …⟩.)

3.2. Lattice cohomology associated with ℤ^{s} and a system of weights. [16] We fix a free ℤ–module, with a fixed basis {E_{j}}_{j}, denoted by ℤ^{s}. It is also convenient to fix a total ordering of the index set J, which in the sequel will be denoted by {1, …, s}. Using the pair (ℤ^{s}, {E_{j}}_{j}) and a system of weights we determine a cochain complex whose cohomology is our central object.

3.2.1. The cochain complex. ℤ^{s} ⊗ ℝ has a natural cellular decomposition into cubes. The set of zero–dimensional cubes is provided by the lattice points ℤ^{s}. Any l ∈ ℤ^{s} and subset I ⊂ J of cardinality q defines a q–dimensional cube, which has its vertices in the lattice points (l+∑_{j∈I′} E_{j})_{I′}, where I′ runs over all subsets of I. On each such cube we fix an orientation. This can be determined, e.g., by the order (E_{j_{1}}, …, E_{j_{q}}), where j_{1} < ⋯ < j_{q}, of the involved base elements {E_{j}}_{j∈I}. The set of oriented q–dimensional cubes defined in this way is denoted by Q_{q} (0 ≤ q ≤ s).

Let C_{q} be the free ℤ–module generated by oriented cubes □_{q} ∈ Q_{q}. Clearly, for each □_{q} ∈ Q_{q}, the oriented boundary ∂□_{q} has the form ∑_{k} ε_{k} □_{q−1}^{k} for some ε_{k} ∈ {−1, +1}. Here, in this sum, we write only those (q − 1)–cubes which appear with non–zero coefficient. These are called faces of □_{q}.

Although ∂ ∘ ∂ = 0, the homology of the chain complex (C_{∗}, ∂) (or, of the cochain complex (Hom_{ℤ}(C_{∗}, ℤ), δ)) is not very interesting: it is just the (co)homology of ℝ^{s}. In order to get a more interesting object, we consider a set of weight functions w_{q} : Q_{q} → ℤ (0 ≤ q ≤ s).

3.2.2. Definition. A set of functions w_{q} : Q_{q} → ℤ (0 ≤ q ≤ s) is called a set of compatible weight functions if the following hold:

(a) For any integer k ∈ ℤ, the set w_{0}^{−1}( (−∞, k] ) is finite;

(b) for any □_{q} ∈ Q_{q} and for any face □_{q−1} ∈ Q_{q−1} of it one has w_{q}(□_{q}) ≥ w_{q−1}(□_{q−1}).

Example 3.2.3. Assume that some w_{0} : Q_{0} → ℤ satisfies (a) for all k ∈ ℤ. For any q ≥ 1 set

w_{q}(□_{q}) := max{w_{0}(v) : v is a vertex of □_{q}}.

Then {w_{q}}_{q} is a set of compatible weight functions.

In the presence of a compatible weight functions {w_{q}}_{q}, one sets F^{q} := Hom_{ℤ}(C_{q}, T_{0}^{+}). Then F^{q} is, in fact, a ℤ[U]–module by (p ∗ ϕ)(□_{q}) := p(ϕ(□_{q})) (p ∈ ℤ[U]), and F^{q} has a ℤ–grading: ϕ ∈ F^{q} is homogeneous of degree d ∈ ℤ if for each □_{q} ∈ Q_{q} with ϕ(□_{q}) ≠ 0, ϕ(□_{q}) is a homogeneous element of T_{0}^{+} of degree d − 2 · w(□_{q}). (In the sequel sometimes we will omit the index q of w_{q}.)

Next, one defines δ_{w} : F^{q} → F^{q+1}. For this, fix ϕ ∈ F^{q} and we show how δ_{w}ϕ acts on a cube □_{q+1} ∈ Q_{q+1}. First write ∂□_{q+1} = ∑_{k} ε_{k}□_{q}^{k}, then set

(δ_{w}ϕ)(□_{q+1}) := ∑_{k} ε_{k} U^{w(□_{q+1})−w(□_{q}^{k})} ϕ(□_{q}^{k}).

Then δ_{w} ∘ δ_{w} = 0, hence (F^{∗}, δ_{w}) is a cochain complex. Moreover, (F^{∗}, δ_{w}) has an augmentation. Set 𝔪_{w} := min_{l∈ℤ^{s}} w_{0}(l) and choose l_{w} ∈ ℤ^{s} such that w_{0}(l_{w}) = 𝔪_{w}. Then one defines the ℤ[U]– linear map ϵ_{w} : T_{2𝔪_{w}}^{+} ⟶ F^{0} such that ϵ_{w}(U^{−𝔪_{w}−n})(l) is the class of U^{−𝔪_{w}+w_{0}(l)−n} in T_{0}^{+} for any n ∈ ℤ_{≥0}. Then, ϵ_{w} is injective, and δ_{w} ∘ ϵ_{w} = 0. Both ϵ_{w} and δ_{w} are homogeneous (degree zero) morphisms of ℤ[U]–modules.

3.2.4. Definitions. The homology of the cochain complex (F^{∗}, δ_{w}) is called the lattice cohomology of the pair (ℝ^{s}, w), and it is denoted by ℍ^{∗}(ℝ^{s}, w). The homology of the augmented cochain complex

0 ⟶ T_{2𝔪_{w}}^{+} ⟶^{ϵ_{w}} F^{0} ⟶^{δ_{w}} F^{1} ⟶^{δ_{w}} …

is called the reduced lattice cohomology of the pair (ℝ^{s}, w), and it is denoted by ℍ_{red}^{∗}(ℝ^{s}, w). For any q ≥ 0, both ℍ^{q} and ℍ_{red}^{q} admit an induced graded ℤ[U]–module structure, and one has graded ℤ[U]–module isomorphisms ℍ^{q} = ℍ_{red}^{q} (for q > 0) and ℍ^{0} = T_{2𝔪_{w}}^{+} ⊕ ℍ_{red}^{0}. Moreover, the ℤ–grading of F^{q} induces a ℤ–grading on ℍ^{q} and ℍ_{red}^{q}; the d–homogeneous part is denoted by ℍ_{d}^{q}, or ℍ_{red,d}^{q}.

If each ℍ_{red}^{q} has finite ℤ–rank, then one can define the ‘normalized Euler characteristic’

(3.2.5)  eu(ℍ^{∗}(ℝ^{s}, w)) := −𝔪_{w} +_{∑q}(−1)^{q} rank_{ℤ}(ℍ_{red}^{q}(ℝ^{s}, w)).

3.2.6. Modification. Clearly, instead of all the cubes of ℝ^{s} we can consider only those ones which sit in [0, ∞)^{s}, or only in the rectangle R := [0, T_{1}] × ⋯ × [0, T_{s}] (for some T_{i} ∈ ℤ_{≥0}). In such a case, we write ℍ^{∗}([0, ∞)^{s}, w) or ℍ^{∗}(R, w) for the corresponding lattice cohomologies.

3.3. The 𝕊^{∗}–representation. Next, we present a more geometric realization of the modules ℍ^{∗}.

3.3.1. Definitions. For each n ∈ ℤ, define S_{n} = S_{n}(w) ⊂ ℝ^{s} as the union of all the cubes □_{q} (of any dimension) with w(□_{q}) ≤ n. Clearly, S_{n} = ∅, whenever n < 𝔪_{w}. For any q ≥ 0, set

𝕊^{q}(ℝ^{s}, w) := ⊕_{n≥𝔪_{w}}H^{q}(S_{n}, ℤ).

Then 𝕊^{q} is ℤ (in fact, 2ℤ)–graded, the d = 2n–homogeneous elements 𝕊_{d}^{q} consists of H^{q}(S_{n}, ℤ). Also, 𝕊^{q} is a ℤ[U]–module. The U–action is given by the restriction map r_{n+1} : H^{q}(S_{n+1}, ℤ) ⟶ H^{q}(S_{n}, ℤ), namely, U ∗ (α_{n})_{n} := (r_{n+1}α_{n+1})_{n}. Moreover, for q = 0 one has an augmentation H^{0}(S_{n}, ℤ) = ℤ ⊕ H̃^{0}(S_{n}, ℤ), hence an augmentation of the graded ℤ[U]–modules

𝕊^{0} = (⊕_{n≥𝔪_{w}}ℤ) ⊕ (⊕_{n≥𝔪_{w}} H̃^{0}(S_{n}, ℤ)) = T_{2𝔪_{w}}^{+} ⊕ 𝕊_{red}^{0}.

The point is that this ℤ[U]–module 𝕊^{∗} coincides with the lattice cohomology ℍ^{∗}. More precisely, we have the following theorem whose proof (together with many other properties of the lattice cohomology) can be found in [16].

Theorem 3.3.2. There exists a graded ℤ[U]–module isomorphism, compatible with the augmentations, between ℍ^{∗}(ℝ^{s}, w) and 𝕊^{∗}(ℝ^{s}, w). Similar statement is valid for ℍ^{∗}([0, ∞)^{s}, w), or for ℍ^{∗}(∏_{i}[0, T_{i}], w) too.

From now on we will denote both modules with the same symbol ℍ^{∗}, no matter which representation we use.

3.3.3. The above description allows us to reinterpret classical homological exact sequences in lattice cohomology. For example, the Mayer–Vietoris exact sequences will have the following consequence. For any integer T consider the half spaces ℝ_{x_{s}≤T}^{s} = {x ∈ ℝ^{s} : x_{s} ≤ T } and ℝ_{x_{s}≥T}^{s} = {x ∈ ℝ^{s} : x_{s} ≥ T } separated by the hyperplane ℝ_{x_{s}=T}^{s} = {x ∈ ℝ^{s} : x_{s} = T }. The restriction of the weight function on these spaces determines lattice cohomologies.

Proposition 3.3.4. (Mayer Vietoris formula for eu) With the above notations one has:

eu(ℍ^{∗}(ℝ^{s}, w)) = eu(ℍ^{∗}(ℝ_{x_{s}≤T}^{s} , w) + eu(ℍ^{∗}(ℝ_{x_{s}≥T}^{s} , w) − eu(ℍ^{∗}(ℝ_{x_{s}=T}^{s} , w).

Similar statement is valid for different rectangles instead of the whole space ℝ^{s}.

Proof. Use the classical Mayer–Vietoris exact sequence and (3.3.2). □

3.4. The lattice cohomology associated with a plumbing graph. Let G be a negative definite plumbing graph as in (2.4.1). Let s be the number of vertices. Then we can associate to L = ℤ^{s} the the free ℤ–module C_{q} generated by oriented cubes □_{q} ∈ Q_{q}, as in (3.2.1).

To any k ∈ Char we associate weight functions {w_{q}}_{q} as follows. First, we define χ_{k} : L → ℤ by

χ_{k}(l) = −(l, l + k)/2.

We also write 𝔪_{k} := min { χ_{k}(l) : l ∈ L}. Then the weight functions are defined as in (3.2.3) by:

w_{q}(□_{q}) = max{χ_{k}(v) : v is a vertex of □_{q}}.

The associated lattice cohomology (resp. reduced lattice cohomology) will be denoted by ℍ^{∗}(G, k) (resp. ℍ_{red}^{∗}(G, k)). It is proved (cf. [16]) that ℍ_{red}^{∗}(G, k) is finitely generated over ℤ, hence eu(ℍ^{∗}(G, k)) := eu(ℍ^{∗}(ℝ^{s}, w)) is well–defined, cf. (3.2.5).

Remark 3.4.1. Although each k provides a different cohomology module, there are only |L′/L| essentially different ones. Indeed, assume that [k] = [k′], hence k′ = k + 2l for some l ∈ L. Then

(3.4.2)  χ_{k′}(x − l) = χ_{k}(x) − χ_{k}(l) for any x ∈ L.

Therefore, the transformation x ↦ x′ := x − l realizes the following identification:

ℍ^{∗}(G, k′) = ℍ^{∗}(G, k)[−2χ_{k}(l)].

3.5. The distinguished representatives k_{r}. We fix a spin^{c}–structure [k]. Recall, see (2.4.1), that [k] has the form k_{can} + 2(l′ + L) for some l′ ∈ L′. Among all the characteristic elements in [k] we will choose a very special one. Consider the (Lipman, or anti–nef) cone

S′ := {l′ ∈ L′ : (l′, E_{v}) ≤ 0 for any vertex v}.

3.5.1. Definition. [13, (5.4–5.5)] We denote by l′_{[k]} ∈ L′ the unique minimal element of (l′ + L) ∩ S′ and we call k_{r} := k_{can} + 2l′_{[k]} the distinguished representative of the class [k].

For example, since the minimal element of L ∩ S′ is the zero cycle, we get l′_{[k_{can}]} = 0, and the distinguished representative in [k_{can}] is the canonical cycle k_{can} itself.

The classes k_{r} generalize the canonical cycle for different spin^{c}–structures. For some applications (showing their importance) see [13, 16, 18]. The next properties are proved in [16]:

Lemma 3.5.2. (a) ℍ^{∗}(G, k_{r}) ≅ ℍ^{∗}([0, ∞)^{s}, k_{r}).

(b) The set {ℍ^{∗}(G, k_{r})}_{[k_{r}]} is independent on the plumbing representation G of the 3–manifold M, hence it associates a ℤ[U]–module to any pair (M, k_{r}), where [k_{r}] ∈ Spin^{c}(M).

4. The lattice reduction

4.1. The strategy. Our goal is the computation of the lattice cohomologies of the 3–manifold S_{−d}^{3}(K). First, we need some preparations. In the first step we reduce the problem from the level of the lattice of rank |J | associated with G to another lattice of rank ν. This reduction is based on a general principle proved in [8]. Then, the lattice cohomologies of the reduced lattice will be described in terms of the semigroups {Γ_{ℓ}}_{ℓ=1}^{ν}. In this way, the cohomology modules will codify the very subtle distribution properties of the semigroup elements of Γ_{ℓ} (as subsemigroups of ℤ_{≥0}).

4.2. The lattice reduction. The general statement. In [16] is proved that the lattice cohomologies of rational graphs (for definition see below) are trivial. On the other hand, any non–rational graph can be transformed into a rational graph by surgery — decreasing some of the decorations — along some of its vertices. It turns out that all the lattice cohomology information is ‘concentrated in these vertices’. They are called bad vertices (cf. [25, 13, 16, 17]).

4.2.1. Definitions. We say that a negative definite graph is rational if it is the resolution graph of a rational singularity. They were characterized combinatorially by Artin, for more details see [2, 19, 13], or (4.2.6) below. Next, we say that a subset of vertices \{\overline{v}_{j}\}_{j=1}^{ν} of a negative definite plumbing graph are bad vertices if replacing their decorations e_{\overline{v}_{j}} := (E_{\overline{v}_{j}}, E_{\overline{v}_{j}}) by some more negative integers e′_{\overline{v}_{j}} ≤ e_{\overline{v}_{j}} we get a rational graph.

For any graph there is a family of bad vertices with smallest cardinality. If this cardinality is less than or equal to one, then the graph is called almost rational (cf. [13]).

The idea of the reduction theorem is present already in [13]. The new lattice of rank ν will be associated with the bad vertices, the lattice points are associated with some important cycles of L (as distinguished members of Laufer–type computational sequences of L). In the next paragraphs we recall first the definitions of these cycles.

4.2.2. The definition of the lattice points x(i_{1}, …, i_{ν}). Suppose we have a (minimal) family of bad vertices \overline{J} := \{\overline{v}_{j}\}_{j=1}^{ν} ⊂ J. Then split the set of vertices J into the disjoint union \overline{J} ⊔ J^{∗}. Furthermore, for any given vertex v and any (rational) cycle x, we write m_{v}(x) for the v–coefficient −(x, E_{v}^{∗}) of x. (In other words, x = ∑_{v} m_{v}(x)E_{v}).

Then the cycles x(i_{1}, …, i_{ν}) are defined via the next lemma.

Lemma 4.2.3. Fix [k] as above. For any integer coordinates (i_{1}, …, i_{ν}) ∈ (ℤ_{≥0})^{ν}, there exists a unique cycle x(i_{1}, …, i_{ν}) ∈ L satisfying the following properties:

(a) m_{\overline{v}_{j}}(x(i_{1}, …, i_{ν})) = i_{j} for any bad vertex \overline{v}_{j};

(b) (x(i_{1}, …, i_{ν}) + l′_{[k]}, E_{v}) ≤ 0 for every ‘non-bad vertex’ v ∈ J^{∗};

(c) x(i_{1}, …, i_{ν}) is minimal with the two previous properties. Moreover, x(0, …, 0) = 0 and x(i_{1}, …, i_{ν}) ≥ 0.

The proof is almost word–by–word the proof of Lemma 7.6. of [13], valid for ν = 1, see also [8].

Remark 4.2.4. Since χ_{k_{r}}(x + E_{v}) − χ_{k_{r}}(x) = χ_{k_{r}}(E_{v}) − (x, E_{v}) = 1 − (l′_{[k]} + x, E_{v}), property (b) is equivalent to χ_{k_{r}}(x(i_{1}, …, i_{ν}) + E_{v}) > χ_{k_{r}}(x(i_{1}, …, i_{ν})) for every v ∈ J^{∗}. In fact (see [13, (9.1)] for ν = 1, or [8]), the smallest χ_{k_{r}}–value on all the lattice points \{x ∈ L : m_{\overline{v}_{j}}(x) = i_{j} for every \overline{v}_{j}\} is realized by x(i_{1}, …, i_{ν}).

The point is that in our main applications, we do not really need the cycles x(i_{1}, …, i_{ν}) themselves, but only the values χ_{k_{r}}(x(i_{1}, …, i_{ν})). These can be computed inductively thanks to the following proposition.

In order to simplify the notation we set i := (i_{1}, …, i_{j}, …, i_{ν}) and i+1_{j} := (i_{1}, …, i_{j} +1, …, i_{ν}).

Proposition 4.2.5. For any distinguished representative k_{r} ∈ Char, any i = (i_{1}, …, i_{ν}) ∈ (ℤ_{≥0})^{ν} and j ∈ \overline{J} one has

χ_{k_{r}}(x(i + 1_{j})) = χ_{k_{r}}(x(i)) + 1 − (x(i) + l′_{[k]}, E_{\overline{v}_{j}}).

Moreover, χ_{k_{r}}(x(0, …, 0)) = 0.

Proof. We construct the following computation sequence {x_{n}}_{n=1}^{N} (as a generalization of Laufer’s computation sequence) connecting x(i) + E_{\overline{v}_{j}} and x(i + 1_{j}), that is x_{1} = x(i) + E_{\overline{v}_{j}} and x_{N} = x(i + 1_{j}) and x_{n+1} = x_{n} + E_{v_{j(n)}} for some j(n) ∈ J^{∗}. The construction runs as follows: one start with x_{1} := x(i) + E_{\overline{v}_{j}}. Then, assume that x_{n} is already constructed. If x_{n} satisfies the property (b) of (4.2.3) then we stop and we take n = N. Otherwise, there is at least one j(n) ∈ J^{∗} such that (x_{n} + l′_{[k]}, E_{v_{j(n)}}) > 0. Then write x_{n+1} := x_{n} + E_{v_{j(n)}}.

We claim that this procedure is finite and its final term x_{N} is precisely x(i + 1_{j}).

To prove this, it is enough to show that x_{n} ≤ x(i + 1_{j}) for any 1 ≤ n ≤ N; the minimality property (c) of x(i + 1_{j}) will do the rest. For n = 1 it is clear, so assume it is true for x_{n} and take x_{n+1} = x_{n} + E_{v_{j(n)}}. Then we have to verify that m_{v_{j(n)}}(x_{n+1}) ≤ m_{v_{j(n)}}(x(i + 1_{j})) or equivalently m_{v_{j(n)}}(x_{n}) < m_{v_{j(n)}}(x(i + 1_{j})). Suppose that this is not true, i.e. m_{v_{j(n)}}(x(i + 1_{j}) − x_{n}) = 0. Then (x_{n} + l′_{[k]}, E_{v_{j(n)}}) = (x(i + 1_{j}) + l′_{[k]}, E_{v_{j(n)}}) − (x(i + 1_{j}) − x_{n}, E_{v_{j(n)}}) ≤ 0, a contradiction.

Since in every step of the construction of the sequence we imposed that (x_{n} + l′_{[k]}, E_{v_{j(n)}}) > 0, we have χ_{k_{r}}(x_{n+1}) ≤ χ_{k_{r}}(x_{n}) for any 1 ≤ n < N. What we will prove next is that (x_{n} + l′_{[k]}, E_{v_{j(n)}}) is exactly 1 for any 1 ≤ n < N, hence χ_{k}(x_{n+1}) = χ_{k}(x_{n}), which proves the proposition.

The last claim is based on the following well–know fact proved by Laufer [9]

4.2.6. [Laufer’s Criterion] Let {z_{n}}_{n=1}^{M} be a computation sequence provided by Laufer’s algorithm [9] (similar as above with [k] = [k_{can}]) connecting z_{1} = E_{v_{j}} (for some j ∈ J) and the Artin’s fundamental cycle z_{M} = z_{min}, the minimal non–zero cycle of S′ ∩ L. (This means that z_{n+1} = z_{n} + E_{v_{n}} for some v_{n}, where (z_{n}, E_{v_{n}}) > 0.) Then the graph is rational if and only if in all steps 1 ≤ n < M one has (z_{n+1} − z_{n}, z_{n}) = 1.

Now we can finish the proof; we just have to take z_{n} := x_{n} − x(i) for 1 ≤ n ≤ N. It is easy to see that \{z_{n}\}_{n=1}^{N} is the beginning of a Laufer sequence \{z_{n}\}_{n=1}^{M} connecting E_{\overline{v}_{j}} with z_{min}. Moreover, the values (z_{n}, E_{v_{j}(n)}) will stay unmodified for every n if we replace our graph G with the rational graph G̃ by decreasing the decorations of the bad vertices. Therefore, by Laufer’s Criterion, (z_{n}, E_{v_{j(n)}}) = 1 in G̃, and consequently in G too. This shows that

1 = (x_{n} − x(i), E_{v_{j(n)}}) = (x_{n} + l′_{[k]}, E_{v_{j(n)}}) − (x(i) + l′_{[k]}, E_{v_{j(n)}}) ≥ (x_{n} + l′_{[k]}, E_{v_{j(n)}}).

Since (x_{n} + l′_{[k]}, E_{v_{j(n)}}) > 0, it must equal 1. □

4.2.7. Definition of the new lattice \overline{L}. Let us fix [k] and assume that the graph G admits ν bad vertices as above. Then define \overline{L} = (ℤ_{≥0})^{ν}, and the function \overline{w}_{0} : (ℤ_{≥0})^{ν} → ℤ by

(4.2.8)  \overline{w}_{0}(i_{1}, …, i_{ν}) := χ_{k_{r}}(x(i_{1}, …, i_{ν})).

Then \overline{w}_{0} defines a set \{\overline{w}_{q}\}_{q=0}^{ν} of compatible weight functions depending on [k], defined similarly as in (3.2.3), denoted by \overline{w}[k].

Theorem 4.2.9 (Reduction Theorem [8]). Let G be a negative definite connected graph and let k_{r} be the distinguished representative of a characteristic class. Suppose \overline{J} = \{\overline{v}_{j}\}_{j=1}^{ν} is a (minimal) family of bad vertices and (\overline{L}, \overline{w}[k]) is the new weighted lattice associated with \overline{J} and k_{r}. Then there is a graded ℤ[U]–module isomorphism

(4.2.10)  ℍ^{∗}(G, k_{r}) ≅ ℍ^{∗}(\overline{L}, \overline{w}[k]).

This and (3.3.2) imply (see also [17]) the following:

Corollary 4.2.11. Fix ν ≥ 1. If a graph G has a family of ν bad vertices then ℍ^{q}(G, k) = 0 for q ≥ ν and any k ∈ Char.

5. The lattice reduction for S_{−d}^{3}(K)

5.1. In this subsection we consider the graph G = G(M) determined in (2.3) and we apply the general machinery of the previous subsection (4.2).

We first identify the bad vertices of the graph G = G(M).

Proposition 5.1.1. The smallest family of bad vertices of G is exactly the family of (−1)– vertices \overline{J} = \{v_{0,ℓ}\}_{ℓ=1}^{ν}. In fact, if we replace all the (−1) decorations by (−2), then we get a rational graph.

Proof. It is not difficult to see (using e.g. Laufer’s Criterion (4.2.6)) that a rational graph can never have a vertex v with decoration e_{v} greater than or equal to 2 − ∂_{v}, where ∂_{v} is the valency of v (the number of adjacent vertices to v). Therefore, all the (−1)–vertices v_{0,ℓ} of G must be included in any family of bad vertices. We just have to show that decreasing their decoration we obtain a rational graph. In order to prove this we will use Laufer’s Criterion (4.2.6).

Let us replace all the −1 decorations by −2. In this way the graphs G_{ℓ}^{∗} (1 ≤ ℓ ≤ ν) and G are replaced by G̃_{ℓ}^{∗} and G̃ respectively. Let Ḡ_{ℓ}^{∗} be the subgraph of G̃ consisting of G̃_{ℓ}^{∗} and E_{+} and the connecting edge. In [15, (2.4.2)] is proved that Ḡ_{ℓ}^{∗} is a sandwiched graph, hence a rational graph. We wish to show that G̃ is rational too.

Let Z be the cycle on G̃ whose restriction on each G̃_{ℓ}^{∗} is Z_{ℓ} (for its definition see (2.3)), and the multiplicity of E_{+} is 1. Let z̄_{ℓ,min}^{∗} and z̃_{min} be the Artin’s fundamental cycle in Ḡ_{ℓ}^{∗} and G̃ respectively. Notice that for any E_{v} one has (E_{v}, Z) ≤ 0, hence Z ≥ z̃_{min}, hence m_{v_{+}}(z̃_{min}) = m_{v_{+}}(z̄_{ℓ,min}^{∗}) = 1. In particular, the Laufer computation sequence associated with the fundamental cycle z̃_{min} breaks into the computation sequences associated with the graphs Ḡ_{ℓ}^{∗}. Since this graphs are rational, at any step (z_{n+1} − z_{n}, z_{n}) = 1, cf. (4.2.6). Hence, this is true for the graph G̃ too, which by Laufer’s Criterion shows that G̃ is rational too. □

5.2. Next, we wish to determine the cycles l′_{[k]}. As we already mentioned in (2.4.3), L′/L is the cyclic group of order d generated by [E_{+}^{∗}].

Proposition 5.2.1. Fix a ∈ {0, 1, …, d − 1}, and set [k] = [k_{can} + 2aE_{+}^{∗}]. Then l′_{[k]} = aE_{+}^{∗}.

Proof. Note that S′ is generated over ℤ_{≥0} by the cycles E_{j}^{∗}. Write l′_{[k]} as ∑_{j} a_{j}E_{j}^{∗} for some a_{j} ∈ ℤ_{≥0}. Then aE_{+}^{∗} − ∑_{j} a_{j}E_{j}^{∗} = l ∈ L with l ≥ 0. We have to show that l = 0.

Let l_{+} be the coefficient of E_{+} in l; hence l_{+} ≥ 0. Since all the graphs G_{ℓ}^{∗} are unimodular, the coefficient of E_{+} in E_{+}^{∗} is 1/d. Hence, l can be written as l_{+}dE_{+}^{∗} + l̄ for some l̄ ∈ L supported on ∪_{ℓ}G_{ℓ}^{∗}. Note that for any j ≠ j_{+} one has 0 ≤ a_{j} = (l, E_{j}) = (l̄, E_{j}), hence, by the negative definiteness of the form we get (†) l̄≤ 0.

Next, for j = j_{+} we get 0 ≤ a_{j_{+}} = (l − aE_{+}^{∗}, E_{+}) = (l_{+}dE_{+}^{∗} − aE_{+}^{∗} + l̄, E_{+}) = −l_{+}d + a + (l̄, E_{+}). Since l̄ is supported on ∪_{ℓ}G_{ℓ}^{∗}, (†) implies that (l̄, E_{+}) ≤ 0. Hence a ≥ l_{+}d, which is possible (since l_{+} ≥ 0 and a ∈ {0, 1, …, d − 1}) only if l_{+} = 0. But then l̄= l, and since l ≥ 0 we get l̄≥ 0. This with (†) implies l̄= 0, hence l = 0 too. This shows the minimality of aE_{+}^{∗}. □

5.3. The cycles x(i). In the sequel we fix a and [k] as in (5.2.1), and we start to determine the cycles x(i) associated with [k], or rather their χ_{k_{r}}–values (using the recursive identities of (4.2.5)). According to this, we need to determine (x(i) + aE_{+}^{∗}, E_{0,ℓ}) for any of the bad vertices v_{0,ℓ} ∈ \overline{J}.

Obviously, the part of the cycle x(i_{1}, …, i_{ν}) contained in G_{ℓ}^{∗} only depends on i_{ℓ}, and we will denote it by x_{ℓ}(i_{ℓ}). Hence, for some m_{+}(i) ∈ ℤ_{≥0} one has

(5.3.1)  x(i_{1}, …, i_{ν}) = _{∑ℓ} x_{ℓ}(i_{ℓ}) + m_{+}(i)E_{+}.

The cycle x_{ℓ}(i_{ℓ}) is the same as for the ν = 1 case, hence we can read all its properties from [18].

For any ℓ, let Z_{ℓ} be an integral cycle supported on G_{ℓ}^{∗} defined by Z_{ℓ} := E_{0,ℓ}^{∗}(G_{ℓ}^{∗}) (i.e. the dual cycle of E_{0,ℓ} considered in the graph G_{ℓ}^{∗} embedded into L); it is also the compact part of the pull–back divisor of the germ f_{ℓ}, cf. (2.3).

Lemma 5.3.2. [18] Write i_{ℓ} = α_{ℓ}m_{ℓ} + β_{ℓ} with α_{ℓ} ∈ ℤ_{≥0} and 0 ≤ β_{ℓ} ≤ m_{ℓ} − 1. Then
x_{ℓ}(i_{ℓ}) = α_{ℓ}Z_{ℓ} + x_{ℓ}(β_{ℓ}).

Moreover,

(x_{ℓ}(i_{ℓ}), E_{0,ℓ}) = −α_{ℓ} + (x_{ℓ}(β_{ℓ}), E_{0,ℓ}) = −α_{ℓ} + \{ \begin{matrix}1 & if β_{ℓ} ∉ Γ_{ℓ}, \\ 0 & if β_{ℓ} ∈ Γ_{ℓ}. \end{matrix}

Next, we compute m_{+}(i). By definition, x(i) is the minimal cycle satisfying properties (a) and (b) of (4.2.3), therefore m_{+}(i) is the smallest integer satisfying:

0 ≥^{(b)} (x(i) + aE_{+}^{∗}, E_{+}) =^{(a)} ∑_{ℓ} i_{ℓ} − a − (d + ∑_{ℓ} m_{ℓ}) · m_{+}(i).

Hence, with the notation i := ∑_{ℓ} i_{ℓ} and M := ∑_{ℓ} m_{ℓ}:

(5.3.3)  m_{+}(i) = ⌈\frac{i − a}{d + M} ⌉ ,

where ⌈x⌉ represents the smallest integer greater than or equal to x. Summarized all this together, and using (4.2.5) and (4.2.9), we obtain the following facts.

Proposition 5.3.4. For M = S_{−d}^{3}(K) and any fixed a = 0, …, d − 1, the weights of the (first quadrant) lattice \overline{L} = (ℤ_{≥0})^{ν} associated with the bad vertices \{v_{0,ℓ}\}_{ℓ=1}^{ν} of the graph G(M) are recursively determined by \overline{w}(0) = 0 and

(5.3.5)  \overline{w}(i + 1_{ℓ}) − \overline{w}(i) = 1 + α_{ℓ} − ⌈\frac{i − a}{d + M} ⌉ − \{ \begin{matrix}1 & if β_{ℓ} ∉ Γ_{ℓ}, \\ 0 & if β_{ℓ} ∈ Γ_{ℓ} \end{matrix}

for any i = (i_{1}, …, i_{ℓ}, …, i_{ν}) ∈ (ℤ_{≥0})^{ν}. Using this weight, one has a ℤ[U]–module isomorphism:

(5.3.6)  ℍ^{∗}(G, k_{r}) ≅ ℍ^{∗}(\overline{L}, \overline{w}[k]).

In fact, even \overline{L} can be reduced more. First, we reduce it to a finite multi–rectangle:

Corollary 5.3.7. Set min := min_{ℓ}{m_{ℓ} − μ_{ℓ}}. Then the following facts hold.

(a) Set

α_{0} := ⌈\frac{M − min −a − ν}{d} ⌉ .

Let α ≥ α_{0} be any integer, and consider i such that

(5.3.8)  (αm_{1}, …, αm_{ν}) ≤ i < ((α + 1)m_{1}, …, (α + 1)m_{ν}).

Then \overline{w}(i + 1_{ℓ}) ≥ \overline{w}(i) for any ℓ ∈ \{1, …, ν\}.

(b)

ℍ^{∗}(G, k_{r}) ≅ ℍ^{∗}([0, α_{0}m_{1}] × ⋯ × [0, α_{0}m_{ν}], \overline{w}).

Proof. (a) Assume that i ≤ (α+1)M −min −ν (†). Since α ≥ α_{0} we get dα ≥ M −min −a−ν. This together with (†) implies (i − a)/(d + M) ≤ α, hence (a) follows from (5.3.5). Similarly, if i > (α + 1)M − min −ν, that is if ∑_{ℓ} β_{ℓ} > M − min −ν, then (since β_{ℓ} ≤ m_{ℓ} − 1) necessarily β_{ℓ} ∈ Γ_{ℓ} for all ℓ, hence the last contribution in (5.3.5) is zero. On the other hand, i ≤ (α + 1)M − 1, hence (i − a)/(d + M) ≤ α + 1.

(b) Let [0, i] be the multi–rectangle in \overline{L} consisting of the lattice points j with 0 ≤ j ≤ i and all the cubes with these vertices. We show that if i is as in (5.3.8), then for any ℓ one has:

ℍ^{∗}([0, i], \overline{w}[k]) = ℍ^{∗}([0, i + 1_{ℓ}], \overline{w}[k]).

In order to see this consider the inclusion ι : [0, i] → [0, i + 1_{ℓ}] and its retract ρ : [0, i + 1_{ℓ}] → [0, i] (i.e. ρ|[0, i]=identity) with ρ(j + 1_{ℓ}) = j for any j satisfying 0 ≤ j ≤ i and j_{ℓ} = i_{ℓ}. Using (4.2.5) for both j and i, we get for any j satisfying 0 ≤ j ≤ i and j_{ℓ} = i_{ℓ} that

\overline{w}(j + 1_{ℓ}) − \overline{w}(j) = \overline{w}(i + 1_{ℓ}) − \overline{w}(i) + (x(i) − x(j), E_{0,v_{ℓ}}).

Since i ≥ j and i_{ℓ} = j_{ℓ}, the cycle x(i) − x(j) is effective and it is supported on the complement of E_{0,v_{ℓ}}, hence (x(i) − x(j), E_{0,v_{ℓ}}) ≥ 0. This together with the inequality from part (a) provides \overline{w}(j + 1_{ℓ}) ≥ \overline{w}(j) for any j.

Hence, using the notation of (3.3.1), for any n the inclusion ι : S_{n} ∩[0, i] → S_{n} ∩[0, i+1_{ℓ}] and the retract ρ : S_{n} ∩ [0, i + 1_{ℓ}] → S_{n} ∩ [0, i] induce isomorphisms at the level of (simplicial) cohomology, hence the result follows from (3.3.2).

Finally, by induction and part (a), we get ℍ^{∗}([0, α_{0}m_{1}] × ⋯ × [0, α_{0}m_{ν}], \overline{w}) ≅ ℍ^{∗}(\overline{L}, \overline{w}). □

The multi–rectangle [0, α_{0}m_{1}] × ⋯ × [0, α_{0}m_{ν}] can be divided further in smaller parts: ‘small rectangles’ and ‘stripes’. A ‘small rectangle’ has the form

R_{α_{1},...,α_{ν}} := { x ∈ ℝ^{ν} : α_{ℓ}m_{ℓ} ≤ x_{ℓ} ≤ (α_{ℓ} + 1)m_{ℓ} for any ℓ }.

For simplicity, we also write R_{α} := R_{α,...,α}. Clearly, [0, α_{0}m_{1}] × ⋯ × [0, α_{0}m_{ν}] = ∪ R_{α_{1},...,α_{ν}}, where the union is over 0 ≤ α_{ℓ} < α_{0} for all ℓ. On the other hand, we also consider the following subsets for any α ∈ ℤ_{≥0} and x ∈ ℝ^{ν} with x := ∑ x_{ℓ}:

B_{α} := \{ x ≥ 0 : ⌈\frac{x − a}{d + M} ⌉ = α\},

T_{α}^{+} := \{ x ≥ 0 : \frac{x − a}{d + M} = α\},

T_{α}^{−} := \{ x ≥ 0 : \frac{x − a − 1}{d + M} = α − 1\}.

Obviously, T_{α}^{−} ∪ T_{α}^{+} ⊂ B_{α} are the two limiting planes of the α–stripe B_{α} (with T_{0}^{−} = ∅). For any i ∈ B_{α} the coefficient m_{+}(i) takes constant value m_{+}(i) = α. A direct computation shows:

Lemma 5.3.9. For any α ≥ 0 one has:

(a) αm ∈ ∪_{α′≤α} B_{α′};

(b) if (α + 1)m ∈ ∪_{α′≤α} B_{α′}, then α ≥ α_{0};

(c) if αm ∈ T_{α}^{−} then α > α_{0}.

In particular, if 0 ≤ α ≤ α_{0}, then αm ∈ B_{α} \ T_{α}^{−}; and αm ∈ T_{α}^{+} if and only if a = α = 0. Moreover, for 0 ≤ α, α′ ≤ α_{0} one has T_{α}^{+} ∩ R_{α′} ≠ ∅ if and only if α = α′.

Corollary 5.3.10. Consider a small rectangle R = R_{α_{1},...,α_{ν}} such that α := max_{ℓ}{α_{ℓ}} < α_{0}. Then:

(i) if i ∈ R then i ≤ (α + 1)m, hence R ⊂ ∪_{α′≤α+1}B_{α′};

(ii) if R ∩ B_{α+1} ≠ ∅, then R ⊂ B_{α} ∪ B_{α+1}; (iii) if R ∩ B_{α+1} = ∅, then R ⊂ ∪_{α′≤α}B_{α′}.

Proof. Use (5.3.9) and direct verification. Notice that the difference between the two i–values of the two opposite corners of R is M, which is less than d + M, showing (ii). □

The next proposition shows that in the right hand side of (5.3.7)(b) one can omit all the small rectangles except those of type R_{α}.

Proposition 5.3.11. ℍ^{∗}(G, k_{r}) ≅ ℍ^{∗}(∪_{0≤α<α_{0}}R_{α}, \overline{w}).

Proof. We proceed by induction. Assume that we already reduced the rectangle [0, α_{0}m_{1}] × ⋯ × [0, α_{0}m_{ν}] by suitable deformation retracts to

(∪_{α_{ℓ}≤α}R_{α_{1},...,α_{ν}}) ∪ (∪_{α<α′<α_{0}}R_{α′}).

Next, we wish to eliminate all the small rectangles not in ∪_{α_{ℓ}<α}R_{α_{1},...,α_{ν}}, and not equal to R_{α}. Consider such a small rectangle. Then max_{ℓ}{α_{ℓ}} = α, and ᾱ = min_{ℓ}{α_{ℓ}} < α. We consider the different cases of (5.3.10).

In the case of (ii), ⌈(i − a)/(d + M)⌉ ≥ α for all i ∈ R, hence for any i ∈ R and any index j with α_{j} < α one has \overline{w}(i + 1_{j}) ≤ \overline{w}(i). Hence R can be contracted in the (increasing) direction of the j–coordinate similarly as in the proof of (5.3.7)(b).

In the case of (iii), ⌈(i − a)/(d + M)⌉ ≤ α for all i ∈ R, hence for any i ∈ R and any index j with α_{j} = α one has \overline{w}(i + 1_{j}) ≥ \overline{w}(i). Hence R can be contracted in the (decreasing) direction of the j–coordinate similarly as above.

The order how we eliminate the small rectangles is the following. In the first series we eliminate those one which satisfy (iii) (i.e. R ∩ B_{α+1} = ∅) in the following order: first those with α_{1} = α (in decreasing direction of the first coordinate), then those from the remaining ones with α_{2} = α, and continuing in this way, finally all those remaining rectangles satisfying R ∩ B_{α+1} = ∅ with α_{ν} = α.

After this series of contractions, all those which still have to be contracted satisfy (ii), hence any coordinate j with α_{j} < α is a good candidate for a contraction in increasing direction. Then we contract in increasing direction of 1_{1} all the rectangles with α_{1} < α, then all the remaining ones with α_{2} < α in direction 1_{2}, and continuing in this way, at the end all those remaining one with α_{ν} < α in direction 1_{ν}. □

6. The lattice cohomology of S_{−d}^{3}(K) via the semigroups Γ_{ℓ}.

6.1. In this section we determine the weights of lattice points sitting in ∪_{0≤α<α_{0}}R_{α} in terms of the semigroups {Γ_{ℓ}}_{ℓ}. As a consequence, we get a description of the lattice cohomology of M := S_{−d}^{3}(K) in terms of these semigroups.

Lemma 6.1.1. For any 0 ≤ α ≤ α_{0} one has

\overline{w}(αm) = α(1 + a − δ) + dα(α − 1)/2.

In particular, these values depend only on δ (and a), but otherwise are independent on the structure of the semigroups Γ_{ℓ}.

Proof. First we verify that

(6.1.2)  χ_{can}(Z_{ℓ}) = m_{ℓ} − δ_{ℓ}.

In the graph G_{ℓ} let K be the canonical cycle, E the union of all exceptional divisors, and ∂_{v} the valency of a vertex v (in G_{ℓ} with arrowhead). Then, by adjunction relations, K + E = E_{0,ℓ}^{∗} + ∑_{v∈V(G_{ℓ})}(2 − ∂_{v})E_{v}^{∗}. Hence −1 + (K, E_{0,ℓ}^{∗}) = (K + E, E_{0,ℓ}^{∗}) = (E_{0,ℓ}^{∗}, E_{0,ℓ}^{∗}) + ∑_{v}(2 − ∂_{v})(E_{v}^{∗}, E_{0,ℓ}^{∗}). Note that (E_{0,ℓ}^{∗}, E_{0,ℓ}^{∗}) = (Z_{ℓ}, Z_{ℓ}) = −m_{ℓ} and (E_{v}^{∗}, E_{0,ℓ}^{∗}) = −m_{v}(Z_{ℓ}). Therefore, by A’Campo’s theorem [1] one gets (K, E_{0,ℓ}^{∗}) = 1 − m_{ℓ} − ∑_{v}(2 − ∂_{v})m_{v}(Z_{ℓ}) = −m_{ℓ} + μ_{ℓ}. Hence χ_{can}(Z_{ℓ}) = −(E_{0,ℓ}^{∗}, E_{0,ℓ}^{∗} + K)/2 = m_{ℓ}/2 − (−m_{ℓ} + μ_{ℓ})/2 = m_{ℓ} − δ_{ℓ}.

Next, we prove the statement of the lemma.

By (5.3.1) and (5.3.2) we have \overline{w}(αm) = χ_{k_{r}}(α ∑_{ℓ} Z_{ℓ} + m_{+}E_{+}). By (5.3.9) m_{+} = α. Then the result follows by induction via the identities χ_{k_{r}}(A) = χ_{can}(A) − a(A, E_{+}^{∗}), χ(A + B) = χ(A) + χ(B) − (A, B), χ_{can}(Z_{ℓ}) = m_{ℓ} − δ_{ℓ} and Z_{ℓ}^{2} = −m_{ℓ}. □

Lemma 6.1.3. Fix 0 ≤ α < α_{0}.

(a) Assume that i ∈ R_{α} ∩ B_{α}. Then for all ℓ one has

\overline{w}(i + 1_{ℓ}) − \overline{w}(i) = \{ \begin{matrix}0 & if β_{ℓ} ∉ Γ_{ℓ}, \\ 1 & if β_{ℓ} ∈ Γ_{ℓ}. \end{matrix}

Therefore, for any i ∈ R_{α}, i_{ℓ} = αm_{ℓ} + β_{ℓ}, with i ≤ α(d + M) + a + 1 (in particular for any i ∈ T_{α+1}^{−} ∩ R_{α} too), one has:

(6.1.4)  \overline{w}(i) − \overline{w}(αm) = ∑_{ℓ} #\{γ ∈ Γ_{ℓ} : γ ≤ β_{ℓ} − 1\}.
(b) Assume that i ∈ R_{α} ∩ B_{α+1}. Then for all ℓ with i + 1_{ℓ} ∈ B_{α+1} one has

\overline{w}(i + 1_{ℓ}) − \overline{w}(i) = − \{ \begin{matrix}1 & if β_{ℓ} ∉ Γ_{ℓ}, \\ 0 & if β_{ℓ} ∈ Γ_{ℓ}. \end{matrix}

Therefore, for any i ∈ R_{α}, i_{ℓ} = αm_{ℓ} + β_{ℓ}, with i ≥ α(d + M) + a + 1 (in particular for any i ∈ T_{α+1}^{−} ∩ R_{α} too), one has:

(6.1.5)  \overline{w}(i) − \overline{w}((α + 1)m) = ∑_{ℓ} #\{γ ∉ Γ_{ℓ} : γ ≥ β_{ℓ}\}.

All these and the results of the previous section summarized provide the following statements. In this final version, we decrease even the bound α_{0} (notice that M − min −ν ≥ μ − 1).

Theorem 6.1.6. (Final reduction theorem) Set

α̃_{0} := ⌈\frac{μ − a − 1}{d}⌉ ,

and for any 0 ≤ α < α̃_{0} set min T_{α+1}^{−} := min \{ \overline{w}(i) : i ∈ T_{α+1}^{−} ∩ R_{α}\}. Then the following facts hold:

(a) \overline{w}(αm) ≤ min T_{α+1}^{−}, \overline{w}((α + 1)m) ≤ min T_{α+1}^{−}.
(b) 𝔪_{k_{r}} = min χ_{k_{r}} = min_{0≤α≤α̃_{0}}\{ \overline{w}(αm) \}.
(c) Let α_{𝔪} be the smallest integer satisfying \overline{w}(α_{𝔪}m) = 𝔪_{k_{r}}. Then

ℍ_{red}^{0}(G, k_{r}) = ⨁_{0≤α<α_{𝔪}} T_{2\overline{w}(αm)}( min T_{α+1}^{−}−\overline{w}(αm))⊕ ⨁_{α_{𝔪}≤α<α̃_{0}} T_{2\overline{w}((α+1)m)}( min T_{α+1}^{−}−\overline{w}((α+1)m)).

(d)

rank_{ℤ} ℍ_{red}^{0}(G, k_{r}) = ∑_{0≤α<α_{𝔪}} ( min T_{α+1}^{−} − \overline{w}(αm)) + ∑_{α_{𝔪}≤α<α̃_{0}} ( min T_{α+1}^{−} − \overline{w}((α + 1)m)),

or

rank_{ℤ} ℍ_{red}^{0}(G, k_{r}) − 𝔪_{k_{r}} = ∑_{0≤α<α̃_{0}} ( min T_{α+1}^{−} − \overline{w}((α + 1)m)).

(e) For any q > 0 one has

ℍ^{q}(G, k_{r}) = ⨁_{0≤α<α̃_{0}} ℍ^{q}(R_{α}, \overline{w}).

Proof. First we prove statements (a)–(e) for α_{0} instead of α̃_{0}.

(a) and (b) follow from (6.1.3). (c) follows by a direct verification (or, using the general statement [13, (3.6)]). The first part of (d) is based on (c), while the second one uses the general formula [13, (3.8)], or can be determined from the first identity and from the fact: \overline{w}(α_{1}m) > \overline{w}(α_{2}m) for all α_{1} > α_{2} ≥ α_{𝔪}. For (e) use (3.3.2) and a Mayer–Vietoris argument.

Next, we prove that all the small rectangles R_{α} for α ≥ α̃_{0} can also be eliminated. We prove this by increasing induction: we verify that for any α′_{0} > α̃_{0} one has

(6.1.7)  ⨁_{0≤α<α′_{0}} ℍ^{q}(R_{α}, \overline{w}) = ⨁_{0≤α<α′_{0}−1} ℍ^{q}(R_{α}, \overline{w}),

i.e., the last small rectangle can be contracted into (α′_{0} − 1)m compatibly with the weights. By a direct verification,

(6.1.8)  α ≥ α̃_{0} if and only if for all i ∈ T_{α+1}^{−} ∩ R_{α} one has ∑_{ℓ} β_{ℓ} ≥ μ,

or, if there is an i ∈ T_{α+1}^{−} ∩ R_{α} with β_{ℓ} ≥ μ_{ℓ} for any ℓ. Since the largest element of ℤ \ Γ_{ℓ} is μ_{ℓ} − 1, such an i satisfies \overline{w}(i) = \overline{w}((α + 1)m) by (6.1.5). Hence, in (c), proved for the bound α_{0}, all the contribution in ℍ_{red}^{0}(G, k_{r}) corresponding to α ≥ α̃_{0} are zero, since min T_{α+1}^{−} = \overline{w}((α + 1)m).

For q > 0 we need a better argument, a sequence of contraction (which reproves the q = 0 case too). For any 0 ≤ β_{ℓ} ≤ m_{ℓ} set the rectangle

R_{α}^{β_{1},...,β_{ν}} := {x ∈ ℝ^{ν} : αm_{ℓ} ≤ x_{ℓ} ≤ αm_{ℓ} + β_{ℓ}}.

E.g., R_{α}^{m_{1},...,m_{ν}} = R_{α} and R_{α}^{0,...,0} = { αm}.

Fix β_{ℓ}^{o} ≥ μ_{ℓ} for all ℓ such that ∑_{ℓ} β_{ℓ}^{o} = αd + a + 1, i.e. the corresponding i is on T_{α+1}^{−}. Then we show that R_{α} contracts onto R_{α}^{β_{1}^{o},...,β_{ν}^{o}}. Assume that R_{α} is already contracted onto R_{α}^{β̄_{1},...,β̄_{ν}} with β̄_{ℓ} ≥ β_{ℓ}^{o} for all ℓ and β̄_{ℓ} > β_{ℓ}^{o} for some ℓ. Fix such an ℓ. We show that for any i ∈ R_{α}^{β̄_{1},...,β̄_{ν}} \ R_{α}^{β̄_{1},...,β̄_{ℓ}−1,...,β̄_{ν}} the inequality \overline{w}(i) ≥ \overline{w}(i − 1_{ℓ}) holds. This defines a contraction of R_{α}^{β̄_{1},...,β̄_{ν}} onto R_{α}^{β̄_{1},...,β̄_{ℓ}−1,...,β̄_{ν}} compatibly with the weights.

Indeed, if i > α(d + M) + a + 1 then \overline{w}(i) = \overline{w}(i − 1_{ℓ}) from (6.1.3)(b). If i ≤ α(d + M) + a + 1 then \overline{w}(i) = \overline{w}(i − 1_{ℓ}) + 1 from (6.1.3)(a). Hence R_{α} contracts onto R_{α}^{β_{1}^{o},...,β_{ν}^{o}}.

Finally R_{α}^{β_{1}^{o},...,β_{ν}^{o}} contracts onto the point αm using same type of contraction (but now in arbitrary order of the coordinates) via (6.1.3)(a) (repeating the case i ≤ α(d + M) + a + 1). □

Remark 6.1.9.

(a) The bound α̃_{0} in Theorem 6.1.6 is optimal. This can be seen already at the ℍ^{0}–level, cf. part (d), the statement (6.1.8) from the proof of the ‘final reduction’ and the comments after it.

(b) The cohomology ℍ^{q}(R_{α}, \overline{w}[k]) depends only on the \overline{w}–values on αm, on (α+1)m and along T_{α+1}^{−}.

Indeed, for any n consider S_{n} as in (3.3.1). Then for n < min T_{α+1}^{−} the space S_{n} ∩ R_{α} has the same homotopy type as S_{n} ∩ {αm, (α + 1)m}; while for n ≥ min T_{α+1}^{−} it has the homotopy type of the suspension of S_{n} ∩ T_{α+1}^{−}. In particular, all the nontrivial homogeneous elements of ℍ^{q}(R_{α}, \overline{w}[k]) (q > 0) have degree ≥ min T_{α+1}^{−}.

In fact, one can define on each T_{α}^{−} a ‘generalized lattice structure’ and the corresponding lattice cohomology. This can be done, similarly as in (3.3.1) in two different ways. The simplest definition is via the 𝕊^{∗}–representation: one defines

(6.1.10)  ℍ^{∗}(T_{α}^{−}, \overline{w}) := ⊕_{n≥min T_{α}^{−}}H^{∗}(S_{n} ∩ T_{α}^{−}, ℤ).

Equivalently, one can define a lattice decomposition of T_{α}^{−} (or, just of T_{α}^{−} ∩ R_{α−1}) as follows: the ‘generalized q–cubes’ of T_{α}^{−} are the intersection of (q + 1)–cubes (as in (3.2.1)) with T_{α}^{−}. Lemma (6.1.3) guarantees that if □_{q+1} ∩ T_{α}^{−} ≠ ∅, then

max{w(v) : vertex v of □_{q+1}} = max{w(v) : vertex v of □_{q+1} ∩ T_{α}^{−}}.

Therefore, the right hand side of this identity defines an intrinsic set of weight functions on the generalized cubes of T_{α}^{−}, hence a cochain complex too determining the corresponding lattice cohomology.

With this ‘generalized lattice structure’ of T_{α}^{−}, one has

(6.1.11)  ℍ^{q}(R_{α}, \overline{w}) = ℍ_{red}^{q−1}(T_{α+1}^{−}, \overline{w}) for q > 0.

(c) In most of the notations above, we have omitted the symbol a codifying the characteristic element k_{r}. In fact, for any α ≥ 0 and a ∈ {0, …, d − 1}, T_{α+1}^{−} is

T_{α+1,a}^{−} := { i : i_{ℓ} = αm_{ℓ} + β_{ℓ}; ∑_{ℓ} β_{ℓ} = αd + a + 1}.

Note that when α runs over ℤ_{≥0} and a ∈ {0, …, d − 1}, the integer n = αd + a runs over ℤ_{≥0}. This motivates to consider for any n ∈ ℤ_{≥0}

(6.1.12)  𝕋_{n} := {(β_{1}, …, β_{ν}) ∈ [0, m_{1}] × ⋯ × [0, m_{ν}] : ∑_{ℓ} β_{ℓ} = n + 1}.

Then for d and a fixed 𝕋_{αd+a} = T_{α+1,a}^{−}. Up to the shift \overline{w}(α𝔪) (whose values is given in (6.1.1)), and which is constant on each 𝕋_{n}, but otherwise depends on α = ⌊n/d⌋, the weights on 𝕋_{n} ∩ ℤ^{ν} are given by the right hand side of (6.1.4), or (6.1.5). The second one, namely (6.1.5), in which case the shift is \overline{w}((α + 1)m), gives weights:

(6.1.13)  \overline{W} ((β_{1}, …, β_{ν})) = ∑_{ℓ} #\{γ ∉ Γ_{ℓ} : γ ≥ β_{ℓ}\}.

Then the weight function \overline{W} restricted on all the level sets \{𝕋_{n}\}_{n≥0} of (ℤ_{≥0})^{ν} measures the very subtle distribution properties of the semigroups {Γ_{ℓ}}_{ℓ}, and they provide, as a universal object all the lattice cohomologies ℍ^{q}(G(d), k_{r}) for all q > 0 (and with the additional data of the shifts \overline{w}((α+1)𝔪), all the lattice cohomologies ℍ^{0}(G(d), k_{r}) too) for all the possible values d and a. Here, and in the next discussion, we denote the dependence of G on d by G(d).

More precisely, for any d and a ∈ {0, …, d − 1}, one has:

(6.1.14)  ℍ^{q}(G(d), k_{can} + 2aE_{+}^{∗}) = ⨁_{{n≡a (mod d) \atop n≥0}} ℍ_{red}^{q−1}(𝕋_{n}, \overline{W})[s_{n,d}]  (q > 0),

where s_{n,d} indicates a value of the shift 2\overline{w}((α + 1)m) = 2(α + 1)(1 + a − δ) + d(α + 1)α (with α = ⌊n/d⌋). Moreover, the values \{min \overline{W} |𝕋_{n}\}_{n} and s_{n,d} determine all the cohomology groups ℍ^{0}(G(d), k_{r}) too. The second identity of (6.1.6)(d) together with (6.1.5) reads as:

(6.1.15)  −𝔪_{k_{r}} + rank ℍ_{red}^{0}(G(d), k_{can} + 2aE_{+}^{∗}) = ∑_{{n≡a (mod d) \atop n≥0}} min\{\overline{W} restricted to 𝕋_{n}\}.

In particular (see (3.2.5) for the definition of eu(ℍ^{∗})), for any fixed d > 0 and a ∈ {0, …, d − 1} one has:

(6.1.16)  −eu(ℍ^{∗}(G(d), k_{can} + 2aE_{+}^{∗})) = ⨁_{{n≡a (mod d) \atop n≥0}} eu(ℍ^{∗}(𝕋_{n}, \overline{W})).

(d) Summing up (6.1.14) over a, for any fixed d > 0 and q > 0,

(6.1.17)  ℍ^{q}(G(d)) = ⨁_{a=0}^{d−1} ℍ^{q}(G(d), k_{can} + 2aE_{+}^{∗}) = ⨁_{n≥0} ℍ_{red}^{q−1}(𝕋_{n}, \overline{W})[s_{n,d}].

On the right hand side of (6.1.17) the numbers s_{n,d} depend on d, but the rank of the right hand side is independent of d. In particular, up to shifts of different direct sum blocks, ⊕_{q>0}ℍ^{q}(G(d)) is independent on the choice of the integer d. This can also be deduced from the surgery exact sequences (Theorem A and Theorem B) of [17] (applied for the special vertex v_{+}, and using the fact that G \ v_{+} is rational).

Example 6.1.18.

(a) Assume that for some d and a one gets α̃_{0} = 0. Then ℍ_{red}^{∗}(G, k_{r}) = 0, and ℍ^{0}(G, k_{r}) = T_{0}^{+}.

(b) Assume that for some d and a one gets α̃_{0} = 1. Then ℍ^{∗}(G, k_{r}) = ℍ^{∗}(R_{0}, \overline{w}), hence everything is determined by T_{1,a}^{−}. Indeed,

min T_{1,a}^{−} = min { ∑_{ℓ} #{γ ∈ Γ_{ℓ} : γ ≤ i_{ℓ} − 1}, where ∑_{ℓ} i_{ℓ} = a + 1 }

= min { ∑_{ℓ} #{γ ∉ Γ_{ℓ} : γ ≥ i_{ℓ}}, where ∑_{ℓ} i_{ℓ} = a + 1 } + 1 + a − δ,

𝔪_{k_{r}} = min{0, 1 + a − δ}, ℍ_{red}^{0}(G, k_{r}) is generated by one element of degree 2 max{0, 1 + a − δ}, rank ℍ_{red}^{0}(G, k_{r}) = min T_{1,a}^{−} − max{0, 1 + a − δ}, and finally for q > 0 one has ℍ^{q}(G, k_{r}) = ℍ_{red}^{q−1}(T_{1,a}^{−}, \overline{w}).

(c) If d ≥ μ − 1 then α̃_{0} = 1 for a < μ − 1, and α̃_{0} = 0 for a ≥ μ − 1.

Remark 6.1.19. Assume that we know all the cohomology groups {ℍ^{∗}(G(d), k_{r})}_{k_{r}} for some specific, very large d. Then using them, and also the values \overline{w}(αm) = α(1 + a − δ) + dα(α − 1)/2 for all d > 0, we can recover all the lattice cohomologies {ℍ^{∗}(G(d), k_{r})}_{k_{r}} for any d > 0. Indeed, e.g. for any n ≥ 0 with the choice of d > max{n, ν} one has

ℍ_{red}^{q−1}(𝕋_{n}, \overline{W}) = ℍ^{q}(G(d), k_{can} + 2nE_{+}^{∗})[−s_{n,d}] (q > 0);

and

min\{ \overline{W} |𝕋_{n}\} = rank_{ℤ} ℍ_{red}^{0}(G(d), k_{can} + 2nE_{+}^{∗}) + max\{0, 1 + n − δ\}.

Remark 6.1.20. In the above discussion, e.g. in (6.1.9), the space 𝕋_{n} (intersection of a simplex with a rectangle) can be replaced by the supporting simplex. Indeed, set

(6.1.21)  Σ_{n} := {(β_{1}, …, β_{ν}) ∈ (ℝ_{≥0})^{ν} : ∑ β_{ℓ} = n + 1}.

The simplex Σ_{n} has a natural ‘generalized cube–decomposition’ similarly as 𝕋_{n}, cf. (6.1.9)(b), where the 0–cubes are the lattice points Σ_{n} ∩ ℤ^{ν}. Moreover, we take the same weight function on Σ_{n} ∩ ℤ^{ν} as for 𝕋_{n} ∩ ℤ^{ν} given in (6.1.13).

First assume that n + 1 ≤ M, i.e. 𝕋_{n} = Σ_{n} ∩ [0, m_{1}] × ⋯ × [0, m_{ν}] is non–empty. (In fact, if α < α̃_{0} then n = αd + a < μ − 1 < M.) We claim that the inclusion (𝕋_{n}, \overline{W}) ↪ (Σ_{n}, \overline{W}) induces an isomorphism of the corresponding lattice cohomologies ℍ^{∗}(Σ_{n}, \overline{W}) → ℍ^{∗}(𝕋_{n}, \overline{W}).

This can be verify as follows. Let us define an application φ : Σ_{n} ∩ ℤ^{ν} → Σ_{n} ∩ ℤ^{ν} as follows. If β = (β_{1}, …, β_{ν}) ∈ 𝕋_{n} ∩ ℤ^{ν} then φ(β) = β. Otherwise, let i be the smallest index for which β_{i} > M_{i}, and j the largest index for which β_{j} < M_{j}. Then φ(β) := (β_{1}, …, β_{i} − 1, …, β_{j} + 1, …, β_{ν}). One can check that \overline{W} (φ(β)) ≤ \overline{W} (β), repeated application of φ maps Σ_{n} ∩ ℤ^{ν} into 𝕋_{n} ∩ ℤ^{ν}, φ extends to the level of arbitrary dimensional cubes, hence induces the desired isomorphism at the level of lattice cohomology.

If n + 1 ≥ μ, then for a certain β with all β_{ℓ} ≥ μ_{ℓ} one has \overline{W} (β) = 0, hence min \overline{W} = 0. Moreover, by similar retraction as above one can show that ℍ_{red}^{∗}(Σ_{n}, \overline{W}) = 0.

In particular, in all the discussion above, H_{red}^{∗}(𝕋_{n}, \overline{W}) can be replaced by H_{red}^{∗}(Σ_{n}, \overline{W}) for all n ≥ 0.

Remark 6.1.22. Let ℍ_{r}^{q} denote the r–degree part of ℍ^{q}. We expect that the generating function ∑_{n,q,r} rank ℍ^{q}(𝕋_{n})_{r}h^{q}t^{n}s^{r} ∈ ℤ[[h, t, s]] can be organized into a ‘nice’ expression in terms of the semigroups (and similar result is valid for ∑_{q,r} rank ℍ^{q}(G(d))_{r}h^{q}s^{r} ∈ ℤ[[h, s]] too).

7. The normalized Euler–characteristic and the Seiberg–Witten invariant

7.1. The Euler–characteristic formula. Consider the graphs {G_{ℓ}^{∗}}_{ℓ=1}^{ν}, d > 0 and the graph G(d) as in (2.3). Let s be the number of vertices of G(d). The main result of this section identified the ‘normalized Euler characteristic’ of the lattice cohomology with the ‘normalized Seiberg–Witten invariant’:

Theorem 7.1.1. For any a ∈ {0, …, d − 1} one has:

(7.1.2)  sw(S_{−d}^{3}(K), [k_{can} + 2aE_{+}^{∗}]) + \frac{(k_{can} + 2aE_{+}^{∗})^{2} + s}{8} = −eu(ℍ^{∗}(G(d), k_{can} + 2aE_{+}^{∗})).

Proof. Consider the polynomial Q(t) = ∑_{i} α_{i}t^{i} as in (2.4.3). Then by (2.4.5)

sw(S_{−d}^{3}(K), [k_{can} + 2aE_{+}^{∗}]) + \frac{(k_{can} + 2aE_{+}^{∗})^{2} + s}{8} = − ∑_{{n≡a (mod d) \atop n≥0}} α_{n}.

This compared with (6.1.16) shows that it is enough to prove α_{n} = −eu(ℍ^{∗}(𝕋_{n}, \overline{W})) for any n ≥ 0. In particular, the identity (7.1.2) is independent of d, it is enough to prove it for d very large compared with μ (determined by the semigroups Γ_{ℓ}), and any 0 ≤ a < d. In that case, we are in the situation of Example 6.1.18.

The point is that the wanted identity depends merely on the semigroups {Γ_{ℓ}}_{ℓ}, and in fact, it is not even important that they are semigroups of ℤ_{≥0}. Indeed, the statement follows from the following combinatorial property of finite subsets of ℤ_{>0}. □

Proposition 7.1.3. Consider ν finite subsets Γ_{ℓ}^{c} ⊂ ℤ_{>0} (1 ≤ ℓ ≤ ν), and define Γ_{ℓ} := ℤ_{≥0} \ Γ_{ℓ}^{c}. (Note that 0 ∈ Γ_{ℓ}, and even Γ_{ℓ}^{c} = ∅ is allowed). Let δ_{ℓ} be the cardinality of Γ_{ℓ}^{c}, and set δ := ∑_{ℓ} δ_{ℓ}. Consider the polynomials Δ_{l}(t) defined by (1 − t) ∑_{γ∈Γ_{ℓ}} t^{γ}, and set Δ(t) := ∏_{ℓ} Δ_{ℓ}(t). Then Δ(t) can be written as

Δ(t) = 1 + δ(t − 1) + (t − 1)^{2}Q(t)

for some polynomial Q(t). Denote its coefficients by Q(t) = ∑_{n≥0} α_{n}t^{n}.

On the other hand, one fixes for any ℓ an integer μ_{ℓ} > max Γ_{ℓ}^{c} (if Γ_{ℓ}^{c} = ∅, then one takes an arbitrary μ_{ℓ} ≥ 0). Next, for any integer a ≥ 0 one can consider the lattice cohomology ℍ^{∗}(R, w) of R = [0, μ_{1}] × ⋯ × [0, μ_{ν}] associated with the weight

w_{a}(i) = w(i) := ∑_{ℓ} #{γ ∈ Γ_{ℓ}^{c} : γ ≥ i_{l}} − δ + min{i, 1 + a}

= ∑_{ℓ} #{γ ∈ Γ_{ℓ} : γ ≤ i_{l} − 1} + min{0, −i + 1 + a}.

where i = (i_{1}, …, i_{ν}) ∈ R ∩ ℤ^{ν} and i = ∑_{ℓ} i_{ℓ}. Then its normalized Euler characteristic satisfies

eu((ℍ^{∗}(R, w_{a})) = α_{a}.

(In particular, it is independent of the choice of the integers μ_{ℓ}, which can be taken even infinity.)

Proof. It is convenient to denote the coefficient of t^{a} of a polynomial P (t) by P |_{a}. Let us start with some remarks.

(a) If we write χ_{Γ_{ℓ}^{c}}(t) := ∑_{γ∈Γ_{ℓ}^{c}} t^{γ} then

(7.1.4)  Δ_{ℓ} = 1 + (t − 1)χ_{Γ_{ℓ}^{c}} = 1 + (t − 1)δ_{ℓ} + (t − 1)^{2}Q_{ℓ}(t),

where Q_{ℓ} = ∑_{γ∈Γ_{ℓ}^{c}}(1 + t + … + t^{γ−1}). Hence Q_{ℓ}|_{a} = #{γ ∈ Γ_{ℓ}^{c} : γ > a}. Q_{ℓ}|_{0} = δ_{ℓ} since 0 ∉ Γ_{ℓ}^{c}.

(b) Since χ_{Γ_{ℓ}^{c}}|_{0} = 0 and Q = ∑_{ℓ} Q_{ℓ} + ∑_{t} Q̃_{t}, where each Q̃_{t} is multiple of at least one χ_{Γ_{ℓ}^{c}}, one also gets that Q|_{0} = δ.

(c) The proof runs over double induction: we will assume that the property is true for ν − 1 finite subsets, and also for any ν finite subsets with total cardinality δ − 1, and we will verify that it is true for ν subsets with total cardinality δ as well. In order to run the induction, we will extend the property to a = −1 as well. The weight function w_{−1} will be defined by the same formula as in the theorem, while Q|_{−1}, by definition, is δ.

(d) Let us verify the extended property for ν = 1 and any a ≥ −1. In this case w_{a} is (non necessarily strictly) increasing on [0, 1+a] and decreasing on [1+a, μ_{1}], with values 0 at 0, −δ +1+a at μ_{1}, and Q_{1}|_{a} − δ + 1 + a at 1 + a (cf. part (a)). Hence the identity eu(ℍ^{∗}(R, w_{a})) = Q_{1}|_{a} follows.

(e) Next, we verify the other ‘starting case’ too, namely δ = 0, that is, all the sets Γ_{ℓ}^{c} are empty. Then w_{a}(i) = min{i, 1 + a}, hence ℍ_{red}^{∗} = 0, and ℍ^{0} = T_{0}^{+}, hence eu(ℍ^{∗}) = 0. But Q = 0 too.

(f) Now we will verify the inductive step. We start with ν finite sets as in the statement of the theorem. We assume that Γ_{ν}^{c} is non–empty, and set γ_{0} := max Γ_{ν}^{c}. This situation (called ‘old’ situation) will be compared with that case (called ‘new’ situation), when all Γ_{ℓ}^{c} for ℓ < ν are preserved, but Γ_{ν}^{c} is replaced by Γ_{ν}^{c} \ {γ_{0}}.

The corresponding ‘old’ and ‘new’ invariants (with obvious notation) can be compared. Indeed, by (a), Δ_{ν}^{old} − Δ_{ν}^{new} = (t − 1)t^{γ_{0}}. Hence, Δ^{old} − Δ^{new} = (t − 1)t^{γ_{0}} · Δ_{<ν}(t) (where the index < ν indicated the corresponding invariant associated with the first ν − 1 subsets), or

(7.1.5)  Δ^{old} − Δ^{new} = t^{γ_{0}}(t − 1) · (1 + (t − 1)(δ − δ_{ν}) + (t − 1)^{2}Q_{<ν}).

Since in the ‘new’ case the total cardinality is δ − 1, from the definition of the Q–polynomials we get

(7.1.6)  Δ^{old} − Δ^{new} = t − 1 + (t − 1)^{2}(Q^{old} − Q^{new}).

The identities (7.1.5) and (7.1.6) combined provides

(7.1.7)  Q^{old} − Q^{new} = \frac{t^{γ_{0}} − 1}{t − 1} + t^{γ_{0}}(δ − δ_{ν}) + t^{γ_{0}}(t − 1)Q_{<ν}.

Next, we fix a ≥ −1, and we will treat the lattice cohomologies associated with the two situtions: the corresponding weigh functions will be denoted by w^{old}, respectively w^{new}. In order to simplify the notation, we write eu(R, w) instead of eu(ℍ^{∗}(R, w)). For both weight functions, by Mayer–Vietoris argument (cf. (3.3.4)) we have

\begin{matrix}eu(R, w) = eu(R_{i_{ν}≤γ_{0}}, w) + eu(R_{γ_{0}≤i_{ν}≤γ_{0}+1}, w) + eu(R_{i_{ν}≥γ_{0}}, w) \\ −eu(R_{i_{ν}=γ_{0}}, w) − eu(R_{i_{ν}=γ_{0}+1}, w). \end{matrix} ; (7.1.8)

Now we have to notice several facts. First, w^{new}(i) = w^{old}(i) if i_{ν} ≤ γ_{0}, and w^{new}(i) = w^{old}(i) + 1 if i_{ν} ≥ γ_{0} + 1. Note also that if on a rectangle the weight function increases by N then the euler characteristic decreases by N (because of the −𝔪–contribution). Also, the choice of γ_{0} implies that for any i with i_{ν} = γ_{0} one has w^{old}(i) ≥ w^{old}(i + 1_{ν}). Hence (R_{γ_{0}≤i_{ν}≤γ_{0}+1}, w^{old}) contracts to (R_{i_{ν}=γ_{0}+1}, w^{old}), showing that in (7.1.8) for w^{old} one has the cancelation eu(R_{γ_{0}≤i_{ν}≤γ_{0}+1}, w^{old}) = eu(R_{i_{ν}=γ_{0}+1}, w^{old}).

The computation will be separated in several subcases. Case a < γ_{0}. In this case for any i with i_{ν} = γ_{0} one has w^{old}(i) = w^{old}(i + 1_{ν}) + 1, hence w^{new}(i) = w^{new}(i + 1_{ν}), showing by the same contraction argument that eu(R_{γ_{0}≤i_{ν}≤γ_{0}+1}, w^{new}) = eu(R_{i_{ν}=γ_{0}+1}, w^{new}). Hence considering (7.1.8) for w^{old} and w^{new}, their difference provides

eu(R, w^{old}) − eu(R, w^{new}) = 1.

This should be compared with (7.1.7). Since by assumption the statement of the theorem is true for the ‘new’ case, we will have the same fact for the ‘old’ case provided that (Q^{old} − Q^{new})|_{a} = 1. But this is clear from the right hand side of (7.1.7) since a < γ_{0}. Case a ≥ γ_{0}. In this case for any i with i_{ν} = γ_{0} one has w^{new}(i) ≤ w^{new}(i + 1_{ν}), hence eu(R_{γ_{0}≤i_{ν}≤γ_{0}+1}, w^{new}) = eu(R_{i_{ν}=γ_{0}}, w^{new}). Hence we get

(7.1.9)  eu(R, w^{old}) − eu(R, w^{new}) = eu(R_{i_{ν}=γ_{0}+1}, w^{old}) − eu(R_{i_{ν}=γ_{0}}, w^{old}).

In order to determine the right hand side, we will analyze eu(R_{i_{ν}=k}, w) for any integer k ≤ γ_{0} + 1 and w = w^{old}. For any i with i_{ν} = k one has the identity w(i) = w_{i_{ν}=k}(i) + N(k), where

w_{i_{ν}=k}(i) = ∑_{ℓ<ν} #{γ ∈ Γ_{ℓ}^{c} : γ ≥ i_{ℓ}} − (δ − δ_{ν}) + min{i − k, 1 + a − k},

an expression which is the weight function associated with the first ν − 1 subsets and a − k, and N(k) is a constant independent on i, namely #{γ ∈ Γ_{ν}^{c} : γ ≥ k} − δ_{ν} + k. Since (R_{i_{ν}=k}, w) satisfies the statement of the theorem by the inductive assumption, we have:

eu(R_{i_{ν}=k}, w) = Q_{<ν}|_{a−k} − N(k) = Q_{<ν}|_{a−k} − #{γ ∈ Γ_{ν}^{c} : γ ≥ k} + δ_{ν} − k.

Hence the right hand side of (7.1.9) is

(7.1.10)  eu(R_{i_{ν}=γ_{0}+1}, w^{old}) − eu(R_{i_{ν}=γ_{0}}, w^{old}) = Q_{<ν}|_{a−γ_{0}−1} − Q_{<ν}|_{a−γ_{0}}.

Since for a ≥ γ_{0} the a–th coefficient of the right hand side of (7.1.7) equals the right hand side of (7.1.10), the inductive step follows again. □

8. Connection with Heegaard–Floer homology

8.1. Review of Heegaard–Floer homology. Let M be an oriented 3–manifold; for simplicity we assume again that M as a rational homology sphere. The Heegaard–Floer homology HF^{+}(M) was introduced by Ozsváth and Szabó in [23]. It is a ℤ[U]-module with a ℚ-grading compatible with the ℤ[U]-action, where deg(U) = −2. Additionally, HF^{+}(M) also has an (absolute) ℤ_{2}-grading; HF_{even}^{+}(M), respectively HF_{odd}^{+}(M), denote the part of HF^{+}(M) with the corresponding parity. Moreover, HF^{+}(M) has a natural direct sum decomposition of ℤ[U]-modules (compatible with all the gradings) corresponding to the spin^{c}-structures of M:

HF^{+}(M) = ⊕_{[k]∈Spin^{c}(M)} HF^{+}(M, [k]).

For any spin^{c}-structure [k], one has a graded ℤ[U]-module isomorphism

HF^{+}(M, [k]) = T_{d(M,[k])}^{+} ⊕ HF_{red}^{+}(M, [k]),

where HF_{red}^{+}(M, [k]) has a finite ℤ-rank and an induced ℤ_{2}-grading. One also considers

χ(HF^{+}(M, [k])) := rank_{ℤ} HF_{red,even}^{+}(M, [k]) − rank_{ℤ} HF_{red,odd}^{+}(M, [k]).

Then one recovers the Seiberg-Witten topological invariant of (M, [k]) (see [27]) via

(8.1.1)  sw(M, [k]) := χ(HF^{+}(M, [k])) − d(M, [k])/2.

With respect to the change of orientation the above invariants behave as follows: d(M, [k]) = −d(−M, [k]) and χ(HF^{+}(M, [k])) = −χ(HF^{+}(−M, [k])).

If M is an integral homology sphere then for the unique spin^{c}–structure sw(M) equals the Casson invariant λ(M) (normalized as in [10] (4.7)).

8.2. Lattice homology and Heegaard–Floer homology. In the sequel we set again M = S_{−d}^{3}(K). First, assume that ν = 1. Then G is almost rational, hence ℍ^{q}(−M) = 0 for any q > 0, cf. (4.2.11). Moreover, by [25, 13, 15], we have:

Theorem 8.2.1. Assume that ν = 1. For any [k] ∈ Spin^{c}(M)
HF_{odd}^{+}(−M, [k_{r}]) = 0,

and

HF_{even}^{+}(−M, [k_{r}]) = ℍ^{0}(G, k_{r})[ − \frac{k_{r}^{2} + |J |}{4}].

In particular

(8.2.2)  d(M, [k_{r}]) = max_{k′∈[k_{r}]} \frac{(k′)^{2} + |J |}{4} = \frac{k_{r}^{2} + |J |}{4} − 2𝔪_{k_{r}}.

The next theorem generalizes this fact for ν = 2. Again, by (4.2.11) one gets ℍ^{q}(−M) = 0 for q > 1. Additionally:

Theorem 8.2.3. Assume that ν = 2. Then for any [k_{r}] ∈ Spin^{c}(M)
HF_{even}^{+}(−M, [k_{r}]) = ℍ^{0}(G, k_{r})[ −\overline{d}]

and

HF_{odd}^{+}(−M, [k_{r}]) = ℍ^{1}(G, k_{r})[ −\overline{d}],

where \overline{d} := (k_{r}^{2} + |J |)/4. In particular, (8.2.2) is valid in this case too.

Proof. Let G be the plumbing graph of S_{−d}^{3}(K) as above, G^{−} the graph obtained from G via replacing the decoration −1 of v_{0,1} with −2, and finally, G^{†} the graph obtained from G by removing the vertex v_{0,1} and all adjacent edges. From (5.1.1) follows that

(8.2.4)  both graphs G^{−} and G^{†} are almost rational.

Let HF_{e/o}^{+}(−Γ) be the even/odd Heegaard–Floer homology of the −M(Γ), where M(Γ) is the plumbed 3–manifold associated with Γ. Then one has the following commutative diagram:

0 ⟶ HF_{e}^{+}(−G) ⟶ HF_{e}^{+}(−G^{−}) ⟶ HF_{e}^{+}(−G^{†}) ⟶ HF_{o}^{+}(−G) ⟶ 0
↓^{T_{G}}  ↓^{T_{G^{−}}}  ↓^{T_{G^{†}}}
0 ⟶ ℍ^{0}(G) ⟶ ℍ^{0}(G^{−}) ⟶ ℍ^{0}(G^{†}) ⟶ ℍ^{1}(G) ⟶ 0

The first line is exact by [25, §2], and using the fact that HF_{o}^{+}(−G^{−}) = HF_{o}^{+}(−G^{†}) = 0. This type of vanishing is proved in [13] for almost rational graphs, cf. (8.2.4).

The second line is exact by [17], and from the vanishing ℍ^{1}(G^{−}) = ℍ^{1}(G^{†}) (here we use again (8.2.4) and (4.2.11)). The natural maps T_{G}, T_{G^{−}} and T_{G^{†}} are constructed in [25]. Since G^{−} and G^{†} are almost rational, the maps T_{G^{−}} and T_{G^{†}}, are isomorphisms by [13]. Hence, T_{G} is also an isomorphisms, and and T_{G^{†}} induces an isomorphism HF_{o}^{+}(−G) → ℍ^{1}(G) as well. □

Remark 8.2.5.

(a) For any ν ≥ 1 the identities (7.1.1) and (8.1.1) imply that

(8.2.6)  ∑_{q}(−1)^{q} rank_{ℤ} ℍ_{red}^{q}(G, k_{r}) = χ(HF^{+}(−M, [k_{r}])) + \frac{d(M, [k_{r}])}{2} − \frac{k_{r}^{2} + |J |}{8} + 𝔪_{k_{r}}.

We predict, cf. Conjecture 5.2.4 in [16], that

HF_{red,even}^{+}(−M, [k_{r}]) = ⨁_{q even} ℍ_{red}^{q}(G, k_{r})[−\overline{d}], and HF_{red,odd}^{+}(M, [k_{r}]) = ⨁_{q odd} ℍ_{red}^{q}(G, k_{r})[−\overline{d}]

and (8.2.2) is valid for any ν ≥ 1. This is compatible with the above theorems, and with the identity (8.2.6) already proved.

(b) The main obstruction to prove the conjectured isomorphisms connecting the Heegaard– Floer homologies with lattice cohomologies is that at this moment we know no natural application connecting them, except level q = 0 (as in the proof of (8.2.3)). Both isomorphisms obtained in the proof of Theorem 8.2.3 are induced by 0–level morphisms.

(c) Although Theorem 8.2.3 does not answer the arbitrary case, it is still important: it is the first family which identifies completely a non–zero HF_{odd}^{+}(−M, [k]) in terms of the combinatorial lattice cohomology.

9. Example. The case of ν cusps of type (2, 3).

9.1. The setup. Assume that for all ℓ the knot K_{ℓ} is the (2, 3) torus knot, i.e. Γ_{ℓ} = {0, 2, 3, ⋯ }, Δ_{ℓ} = t^{2} − t + 1, δ_{ℓ} = 1. Hence δ = ν.

Our goal is to analyze the lattice cohomology ℍ_{red}^{∗}(Σ_{n}, \overline{W}) for any n ≥ 0, cf. (6.1.20). Recall that Σ_{n} is the real simplex {β ∈ ℝ_{≥0}^{ν} : ∑ β_{ℓ} = n + 1} with lattice points LΣ_{n} := Σ_{n} ∩ ℤ^{ν}.

In the next discussion it is convenient to set f_{2}(β) := #{ℓ : β_{ℓ} ≥ 2} for any β ∈ LΣ_{n}. Then it is easy to see that \overline{W} (β) = ν − f_{2}(β). Note that f_{2}(β) ≤ ν and also f_{2}(β) ≤ ⌊(n + 1)/2⌋, hence max f_{2} = min{ν, ⌊(n + 1)/2⌋. Therefore,

(9.1.1)  min \overline{W} |Σ_{n} = max \{0, ν − ⌊\frac{n + 1}{2}⌋\}.

The function β ↦ #{ℓ : β_{ℓ} ≥ 2} provides a stratification of Σ_{n}, which is finer than the standard skeleton decomposition of the simplex Σ_{n}. Moreover, the associated (generalized) cube decomposition of Σ_{n} also depends on the integer n (i.e. the structure of the lattice points on the simplex).

Let us denote by Sk_{r} = Sk_{r}(Σ_{n}) the real r–skeleton of the ν − 1–dimensional Σ_{n}, namely

Sk_{r} := {β ∈ Σ_{n} : at most for r + 1 indices ℓ one has β_{ℓ} > 0}.

Clearly, Sk_{0} ⊂ ⋯ ⊂ Sk_{ν−2} ⊂ Sk_{ν−1} = Σ_{n}, Sk_{r} = ∅ for r < 0 and Sk_{r} = Σ_{n} for r ≥ ν − 1.

For 0 ≤ r ≤ ν − 1, each Sk_{r} is a simplicial complex of dimension r, and Sk_{ν−2} is homeomorphic with the sphere S^{ν−2}. The lattice point distribution of LΣ_{n} in the skeletons is reflected in the next definition:

Definition 9.1.2. Set LSk_{r} := Sk_{r} ∩ ℤ^{ν} = Sk_{r} ∩ LΣ_{n}.

(a) For any r define ϕ(r) = ϕ_{n}(r) as the largest integer r′ ≤ max{r, ν − 1} such that Sk_{r′} contains all the lattice points LΣ_{n} sitting in Sk_{r}, but no other lattice points.

(b) For any n ≥ 0 and r ∈ ℤ define

K_{r}^{n} := Sk_{ϕ_{n}(r)}.

Lemma 9.1.3. (a)

K_{r}^{n} = \{ \begin{matrix}∅ & if r < 0, \\ Σ_{n} & if r ≥ n or r ≥ ν − 1, \\ Sk_{r} & otherwise. \end{matrix}

(b) Let K̃_{r}^{n} be the union of all generalized cubes with vertices in LSk_{r}(Σ_{n}). Then K_{r}^{n} is the largest skeleton included in K̃_{r}^{n}, and K_{r}^{n} ⊂ K̃_{r}^{n} is a homotopy equivalence.

Proof. (a) If r < 0 then ϕ(r) = −1, hence Sk_{ϕ(r)} = ∅. If r ≥ ν − 1, then ϕ(r) = r, and Sk_{ϕ(r)} = Σ_{n}. Next assume that 0 ≤ r ≤ ν − 2. Then LSk_{r+1} \ LSk_{r} ≠ ∅ if and only if r ≤ n − 1. Therefore, if r ≤ n − 1, then ϕ(r) = r, otherwise ϕ(r) = ν − 1. Part (b) follows from the definitions. □

Example 9.1.4. Both situations K_{r}^{n} = K̃_{r}^{n} and K_{r}^{n} ⊊ K̃_{r}^{n} might happen. First assume that ν = 2 and n = 1. Then K_{1}^{1} = K̃_{1}^{1} = Σ_{1} = Sk_{2}(Σ_{1}). But, if ν = 2 and n = 2, then K_{1}^{2} = Sk_{1}(Σ_{2}) ⊊ K̃_{1}^{2}. Indeed, note that Σ_{2} can be cut into nine small triangles, these are the generalized cubes of Σ_{2}. Then K̃_{1}^{2} contains additionally to K_{1}^{2} the three little triangles, which are neighbors of Sk_{0}(Σ_{2}).

Recall, that the lattice cohomology ℍ_{red}^{∗}(Σ_{n}, \overline{W}) can also be computed via the usual (simplicial) cohomology, cf. (3.3.1), via the identity

(9.1.5)  ℍ_{red}^{∗}(Σ_{n}, \overline{W}) = ∑_{k≥min \overline{W} |Σ_{n}} H̃^{∗}(S_{k}(Σ_{n}), ℤ),

where S_{k}(Σ_{n}) is the union of all generalized cubes in Σ_{n} whose vertices have weight ≤ k. The next proposition provides (the homotopy type of ) this simplicial complex in terms of the skeletons of Σ_{n}.

Proposition 9.1.6. For any k one has the homotopy equivalence

(9.1.7)  S_{k}(Σ_{n}) ∼ K_{n+k−ν}^{n} \ K_{ν−k−2}^{n}.

The right hand side written in this form covers the cases k < min \overline{W} |Σ_{n} too. Indeed,

(9.1.8)  K_{n+k−ν}^{n} \ K_{ν−k−2}^{n} = \{ \begin{matrix}∅ & if k < min \overline{W} \\ K_{n+k−ν}^{n} & if k > ν − 2 \\ Sk_{n+k−ν} \ Sk_{ν−k−2} & if min \overline{W} ≤ k ≤ ν − 2. \end{matrix}

Proof. First we verify (9.1.8). If k < 0 then ν − k − 2 ≥ ν − 1, hence K_{ν−k−2}^{n} = Σ_{n}. Similarly, if k < ν − ⌊(n + 1)/2⌋, then n + k − ν ≤ ν − k − 2 hence K_{n+k−ν}^{n} \ K_{ν−k−2}^{n} = ∅. This proves the first row. If k > ν − 2 then ν − k − 2 < 0 giving the second row. Finally, assume the third case. Then k ≥ min \overline{W} implies ν − k − 2 ≤ min\{ν − 2, n − 1\}, hence K_{ν−k−2}^{n} = Sk_{ν−k−2}, cf. (9.1.3)(a). Furthermore, k ≤ ν − 2 implies n + k − ν ≤ n − 2, hence K_{n+k−ν}^{n} = Sk_{n+k−ν} too by (9.1.3)(a).

Now we start to prove (9.1.7). It is convenient to write sometimes Σ_{n} as Σ_{n}^{ν−1} in order to emphasize both its dimension and its lattice point structure.

For any β ∈ LΣ_{n} write f_{1}(β) := #{ℓ : β_{ℓ} ≥ 1}. One verifies that f_{1} ≤ min{n + 1, ν} and

(9.1.9)  f_{2} ≤ min{f_{1}, n + 1 − f_{1}} ≤ n + 1.

Our goal is to describe S_{k}(Σ_{n}). For this first we have to list those lattice points β ∈ LΣ_{n} with \overline{W} (β) ≤ k, or f_{2}(β) ≥ ν − k.

Assume that for some β ∈ LΣ_{n} one has f_{1}(β) = p + 1. This means exactly that β ∈ LSk_{p}(Σ_{n}) \ LSk_{p−1}(Σ_{n}). Hence β is a lattice point in the interior of one of the open p–faces. The convex closure of such points on a fixed p–face is a p–simplex, and its lattice points have the structure of LΣ_{n−1−p}^{p} (if p > 0 then on any edge there are n − p lattice points). Their union is denoted by Sk_{p}^{o}(Σ_{n}) with lattice points LSk_{p}^{o}(Σ_{n}).

Moreover, for such points f_{2}(β) ≥ ν − k if and only if β ∈ LΣ_{n−1−p}^{p} \ LSk_{ν−k−2}(Σ_{n−1−p}^{p}). Next, we wish to list the possible values p. For this we use (9.1.9). If f_{1}(β) ≤ n + 1 − f_{1}(β), i.e.

p+1 ≤ (n+1)/2, then f_{2}(β) ≤ f_{1}(β), thus ν−k ≤ p+1. If p+1 ≥ (n+1)/2, then f_{2}(β) ≤ n+1−f_{1}(β), or ν − k ≤ n + 1 − p − 1. Hence the possible p–values are ν − k − 1 ≤ p ≤ n + k − ν. In fact, the first inequality as restriction is superfluous; indeed if p ≤ ν−k−2 then LΣ_{n−1−p}^{p}\LSk_{ν−k−2}(Σ_{n−1−p}^{p}) = ∅. Summed up, we get

\begin{matrix}S_{k}(Σ_{n}) ∩ ℤ^{ν} = ⋃_{p≤n+k−ν} LSk_{p}^{o}(Σ_{n}) \ LSk_{ν−k−2}(Sk_{p}^{o}(Σ_{n})) \\ = LSk_{n+k−ν}(Σ_{n}) \ ⋃_{p≤n+k−ν} LSk_{ν−k−2}(Sk_{p}^{o}(Σ_{n})). \end{matrix} ; (9.1.10)

Now we are ready to verify (9.1.7). If k < min \overline{W} then both sides are empty, cf. (9.1.8). If k > ν − 2, then (9.1.10) reads as S_{k}(Σ_{n}) ∩ ℤ^{ν} = LSk_{n−k−ν}, hence S_{k}(Σ_{n}) = K̃_{n−k−ν}^{n}, which by (9.1.3)(b) has the same homotopy type as K_{n−k−ν}^{n} and which agree with (9.1.8). Finally, assume that min \overline{W} ≤ k ≤ ν − 2. Recall from the first paragraph of the proof (the proof of the third row of (9.1.8), ϕ(n + k − ν) = n + k − ν, i.e. the cubes filling LSk_{n+k−ν} provide exactly K̃_{n+k−ν}^{n} whose homotopy type is K_{n+k−ν}^{n} = Sk_{n+k−ν}.

The point is that in the range p ≤ n + k − ν, both inequalities ν − k − 2 ≤ p − 1 and ν − k − 2 < n − 1 − p hold (compare with (9.1.3)(a)), hence each p–simplex Σ_{n−1−p}^{p} contains additional lattice points which are not in LSk_{ν−k−2}(Σ_{n−1−p}^{p}). Hence the union of the cubes missing from Σ_{n−1−p}^{p} due to the lattice points ∪_{p}LSk_{ν−k−2}(Σ_{n−1−p}^{p}) forms a regular tubular neighborhood of Sk_{ν−k−2}. □

We expect that the space Sk_{k}\Sk_{l} was already studied in the literature, nevertheless we were not able to find a reference for its homotopy type or homology; therefore we present the corresponding statement in the next proposition.

Proposition 9.1.11. Let Sk_{k} be the k–skeleton of the (ν − 1)–dimensional simplex Σ = Σ^{ν−1}. Assume that −1 ≤ l < k ≤ ν − 1. Then Sk_{k} \ Sk_{l} has the homotopy type of a bouquet of spheres of dimension k − l − 1:

Sk_{k} \ Sk_{l} ∼ ⋁_{c_{k,l}} S^{k−l−1},

where

c_{k,l} = (−1)^{k−l−1}[ − 1 + ∑_{i=l+1}^{k}(−1)^{i−l−1}\binom{ν}{i + 1}\binom{i}{l + 1}].

Remark 9.1.12. Before we start to prove (9.1.11), let us list some particular cases (which are essential steps in the proof too). Note also that a bouquet of c (c ≥ 0) copies of S^{0} consists of c + 1 points. Furthermore, the next combinatorial identity sometimes is applicable:

(9.1.13)  ∑_{i∈ℤ}(−1)^{i−l−1}\binom{ν}{i + 1}\binom{i}{l + 1} = 1.

It can be proved as follows. Expand (1+z)^{ν} into ∑_{k} \binom{ν}{k}z^{k}, and (1+z)^{−(l+2)} into ∑_{k}(−1)^{k}\binom{l+1+k}{l+1}z^{k}. Then the coefficient of z^{ν−(l+2)} in the product (1 + z)^{ν}(1 + z)^{−(l+2)} is 1.

(a) Assume that 0 ≤ k ≤ ν − 1 and l = −1. Then the dimension of the spheres is k and their number is \binom{ν−1}{k+1}. (Use ∑_{i=0}^{k+1}(−1)^{k+i−1}\binom{ν}{i} = \binom{ν−1}{k+1}.)

(b) Assume that 0 ≤ l ≤ ν − 2 and k = ν − 1. Then Σ \ Sk_{l} is contractible (use (9.1.13)).

(c) Assume that 0 ≤ l ≤ ν − 3 and k = ν − 2. Then the dimension of the spheres is ν − l − 3 and their number is \binom{ν−1}{l+1} (use again (9.1.13)).

Proof. The (ν − 1)–dimensional simplex Σ has a natural cell–decomposition providing the skeleton decomposition too. Let C_{k} be the free ℤ–module generated by the k–cells; it has rank \binom{ν}{k+1}, 0 ≤ k ≤ ν − 1. Let ∂_{k+1} : C_{k+1} → C_{k} be the boundary map, C_{−1} = ℤ, and ∂_{0} : C_{0} → C_{−1} the augmentation. The augmented complex C_{•} is exact. By induction on k one shows that the rank of im ∂_{k} is \binom{ν−1}{k}. The augmented chain–complex associated with Sk_{k} is 0 → C_{k} → ⋯ → C_{−1} → 0, which is exact except at place k, where the homology is im ∂_{k+1}. Since Σ is simply connected , Sk_{2} is simply connected too, hence the homotopy type follows by Whitehead theorem. This provides the case l = −1 corresponding to (9.1.12)(a).

If k = ν − 1 then Σ \ Sk_{l} contracts to the barycenter of Σ, cf. (9.1.12)(b).

Assume that k = ν −2 and l ≥ 0. Then Sk_{ν−2} = S^{ν−2}, hence the homological statement follows from Alexander duality. For the homotopy statement, or for an independent proof one can use the dual cell decomposition of the simplex. This shows the case (9.1.12)(c). The last two cases also prove the following statement: Fact. Let Σ^{r} be an r–dimensional simplex with boundary ∂Σ^{r}. Let l < r − 1. Then the pair (Σ^{r}, ∂Σ^{r} \ Sk_{l}) has the homotopy type of the pair (Cone(∂Σ^{r} \ Sk_{l})), ∂Σ^{r} \ Sk_{l}). Since ∂Σ^{r} \ Sk_{l} has the homotopy type of a bouquet of (r − l − 2)–dimensional spheres, gluing Σ^{r} along ∂Σ^{r} \ Sk_{l} is equivalent by gluing (r − l − 1)–cells. Their number is \binom{r}{l+1}.

Now we prove the general case by increasing induction on k. Set X_{k} := Sk_{k} \ Sk_{l}. Start with k = l + 1. Then X_{l+1} is the collection of open l + 1 faces, hence it has the homotopy type of \binom{ν}{l+2} points. Next, we take X_{l+2}, it contains the space X_{l+1}, and we glue to it the (l + 2) faces Σ^{l+2} along their partial boundaries ∂Σ^{l+2} \ Sk_{l}. Applied Fact for r = l + 2, each face contributes with \binom{l+2}{l+1} copies of 1–cells. The total number of l + 2–faces is \binom{ν}{l+3}. Hence we have to glue \binom{ν}{l+3}\binom{l+2}{l+1} 1–cells.

By similar argument, X_{l+3} is obtained from X_{l+2} by gluing \binom{ν}{l+4}\binom{l+3}{l+1} 2–cells. If we continue in this way finally we get X_{ν−1}, which is contractible, hence when we glue in this inductive procedure the r cells we have to kill totally the r homology (and for r = 2, the fundamental group). Hence at each step we get a bouquet of spheres, and their number can be determined by an Euler characteristic argument. This is exactly the formula of the statement. □

9.2. Particular cases.

Example 9.2.1. Since H̃^{q}(S_{k}(Σ_{n}), ℤ) = 0 for q ≥ ν − 1, we get that ℍ_{red}^{q}(Σ_{n}, \overline{W}) = 0 for q ≥ ν − 1; this also shows that ℍ^{q}(S_{−d}^{3}(K)) = 0 for any q ≥ ν, a fact compatible with (4.2.11).

In this example we wish to investigate H̃^{ν−2}(S_{k}(Σ_{n})) for any fixed ν ≥ 2. This group can be non– zero only if in (9.1.6) we have Sk_{k}(Σ_{n}) = Sk_{ν−2}, or if the following three facts hold simultaneously: k − ν + n = ν − 2, ν − k − 2 < 0 and k − ν + n < n. This system has a unique solution, namely ν = k + 1 = n + 1. Therefore,

rank ℍ^{ν−1}(S_{−d}^{3}(K)) = rank ℍ_{red}^{ν−2}(Σ_{ν−1}, \overline{W}) = 1.

(Compare also with (4.4.1) and (4.4.2) of [16].)

Example 9.2.2. Assume that ν = 3 and d = 2. For a = 0 one has 𝔪_{k_{r}} = −2, ℍ_{red}^{0}(G(d), k_{r}) = T_{0}(1)^{2} ⊕T_{−4}(2) of rank 4, ℍ^{1}(G(d), k_{r}) = 0 and rank ℍ^{2}(G(d), k_{r}) = 1. Hence eu(ℍ^{∗}(G(d), k_{r})) = 7.

On the other hand, for a = 1, one has 𝔪_{k_{r}} = −1, ℍ_{red}^{0}(G(d), k_{r}) = T_{0}(1)^{2} of rank 2, and rank ℍ^{1}(G(d), k_{r}) = 4, while ℍ^{2}(G(d), k_{r}) = 0. Hence eu(ℍ^{∗}(G(d), k_{r})) = −1. This shows that eu(ℍ^{∗}(G(d), k_{r})) < 0 can be realized.

The above data can be compared with the coefficients of the polynomial Q in the light of (7.1.1) and (7.1.3): by an easy computation Q(t) = t^{4} − t^{3} + 3t^{2} + 3. Note that the sum of the coefficients of the monomials with even (respectively odd) exponent is 7 (resp. −1).

10. Another example

Assume that ν = 2, both knots are torus knots, one of them of type (3, 4) the other (2, 7). Hence μ_{1} = μ_{2} = 6 and Γ_{1}^{c} = {1, 2, 5} and Γ_{2}^{c} = {1, 3, 5}. The Alexander polynomials are Δ_{1}(t) = 24 A. Némethi and F. Román

t^{6} − t^{5} + t^{3} − t + 1 and Δ_{2}(t) = t^{6} − t^{5} + t^{4} − t^{3} + t^{2} − t + 1. By a computation one gets Q(t) = ∑_{i} α_{i}t^{i} = t^{10} + t^{8} + t^{7} + 2t^{6} + t^{5} + 3t^{4} + 3t^{3} + 4t^{2} + 4t + 6. In the next example we take d = 3 and a = 0. The plumbing graph G is:

[figure: −2; −1; −29; −1; −3; −4; −2; −2; −2; −2]

The bound α̃_{0} is 4, so we have to analyze four little rectangle. The values \overline{w}(αm) for 0 ≤ α ≤ 4 are 0, −5, −7, −6 and −2 respectively. The next table contains all the information regarding these values the weights along T_{α+1}^{−} and the ranks of ℍ_{red}^{0}(𝕋_{3α}) (note that ℍ_{red}^{q}(𝕋_{3α}) = 0 for q ≥ 1).

α = 0 α = 1 α = 2 α = 3
min \overline{w}|T_{α+1}^{−}  1  −3  −4  −1
min \overline{W} |𝕋_{3α}^{−}  6  4  2  1
\overline{w}((α + 1)m)  −5  −7  −6  −2
rank ℍ_{red}^{0}(𝕋_{3α})  0  1  0  1
−eu(ℍ^{∗}(𝕋_{3α}, \overline{W}) 6  3  2  0
α_{3α}  6  3  2  0

Hence the interested reader might exemplify using these entries the corresponding theoretical results from the body of the paper.

In particular, for G = G(3) and a = 0 we get: min \overline{w} = −7, rank ℍ_{red}^{0}(G, can) = 6, while rank ℍ^{1}(G, can) = 2, hence eu ℍ^{∗}(G, can) = 11.

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A. Rényi Institute of Mathematics, 1053 Budapest, Reáltanoda u. 13-15, Hungary.

    E-mail address: nemethi@renyi.hu

Depart. de Álgebra, Universidad Complutense de Madrid, Plaza de Ciencias s/n, E-28040 Madrid, Spain

    E-mail address: ferchu@mat.ucm.es