SOME PROPERTIES OF THE LATTICE COHOMOLOGY

ANDRÁS NÉMETHI

The present manuscript is based on the author’s plenary lectures on the meeting ‘Geometry Conference’, organized at Yamagata University, Japan, in September 2010. The goal of the series of talks (and of the present manuscript) was to present some basic properties of the lattice cohomology, and to create some connections with other parts of the mathematics, like singularity theory and low– dimensional topology. The statements closely follow the author’s papers [N08, N10, N10b].

The material covers the following facts: the first section reviews the notations related with a plumbing graph and the definition of the lattice cohomology. In the second section we identify the Euler characteristic of the lattice cohomology with the Seiberg–Witten invariant of the plumbed 3–manifold, and also with the periodic constant of a combinatorial multivariable Poincaré series. In section 3 we discuss several exact sequences associated with the lattice cohomology. They correspond to ‘surgery exact sequences’ in low–dimensional topology. Section 4 introduces the ‘relative lattice cohomology’ and the corresponding ‘relative exact sequence’. This section list some additional results connecting the Euler characteristic of the relative lattice cohomology with the periodic constant of some classical series.

The author wishes to thank the organizers of the meeting, especially to Professors T. Ashikaga, T. Okuma, M. Tomari and T. Tomaru, for the invitation, for the financial support, and for the warm hospitality during the visit.

1. Notations and preliminaries. Definition of the lattice cohomology.

1.1. Notations. First we will introduce the needed notations regarding plumbing graphs and we recall the definition of the lattice cohomology from [N08]. Since in [loc.cit.] the graphs were connected and negative definite (as the ‘normal’ plumbing representations of isolated complex surface singularity links), at the beginning we will start with these assumptions.

Let Γ be such a plumbing graph with vertices J and edges E; we set |J | = s, and we fix an order on J. Γ can also be codified in the lattice L, the free ℤ–module generated by {E_{j}}_{j∈J} and the ‘intersection form’ {(E_{i}, E_{j})}_{i,j}, where (E_{i}, E_{j}) for i ≠ j is 1 or 0 corresponding to the fact that (i, j) is an edge or not; and (E_{i}, E_{i}) is the decoration of the vertex i, usually denoted by e_{i}. The graph is negative definite if this form is so. The graph Γ may have cycles, but we will assume that all the genus decorations are zero. The associated plumbed 3–manifold M(Γ) is not necessarily a rational homology sphere, this happens exactly when the graph is a tree. Let L′ be the dual lattice {l′ ∈ L ⊗ ℚ : (l′, L) ⊂ ℤ}; it is generated by {E_{j}^{∗}}_{j}, where (E_{j}^{∗}, E_{i}) = −δ_{ij} (the negative of the Kronecker–delta). Moreover,

Char := \{k ∈ L′ : χ_{k}(l) := −\frac{1}{2}(k + l, l) ∈ ℤ for all l ∈ L\}

denotes the set of characteristic elements of L (or Γ).

Let K = K_{Γ} ∈ L′ denote the canonical characteristic element of Γ defined by the adjunction formulae (K_{Γ} + E_{j}, E_{j}) + 2 = 0 for all j ∈ J.

As usual (following Ozsváth and Szabó), T_{0}^{+} denotes the ℤ[U]–module ℤ[U, U^{−1}]/Uℤ[U] with grading deg(U^{−d}) = 2d (d ≥ 0). More generally, for any r ∈ ℚ one defines T_{r}^{+}, the same module as T_{0}^{+}, but graded (by ℚ) in such a way that the d + r–homogeneous elements of T_{r}^{+} are isomorphic with the d–homogeneous elements of T_{0}^{+}. (E.g., for m ∈ ℤ, T_{2m}^{+} = ℤ[U, U^{−1}]/U^{−m+1}ℤ[U].)

2000 Mathematics Subject Classification. Primary. 14B05, 14J17, 32S25, 57M27, 57R57. Secondary. 14E15, 32S45, 57M25.

Key words and phrases. plumbed manifolds, plumbing graphs, 3-manifolds, lattice cohomology, surface singularities, rational singularities, Seiberg-Witten invariant, Heegaard-Floer homology.

The author was partially supported by the Japanese JSPS Grant, which made his visit to Japan possible.

1.2. The lattice cohomology associated with k ∈ Char [N08]. L ⊗ ℝ = ℤ^{s} ⊗_{ℤ} ℝ has a natural cellular decomposition into cubes. The set of zero–dimensional cubes is provided by the lattice points L. Any l ∈ L 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. One sees that ∂ ∘ ∂ = 0, but, obviously, the homology of the chain complex (C_{∗}, ∂) is trivial: it is just the homology of ℝ^{s}. In order to get a more interesting (co)homology, one needs to consider a weight functions w : Q_{q} → ℤ (0 ≤ q ≤ s). In the present case this will be defined, for each k ∈ Char fixed, by

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

Once the weight function is defined, one considers F^{q}, the set of morphisms Hom_{ℤ}(C_{q}, T_{0}^{+}) with finite support on Q_{q}. Notice that F^{q} is, in fact, a ℤ[U]–module by (p ∗ ϕ)(□_{q}) := p(ϕ(□_{q})) (ϕ ∈ F^{q}, p ∈ ℤ[U]). Moreover, F^{q} has a 2ℤ–grading: ϕ ∈ F^{q} is homogeneous of degree 2d ∈ ℤ if for each □_{q} ∈ Q_{q} with ϕ(□_{q}) ≠ 0, ϕ(□_{q}) is a homogeneous element of T_{0}^{+} of degree 2d − 2 · w(□_{q}).

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

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

One verifies that δ ∘ δ = 0, i.e. (F^{∗}, δ) is a cochain complex (with δ homogeneous of degree zero); its homology is denoted by {ℍ^{q}(Γ, k)}_{q≥0}.

(F^{∗}, δ_{w}) has a natural augmentation too. Indeed, set m_{k} := min_{l∈L}{χ_{k}(l)}. Then one defines the ℤ[U]–linear map ϵ : T_{2m_{k}}^{+} ⟶ F^{0} such that ϵ(U^{−m_{k}−s})(l) is the class of U^{−m_{k}+χ_{k}(l)−s} in T_{0}^{+} for any integer s ≥ 0. Then ϵ is injective and homogeneous of degree zero, δ ∘ ϵ = 0. The homology of the augmented cochain complex

0 ⟶ T_{2m_{k}}^{+} ⟶^{ϵ} F^{0} ⟶^{δ} F^{1} ⟶^{δ} …

is called the reduced lattice cohomology ℍ_{red}^{∗}(Γ, k). For any q ≥ 0, both ℍ^{q} and ℍ_{red}^{q} admit an induced graded ℤ[U]–module structure and ℍ^{q} = ℍ_{red}^{q} for q > 0.

Note that ϵ provides a canonical embedding of T_{2m_{k}}^{+} into ℍ^{0}. Moreover, one has a graded ℤ[U]– module isomorphism ℍ^{0} = T_{2m_{k}}^{+} ⊕ ℍ_{red}^{0}, and ℍ_{red}^{∗} has finite ℤ–rank.

Remark 1.2.2. For the definition of the lattice cohomology for more general weight functions and graphs with non–zero genera, see [N08].

1.3. Reinterpretation of the lattice cohomology. If k′ = k+2l for some l ∈ L then ℍ^{∗}(Γ, k) and ℍ^{∗}(Γ, k′) are isomorphic up to a degree–shift, cf. [N08, (3.3)]. In fact, all the modules {ℍ^{∗}(Γ, k)}_{k} can be packed into only one object, more in the spirit of [OSz03]. This was used in [Gr08] too.

In this way, for any fixed k ∈ Char, L is identified with the sublattice k + 2L ∈ Char, and with the notation l′ := k + 2l one has χ_{k}(l) = −\frac{1}{8}(l′, l′) + \frac{1}{8}(k, k). In particular, up to a shift in degree, for each fixed k, l ↦ χ_{k}(l) and l′ ↦ −\frac{1}{8}(l′, l′) define the same weight–function (on the cubes, see below), hence the same cohomology. In fact, we will modify even this weight function by s/8 (in this way the blowing up will induce a degree preserving isomorphism, cf. §3.2), and set:

(1.3.1)  w : Char → ℚ,  w(k) := −\frac{k^{2} + s}{8} (s = |J |).

The q–cubes □_{q} ∈ Q_{q} are associated with pairs (k, I) ∈ Char × P(J), |I| = q, (here P(J) denotes the power set of J), and have the form {k + 2 ∑_{j∈I′} E_{j})_{I′}, where I′ runs over all subsets of I. The weights are defined by

(1.3.2)  w(□_{q}) = w((k, I)) = max_{I′⊂I} { w(k + 2 ∑_{j∈I′} E_{j}) }.

Moreover, F^{q} are elements of Hom_{ℤ}(Q_{q}, T_{0}^{+}) with finite support. Similarly as above, F^{q} is a ℤ[U]– module with a ℚ–grading: ϕ ∈ F^{q} is homogeneous of degree r if for each □_{q} ∈ Q_{q} with ϕ(□_{q}) ≠ 0, ϕ(□_{q}) is a homogeneous element of T_{0}^{+} of degree r − 2 · w(□_{q}).

It is convenient to consider the module of (infinitely supported) homological cycles too: let F_{q} be the direct product of ℤ_{≥0} × Q_{q} copies of ℤ (considered already in [OSz03] for q = 0). We write the pair (m, □) as U^{m}□. F_{q} becomes a ℤ[U]–module by U(U^{m}□) = U^{m+1}□. Clearly F^{q} = Hom_{ℤ[U]}(F_{q}, T_{0}^{+}), i.e. ϕ(U□) = Uϕ(□) for any ϕ.

δ : F^{q} → F^{q+1} is defined as in (1.2.1) using the new weight–function, or by δ(ϕ)(□) = ϕ(∂(□)), where for □ = (k, I) = (k, {j_{1}, …, j_{q}}) one has:

(1.3.3) ∂(k, I) = ∑_{l=1}^{q}(−1)^{l}( U^{w(k,I)−w(k,I\j_{l})}(k, I \ j_{l}) − U^{w(k,I)−w(k+2E_{j_{l}} ,I\j_{l})}(k + 2E_{j_{l}}, I \ j_{l}) ).

The cohomology of (F^{∗}, δ) is denoted by ℍ^{∗}(Γ). Since the vertices of a cube belong to the same class Char/2L (where a class has the form [k] = {k + 2l}_{l∈L} ⊂ Char), ℍ^{∗}(Γ) has a natural direct sum decomposition

ℍ^{∗}(Γ) = ⊕_{[k]∈Char/2L} ℍ^{∗}(Γ, [k]).

In fact, if [k_{1}] = [k_{2}] then w(k_{1}) − w(k_{2}) ∈ ℤ.

Since Γ is negative definite, for each class [k] ∈ Char/2L one has a well–defined rational number

d[k] := − max_{k∈[k]} \frac{k^{2} + s}{4} = 2 · min_{k∈[k]} w(k).

Then (cf. 1.2) one has a direct sum decomposition:

(1.3.4)  ℍ^{0}(Γ, [k]) = T_{d[k]}^{+} ⊕ ℍ_{red}^{0}(Γ, [k]).

Sometimes we write 𝕋 := ⊕_{[k]}T_{d[k]}^{+} for the canonical submodule of ℍ^{0}, which satisfies ℍ^{0} = 𝕋⊕ℍ_{red}^{0}. Following [N05, N08, N10] we define the Euler characteristic of ℍ^{∗}(Γ, [k]) by

(1.3.5)  eu(ℍ^{∗}(Γ, [k])) := −d[k]/2 + ∑_{q≥0}(−1)^{q} rank_{ℤ} ℍ_{red}^{q}(Γ, [k]).

Remark 1.3.6. ℍ^{∗}(Γ) and ℍ_{red}^{∗}(Γ) depend only on M = M(Γ), and not on the plumbing graph Γ. The involution l′ ↦ −l′ induces an isomorphism ℍ^{∗}(Γ, [k]) = ℍ^{∗}(Γ, [−k]), hence

(1.3.7)  eu(ℍ^{∗}(Γ, [k])) = eu(ℍ^{∗}(Γ, [−k])).

1.4. Generalizations. Graphs which are not negative definite. Since the definition of the lattice cohomology is purely algebraic/combinatorial, its definition can be considered for any graph, or even for any lattice L with fixed base elements {E_{j}}_{j}. Nevertheless, in this note we assume that the graphs are non–degenerate. Of course, doing this generalization, some of the properties of the lattice cohomology associated with connected negative definite graphs will not survive. Here we wish to point out some of the differences.

First, we notice that dropping the connectedness of the graph basically has no effect (this fact already was used in the main inductive proofs of [OSz03] and [N05, (8.3)]). Dropping the negative definiteness is more serious. In order to explain this, we will introduce the following spaces: for any fixed class [k] and any real number r we define S_{r} as the union of all cubes □ ∈ Q_{q} with all vertices in [k] and w(□) ≤ r.

If Γ is negative definite then S_{r} is compact for any r. In particular, the weight function takes its minimum on it, and for each class [k] = Char/2L a decomposition ℍ^{0}(Γ, [k]) = T_{d([k])}^{+} ⊕ ℍ_{red}^{0}(Γ, [k]) can be defined. This property will not survive for general graphs. In fact, if the components of S_{r} are not compact, then we even might have the vanishing ℍ^{0}(Γ) = 0 (and this can happen simultaneously with the non–vanishing of ℍ^{1}). Also, in [N08, Theorem 3.1.12] the lattice cohomology is recovered from the simplicial cohomology of the spaces S_{r}. In the general case, the adapted version of that theorem (with similar proof), valid for any lattice, states that the same formula is valid once if we replace the cohomology groups H^{q}(S_{r}, ℤ) by the cohomology groups with compact support H_{c}^{q}(S_{r}, ℤ). Namely:

ℍ^{∗}(Γ, [k]) ≃ ⨁_{r∈w(k)+ℤ} H_{c}^{∗}(S_{r}, ℤ).

We wish to emphasize, that replacing the cohomology with cohomology with compact support is a conceptual modification: the two groups have different functorial properties.

In fact, exactly this second point of view is determinative in the definition of the lattice cohomology (and in the definition of the maps in the surgery formula treated next): basically we mimic the infinitely supported homology and the (dual) cohomology with compact support, and the corresponding functors associated with them.

Since the surgery exact sequence considered in section 3 is valid for any non–degenerate lattice, it is very convenient to extend the theory to arbitrary graphs (or, at least for non–degenerate ones) in order to have a larger flexibility for computations. Nevertheless, we have to face the following crucial problem. An oriented plumbed 3–manifold M(Γ) has many different plumbing representations Γ. They are connected by the moves of the (oriented) plumbing calculus. For negative definite graphs the only moves are the blowups (and their inverses) by (rational) (−1)-vertices. In [N08, (3.4)] it is proved that the lattice cohomology is stable with respect to these moves. On the other hand, if we enlarge our plumbing graphs, this stability condition will not survive: the same 3–manifold can be represented by many different lattices with rather different lattice cohomologies.

More precisely, for negative definite graphs one conjectures (cf. [N08]) that from the lattice cohomology one can recover (in a combinatorial way) the Heegaard–Floer homology of Ozsváth– Szabó. In particular, the lattice cohomology carries a geometric meaning depending only on M(Γ). This geometric meaning is lost in the context of general graphs (or, at least, it is not so direct).

2. The Seiberg–Witten invariants of negative definite plumbed 3–manifolds

2.1. Preliminaries and golas. Let Γ be a connected negative definite plumbing graph. We assume that it is a tree, and all the plumbed surfaces have genus zero. Hence, the associated oriented plumbed 3–manifold M = M(Γ) is a rational homology sphere. We denote by 𝔰𝔴_{σ}(M) the Seiberg– Witten invariants of M indexed by the spin^{c}–structures σ of M. The spin^{c}–structures on M(Γ) are identified with Char/2L, see e.g. [N05, N08] and (2.9) here.

Our goal is to provide two new combinatorial formulae for 𝔰𝔴_{σ}(M). One of them uses qualitative properties of the coefficients of a combinatorial zeta function associated with Γ, the other is the normalized Euler characteristic of the lattice cohomology of Γ. Both formulae reflect in the most conceptual and optimal way the structure of the graph Γ, and emphasizes the subtle parallelism between these topological invariants and the analytic invariants of normal surface singularities. The main aim is to establish the identity of these three objects: Seiberg–Witten invariant, periodic constant of the zeta function, and the Euler characteristic of the lattice cohomology.

Consider the multi-variable Taylor expansion Z(t) = ∑ p_{l′}t^{l′} at the origin of

(2.1.1)  ∏_{j∈J} (1 − t^{E_{j}^{∗}})^{δ_{j}−2},

where for any l′ = ∑_{j} l_{j}E_{j} ∈ L′ we write t^{l′} = ∏_{j} t_{j}^{l_{j}}, and δ_{j} is the valency of j. This lives in ℤ[[L′]],

the submodule of formal power series ℤ[[t^{±1/d}]] in variables {t_{j}^{±1/d}}_{j}, where d = det(−I). The first identity is the following.

Theorem A. Fix some l′ ∈ L′. Assume that for any j ∈ J the E_{j}^{∗}–coordinate of l′ is larger than or equal to −(E_{j}^{2} + 1). Then the sum

∑_{l∈L, l≱0} p_{l′+l}

equals a multivariable quadratic function on l′, namely

(2.1.2)  −\frac{(K + 2l′)^{2} + |J |}{8} − 𝔰_{[−l′]},

where the constant 𝔰_{[−l′]} depends only on the class [−l′] of −l′ in L′/L = H^{2}(M, ℤ).

Moreover, if ∗ denotes the (torsor) action of L′/L on Spin^{c}(M), one has

𝔰_{[l′]} = 𝔰𝔴_{[l′]∗σ_{can}}(M).

In particular, the normalized Seiberg–Witten invariant appears as the constant term of the ‘combinatorial multivariable Hilbert polynomial’ (2.1.2).

For the second identity let us consider the lattice cohomology ℍ^{∗}(Γ) associated with Γ. It depends only on M, and it has a natural direct sum decomposition indexed by σ ∈ Spin^{c}(M), namely ℍ^{∗}(Γ) = ⊕_{σ}ℍ^{∗}(Γ, σ). Let eu(ℍ^{∗}(Γ, σ)) be the normalized Euler characteristic of the corresponding summand; for more details see (1.3.5). Then one has

Theorem B. For any Γ and σ as above

−eu(ℍ^{∗}(Γ, σ)) = 𝔰𝔴_{σ}(M(Γ)).

In fact, Theorem A was motivated by a similar formula valid for equivariant geometric genera of normal surface singularities, cf. [N08c]. The theorem shows a combinatorial quadratic Hilbertpolynomial type behaviour of the corresponding sum of the coefficients of Z(t). The proof identifies two combinatorial objects:

(2.1.3)  𝔰_{[−l′]} = −eu(ℍ^{∗}(Γ, [K + 2l′])).

The second part of Theorem A rely on a surgery formula for the constant term 𝔰, which fits perfectly with the surgery formula proved for the Seiberg–Witten invariant 𝔰𝔴 in [BN10]. This allows us to prove in subsections (2.9)–(2.10) the identity 𝔰 = 𝔰𝔴 by induction on |J |.

The surgery formula involves in a crucial way the ‘periodic constant’ of a series introduced in [NO09, O08], see (2.7). In fact, via Theorem A, the Seiberg–Witten invariants can be interpreted as the ‘multivariable periodic constants’ of the series Z(t).

The series Z(t) was used in several articles studying invariants of surface singularities [CDG04, CDG08, CHR04, N08b, N08c]. Theorem A puts the results of these articles in a new light. Indeed, as a consequence of the present work, the identity Z(t) with the analytic invariant P(t), in some articles called Campillo–Delgado–Gusein-Zade type identity, implies automatically the Seiberg–Witten Invariant Conjecture of Nicolaescu and the author, cf. [NN02, N03]. This provides a conceptual understanding how the Seiberg–Witten invariants appear in a natural way in the world of singularities, and why they can serve as topological candidates for the equivariant geometric genera.

Theorem B follows from Theorem A and (2.1.3). It also has the following interpretation. It is known that the Seiberg–Witten invariant appears as the normalized Euler characteristic of the Heegaard–Floer theory of Ozsváth and Szabó, see [OSz03b, N05, R04] (or, of the Seiberg–Witten Floer homology). Theorem B says that the normalized Euler characteristics of these cohomology theories and of the lattice cohomology coincide. This supports the conjecture from [N08] which expects precise correspondence between the corresponding cohomology modules and the normalization terms.

In the sequel we list all those results which lead to the proof of the above two theorems.

2.2. Rectangles. In combinatorial enumerations of the weighted q–cubes it is convenient to replace the set of all cubes by only those ones which are supported on a fixed compact subset of L′⊗ℝ. In the simplest case we take rectangles: for any fixed class [k] = k_{0}+2L ⊂ Char, one takes two characteristic elements k_{1}, k_{2} ∈ [k] with k_{1} ≥ k_{2}. We denote by R = R(k_{1}, k_{2}) the rectangle {k ∈ [k] : k_{1} ≥ k ≥ k_{2}}. Similarly, for one fixed element k_{1} ∈ [k], one can take R = R(k_{1}) = {k ∈ [k] : k_{1} ≥ k}. Once such an R is identified, one considers the complex (F^{∗}(R), δ(R)), constructed similarly as (F^{∗}, δ), consisting of all the cubes (k, I) with all vertices in R (this fact will be denoted by (k, I) ∈ R). Using min(w|R) := min_{k∈R} w(k) one also defines the corresponding augmented complex, and one gets the corresponding lattice cohomologies ℍ^{∗}(R, [k]) and ℍ_{red}^{∗}(R, [k]) with ℍ^{0}(R, [k]) = T_{2 min(w|R)}^{+} ⊕ ℍ_{red}^{0}(R, [k]). For more details, see [N08]. We also define the ‘normalized Euler characteristic’ of this lattice cohomology, namely

eu(ℍ^{∗}(R, [k]) := − min(w|R) + ∑_{q≥0}(−1)^{q} rank_{ℤ} ℍ_{red}^{q}(R, [k]).

Let S′_{st} be the strict Lipman cone S′_{st} = {l′ ∈ L′ : (l′, E_{j}) < 0 for all j}.

Proposition 2.2.1. Fix a class [k]. Assume that k_{1} ∈ [k] satisfies k_{1} ∈ −K +S′_{st}, that is (k_{1}, E_{j}) ≤ e_{j} + 1 for any j ∈ J. Then the following facts hold:

(a) For any k ∈ [k], k > k_{1}, there exists some E_{j} in the support |k − k_{1}| of k − k_{1}, so that w(k − 2E_{j}) ≤ w(k).

(b) There exists an increasing (computation) sequence {z_{n}}_{n≥0}, z_{n} ∈ L, with z_{0} = 0, and z_{n+1} = z_{n} + E_{j(n)} for some j(n) ∈ J when n ≥ 0, satisfying:

(i) The coefficients of z_{n} tends to infinity, that is − lim_{n→∞}(z_{n}, E_{j}^{∗}) = ∞ for any j.

(ii) For fixed n ≥ 0, let x ∈ L be such that x ≤ z_{n} and (x, E_{j(n)}^{∗}) = (z_{n}, E_{j(n)}^{∗}). Then w(k_{1}+2x) ≤ w(k_{1} + 2x + 2E_{j(n)}).

(iii) The restriction ℍ^{∗}(R(k_{1} +2z_{n+1}), [k]) → ℍ^{∗}(R(k_{1} +2z_{n}), [k]) is an isomorphism of weighted ℤ[U]–modules compatible with the augmentation.

In particular, ℍ^{∗}(Γ, [k]) = ℍ^{∗}(R(k_{1}), [k]) compatibly with the augmentation.

Moreover, in a similar way, one can find k_{2} (with all its E_{j}–coefficients sufficiently small) such that ℍ^{∗}(Γ, [k]) = ℍ^{∗}(R(k_{1}, k_{2}), [k]).

Proof. (a) Assume that w(k − 2E_{j}) > w(k) for any E_{j} ∈ |k − k_{1}|. This says that (k − 2E_{j})^{2} < k^{2}, that is (k, E_{j}) > E_{j}^{2}, or (k + K, E_{j}) ≥ −1. Since (k_{1} + K, E_{j}) ≤ −1, one gets (k − k_{1}, E_{j}) ≥ 0 for any E_{j}, hence (k − k_{1})^{2} ≥ 0, which is a contradiction.

(b) Fix a D = ∑_{j} d_{j}E_{j} ∈ S′ ∩L with d_{j} ∈ ℤ_{>0} for all j. By (a) we get an increasing computation sequence {y_{n}}_{n=0}^{n_{0}} connecting 0 and D so that w(k_{1} + 2y_{n}) ≤ w(k_{1} + 2y_{n+1}). Then the sequence {z_{n}}_{n≥0} := {mD+y_{n}}_{m≥0; 0≤n≤n_{0}} satisfies (i) and w(k_{1}+2z_{n}) ≤ w(k_{1}+2z_{n+1}). All other properties follow similarly as in [N08, p. 518]. □

2.3. Counting weighted cubes and the Euler–characteristic of the lattice cohomology. The next result generalizes the classical fact that the alternating sum of the number of q–cubes is the Euler characteristic of the cohomology. For any finite set A ⊂ [k] define

E(A) := ∑_{(k,I)∈A}(−1)^{|I|+1}w((k, I)) and M_{A}(t) := ∑_{(k,I)∈A}(−1)^{|I|}t^{w((k,I))}.

Theorem 2.3.1. Let R be a finite rectangle R(k_{1}, k_{2}). Then E(R) = eu(ℍ^{∗}(R, [k])).

Proof. We will reduce the result to the classical case via a certain geometric interpretation of the lattice cohomology from [N08, (3.1.12)]. Note that R can also be interpreted as a real rectangle in L ⊗_{ℤ} ℝ limited by the vertices k_{1} and k_{2}. It has a natural cubic decomposition into the ‘real cubes’ (k, I) with all vertices in the rectangle. For any non–negative integer n define S_{n} as the union of all these real cubes of L ⊗ ℝ with vertices from R, and with weights ≤ n + min(w|R). Let χ(S_{n}) be its (classical) Euler–characteristic and χ_{red}(S_{n}) = χ(S_{n}) − 1 the Euler-characteristic of its reduced simplicial cohomology. Then, by [N08, (3.1.12)], one has for any q ≥ 0 the ℤ–module isomorphisms:

(2.3.2)  ℍ^{q}(R, [k]) = ⨁_{n≥0} H^{q}(S_{n}, ℤ), ℍ_{red}^{q}(R, [k]) = ⨁_{n≥0} H̃^{q}(S_{n}, ℤ).

In particular, if we write M_{R}(t)/(1 − t) as ∑_{n≥0} a_{n}t^{n+min(w|R)}, then

a_{n} = ∑_{{(k,I)∈R \atop w((k,I))≤n+min(w|R)}} (−1)^{|I|} = χ(S_{n}).

Therefore,

\frac{M_{R}(t) − t^{min(w|R)}}{1 − t} = ∑_{n≥0} χ_{red}(S_{n})t^{n+min(w|R)},

hence, by (2.3.2):

lim_{t→1} \frac{M_{R}(t) − t^{min(w|R)}}{1 − t} = ∑_{n≥0} χ_{red}(S_{n}) = ∑_{q≥0}(−1)^{q} rank_{ℤ} ℍ_{red}^{q}(R, [k]),

that is, −\frac{d}{dt}M_{R}(1) = eu(ℍ^{∗}(R, [k])), the wished statement of the theorem. □

2.4. Counting weighted cubes and the zeta function Z(t). The next result provides the key step for the identification of the ‘lattice cohomology package’ and numerical invariants provided by the series Z(t). Before we state it, let us recall that Char = K + 2L′ ⊂ L′, hence (k − K)/2 runs over L′ when k runs over Char.

Theorem 2.4.1. Let Γ be a connected negative definite graph. Then

(2.4.2)  Z(t) = ∑_{k∈Char} ∑_{I∈P(J)} (−1)^{|I|+1} w((k, I)) t^{\frac{1}{2} (k−K)}.

Since ∑_{I}(−1)^{|I|} = 0, here w((k, I)) can be replaced by w((k, I)) + c for any constant c.

Proof. The proof is an induction over the number of vertices of the graph. It start as follows. For each k = ∑_{j} a_{j}E_{j}^{∗} ∈ Char, where a_{j} ≡ e_{j} (mod 2), write z^{a} := ∏_{j} z_{j}^{a_{j}} and set the counting function

U_{Γ}(z) := ∑_{k∈Char} ∑_{I∈P(J)} (−1)^{|I|+1} w((k, I)) z^{a}.

We determine U_{Γ} by induction on |J |. If |J | = 1 and the decoration of the unique vertex is e < 0, then |e|E^{∗} = E, (E^{∗})^{2} = 1/e, and k = aE^{∗} ∈ Char with a ≡ e (mod 2). Hence

(2.4.3)  8 U_{Γ}(z) = ∑_{a≡e (mod 2)} [ − \frac{a^{2}}{|e|} + max \{ \frac{a^{2}}{|e|}, \frac{(a − 2e)^{2}}{|e|} \} ] · z^{a}.

If a ≤ e then a^{2} ≥ (a − 2e)^{2}, hence the coefficient is vanishing. Otherwise we write a = e + 2n for n ∈ ℤ_{>0} and we get that U(z) = ∑_{n≥1} nz^{e+2n} = z^{e+2}/(1 − z^{2})^{2}.

Since the inductive step is slightly long, it is not given here. For details, see [N10]. □

2.5. The definition of the invariant 𝔰 and its relation with ℍ^{∗}(Γ). Let Γ be a graph as above and Z(t) the series defined in the introduction.

Theorem 2.5.1. (a) For any l′ ∈ L′, the expression

(2.5.2)  − ∑_{l∈L, l≱0} p_{l′+l} − \frac{(K + 2l′)^{2} + |J |}{8}

depends only on the class [l′] ∈ L′/L of l′ = ∑_{j∈J} a_{j}E_{j}^{∗}, provided that a_{j} ≥ −e_{j} − 1 for all j ∈ J. (Since Z is supported on S′, the sum in (2.5.2) is finite.)

(b) Consider the map 𝔰 : L′/L → ℚ, [l′] ↦ 𝔰_{[l′]}, where 𝔰_{[−l′]} is the expression (2.5.2). Then the set {𝔰_{[l′]}}_{[l′]} is independent of the negative definite plumbing representation Γ, it depends only on the oriented plumbed 3–manifold M = M(Γ). In fact, for any l′, one has

(2.5.3)  𝔰_{[−l′]} = −eu(ℍ^{∗}(Γ, [K + 2l′])).

Proof. We fix k_{1} = K + 2l′ and k_{2} as in (2.2.1), that is (k_{1}, E_{j}) ≤ e_{j} + 1 and all the coefficients of k_{2} are sufficiently small (k_{2} does not play on essential role, it only assures the finitness of the rectangles), and a computation sequence {z_{n}}_{n≥0} as in (2.2.1)(b). Set

R′ := {k ∈ [k] : k ≥ k_{2}, k − k_{1} = ∑ l_{j}E_{j} so that ∃ l_{j} ≤ 0}.

Although R′ is not finite, R′ ∩ S′ is finite. Fix some ñ so that R′ ∩ S′ ⊂ R(k_{1} + 2zñ, k_{2}) and define R̃ := R′ ∩ R(k_{1} + 2zñ, k_{2}). Take also

∂R̃ := {k ∈ [k] : k_{1} ≤ k ≤ k_{1} + 2zñ, k − k_{1} = ∑ l_{j}E_{j} so that ∃ l_{j} = 0}.

Then, by Theorem 2.4.1

(2.5.4)  ∑_{l∈L, l≱0} p_{l′+l} = E(R̃) − E(∂R̃).

Now, we claim that by combinatorial cancellation in the sum, E(R̃) = E(R), where R = R(k_{1}, k_{2}). This follows by induction using the sequence {z_{n}}_{0≤n≤ñ}, since E(R′ ∩ R(k_{1} + 2z_{n+1}, k_{2})) = E(R′ ∩ R(k_{1} + 2z_{n}, k_{2})). Indeed, for any I containing j(n) and cube (k, I) ∈ R′ ∩ (R(k_{1} + 2z_{n+1}, k_{2}) \ R(k_{1} + 2z_{n}, k_{2})) one has w((k, I)) = w((k + 2E_{v(n)}, I \ j(n))) by (2.2.1)(b)(ii). Similarly, one gets E(∂R̃) = −w(k_{1}). Hence (2.5.4) reads as

(2.5.5)  −w(K + 2l′) + ∑_{l∈L, l≱0} p_{l′+l} = E(R).

The right hand side is eu(ℍ^{∗}(R, [k_{1}]) by Theorem 2.3.1, which equals eu(ℍ^{∗}(Γ, [k_{1}]) by Proposition 2.2.1. In particular, the left hand side too depends only on [l′]. It is invariant under blow up of the graph since the lattice cohomology is so, cf. [N08, N10b]. □

2.6. The additivity formula. Similarly as in the proof of Theorem 2.4.1, we change the variables of the series Z(t). By setting x_{j} := t^{E_{j}^{∗}} for all j ∈ J, Z(t) transforms into Z_{Γ}(x) = ∏_{j∈J} (1−x_{j})^{δ_{j}−2}, whose Taylor series at the origin is denoted by ∑ q_{a}x^{a}, where x^{a} = x_{1}^{a_{1}} ⋯ x_{s}^{a_{s}}. The exponents a_{j} are the coordinates of L′ in the basis {E_{j}^{∗}}_{j}, i.e. if l′ = ∑_{j} a_{j}E_{j}^{∗} then a_{j} = −(l′, E_{j}) and t^{l′} transforms into x^{a}. In particular, q_{a} = p_{l′}, and we also use the notation a for l′ ∈ L′. For any fixed l′ = ∑_{j} a_{j}E_{j}^{∗} we define

(2.6.1)  h_{a}^{Γ} := ∑_{l∈L, l≱0} p_{l′+l} = ∑_{b∈S_{Γ}(a)} q_{b}^{Γ},

where S_{Γ}(a) = {b ∈ L′(Γ) : b = a + ∑ n_{j}E_{j}, n_{j} ∈ ℤ, ∃ n_{j} < 0}.

We assume that s ≥ 2 and we fix an end–vertex j_{0} of Γ. We set

h_{a}^{j_{0}} := ∑ q_{b}^{Γ} (sum over b = a + ∑ n_{j}E_{j}, n_{j} ∈ ℤ, n_{j_{0}} < 0).

The inclusion of the subgraph Γ \ j_{0} induces i : L(Γ \ j_{0}) → L(Γ), i(E_{j}) ↦ E_{j} (the symbol i sometimes is omitted), and its dual, the restriction R : L′(Γ) → L′(Γ \ j_{0}) with

(2.6.2)  \{ \begin{matrix}R(E_{j}^{∗,Γ}) = E_{j}^{∗,Γ\j_{0}} for j ≠ j_{0}, and R(E_{j_{0}}^{∗,Γ}) = 0; \\ R(E_{j}^{Γ}) = E_{j}^{Γ\j_{0}} for j ≠ j_{0}, and R(E_{j_{0}}^{Γ}) = −E_{j′}^{∗,Γ\j_{0}}, \end{matrix}

where j′ is the adjacent vertex of j_{0}. We abridge R(∑_{j} a_{j}E_{j}^{∗}) by R(a).

One has the following combinatorial formula, whose proof again we will omit.

Proposition 2.6.3. Assume that j′, the adjacent vertex of j_{0} in Γ, has valency two. Then, if a_{j_{0}} ≫ 0 (compared with R(a)), one has

h_{a}^{Γ} = h_{a}^{j_{0}} + h_{R(a)}^{Γ\j_{0}}.

Corollary 2.6.4. Fix an end–vertex j_{0} as above. For l′ = ∑_{j} a_{j}E_{j}^{∗,Γ}, with all a_{j} large (and a_{j_{0}} large compared with the others), one has:

h_{a}^{j_{0}} = −𝔰_{[−l′]}^{Γ} − \frac{K_{Γ}^{2} + |J |}{8} + 𝔰_{[−R(l′)]}^{Γ\j_{0}} + \frac{K_{Γ\j_{0}}^{2} + |J \ j_{0}|}{8} − \frac{(l′, E_{j_{0}}^{∗}) · (l′ + K_{Γ}, E_{j_{0}}^{∗})}{2(E_{j_{0}}^{∗}, E_{j_{0}}^{∗})}.

Proof. Theorem (2.5.1) applied for l′ and R(l′), and Proposition 2.6.3 provide

h_{a}^{j_{0}} = −𝔰_{[−l′]}^{Γ} − \frac{(K_{Γ} + 2l′)^{2} + |J |}{8} + 𝔰_{[−R(l′)]}^{Γ\j_{0}} + \frac{(K_{Γ\j_{0}} + 2R(l′))^{2} + |J \ j_{0}|}{8}.

Then use the identity K_{Γ\j_{0}} = R(K_{Γ}), and

l′′ = iR(l′′) + \frac{(l′′, E_{j_{0}}^{∗})}{(E_{j_{0}}^{∗}, E_{j_{0}}^{∗})} · E_{j_{0}}^{∗}

for both l′′ = l′ and l′′ = K_{Γ}, and (i(l), E_{j_{0}}^{∗}) = 0 for any l ∈ L′(Γ \ j_{0}). □

2.7. The series H_{[l′],j_{0}}(t) and its periodic constant. For any series S(t) ∈ ℤ[[L′]], S(t) = ∑_{l′} c_{l′}t^{l′}, we have the natural decomposition

S = ∑_{h∈L′/L} S_{h}, where S_{h} := ∑_{l′ : [l′]=h} c_{l′}t^{l′}.

In particular, for any fixed class [l′] ∈ L′/L, one can consider the component Z_{[l′]}(t) of Z(t). In fact, see e.g. [N08b, (3.1.20)],

(2.7.1)  Z_{[l′]}(t) = \frac{1}{d} ∑ ; ρ∈(L′/L)ρ([l′])^{−1} · ∏_{j∈J} (1 − ρ([E_{j}^{∗}])t^{E_{j}^{∗}})^{δ_{j}−2},

where (L′/L) is the Pontjagin dual of L′/L, and d = det(−I) = |L′/L|.

Definition 2.7.2. For any class [l′] ∈ L′/L and vertex j_{0} ∈ J of Γ set

H_{[l′],j_{0}}(t) := Z_{[l′]}(t)| _{{t_{j_{0}} =t^{d} \atop t_{j} =1 for j≠j_{0}}} ∈ ℤ[[t]].

Definition 2.7.3. Periodic constant [NO09, 3.9], [O08, 4.8(1)]. Let S(t) = ∑_{i≥0} c_{i}t^{i} be a formal power series. Suppose that for some positive integer p, the expression ∑_{i=0}^{pn−1} c_{i} is a polynomial P_{p}(n) in the variable n. Then the constant term of P_{p}(n) is independent of p. We call this constant term the periodic constant of S and denote it by pc(S).

2.8. The surgery formula for 𝔰. The relation between the coefficients h_{a}^{j_{0}} defined in (2.6) and the series H_{[l′],j_{0}}(t) is realized as follows (below, {r} will denote the fractional part of the rational number r):

Theorem 2.8.1. Consider the graph Γ and let j_{0} be one of its end–vertices. Fix l′ = ∑_{j} a_{j}E_{j}^{∗} = ∑_{j} l′_{j}E_{j} with a as in (2.6.4). Abridge l′_{j_{0}} by ℓ. Then

(a) If H_{[l′],j_{0}}(t) = ∑_{i≥0} c_{i}t^{i} then h_{a}^{j_{0}} = ∑_{i<dℓ} c_{i}.

(b) Take l̄′ ∈ L′ such that (l̄′, E_{j_{0}}^{∗}) ∈ (−1, 0]. Then

pc(H_{[l̄′],j_{0}}) = −𝔰_{[−l̄′]}^{Γ} − \frac{(K_{Γ} + 2l̄′)^{2} + |J |}{8} + 𝔰_{[−R(l̄′)]}^{Γ\j_{0}} + \frac{(K_{Γ\j_{0}} + 2R(l̄′))^{2} + |J \ j_{0}|}{8}.

Proof. Write Z_{[l′]}(t) = ∑ c_{l′′}t^{l′′}. Then (a) follows from H_{[l′],j_{0}}(t) = ∑ c_{l′′}t^{dl′′_{j_{0}}} and

h_{a}^{j_{0}} = ∑_{b: n_{j_{0}} <0} q_{b}^{Γ} = ∑_{l′′ : l′′_{j_{0}} <l′_{j_{0}}} c_{l′′}.

(b) For any fixed l̄′ as in the assumption of (b), take l′ such that [l′] = [l̄′], and [R(l′)] = [R(l̄′)] and l′ = ∑_{j} a_{j}E_{j}^{∗} = ∑_{j} l′_{j}E_{j} with a ≫ 0. Since l̄′ − l′ ∈ L, we get that l′_{j_{0}} + (l̄′, E_{j_{0}}^{∗}) = (l̄′ − l′, E_{j_{0}}^{∗}) ∈ ℤ, hence −(l̄′, E_{j_{0}}^{∗}) is the fractional part of l′_{j_{0}}. Write l′_{j_{0}} = ℓ and let n be its integral part. For this l′ we apply part (a) and the identity of (2.6.4).

In order to compute the periodic constant of H_{[l′],j_{0}}(t), notice that if c_{l′′} ≠ 0 then l′′−l′ ∈ L, hence l′′_{j_{0}} − l′_{j_{0}} = (E_{j_{0}}^{∗}, l′ − l′′) ∈ ℤ. Therefore, if the coefficient c_{i} of H_{[l′],j_{0}} is nonzero, then i − d{ℓ} ∈ dℤ. In particular, for c_{i} ≠ 0, i < dℓ if and only if i ≤ dn − 1. This shows that one has to write h_{a}^{j_{0}} as P (n) for some polynomial P, and then, the periodic constant of H_{[l′],j_{0}} is P (0). From (2.6.4), we get that P (n) is the right hand side of that identity after the substitution (l′, E_{j_{0}}^{∗}) = −l′_{j_{0}} = −{ℓ} − n. Therefore:

pc(H_{[l′],j_{0}}) = −𝔰_{[−l′]}^{Γ} − \frac{K_{Γ}^{2} + |J |}{8} + 𝔰_{[−R(l′)]}^{Γ\j_{0}} + \frac{K_{Γ\j_{0}}^{2} + |J \ j_{0}|}{8} − \frac{\{ℓ\} · (\{ℓ\} − (K_{Γ}, E_{j_{0}}^{∗}))}{2(E_{j_{0}}^{∗}, E_{j_{0}}^{∗})}.

Now, this identity provides (b) by a straightforward computation (similar to the computation from the proof of (2.6.4)).

Strictly speaking, the above argument proves the surgery formula (b) only if δ_{j′} = 2 (cf. the assumption of (2.6.3)). In general we argue as follows: let j_{0} be an end–vertex, and let us blow up the unique edge adjacent to j_{0} getting in this way Γ^{b}. Then the newly created vertex j′ has δ_{j′} = 2. Hence in this situation we can apply the above proof for Γ^{b} and Γ^{b} \ j_{0}. Since blowing down the j′–vertex in Γ^{b} \ j_{0} we get Γ \ j_{0}, and all the involved invariants in (b) are stable with respect to blow up/down, the result follows. Indeed,

(K + 2l′)^{2} + |Γ| = (K^{b} + 2π^{∗}(l′))^{2} + |Γ^{b}|,

the 𝔰-terms are stable by (2.5.1)(b), and H_{[l′],j_{0}} by the fact that it depends only on the numbers (E_{j_{1}}^{∗}, E_{j_{2}}^{∗}) where δ_{j_{i}} ≠ 2. □

2.9. The identification of 𝔰 with the Seiberg–Witten invariant. Let Γ be a connected negative definite plumbing graph, and let X̃ be the plumbed 4–manifold constructed from Γ. If Γ is a resolution graph, then the (diffeomorphism type) of the resolution serves for it. Let σ̃_{can} be the canonical spin^{c}–structure on X̃; its first Chern class c_{1}(σ̃_{can}) is −K ∈ L′. The set of spin^{c}–structures Spin^{c}(X̃) is an L′–torsor; if we denote the L′–action by l′ ∗ σ̃, then c_{1}(l′ ∗ σ̃) = c_{1}(σ̃) + 2l′.

If M = M(Γ) is the plumbed 3–manifold associated with Γ, then M = ∂X̃, and all the spin^{c}– structures of M are obtained by restrictions from X̃. Spin^{c}(M) is an L′/L–torsor, compatible with the restriction and the projection L′ → L′/L. The canonical spin^{c}–structure σ_{can} of M is the restriction of σ̃_{can}.

We denote the Seiberg–Witten invariant by 𝔰𝔴 : Spin^{c}(M) → ℚ, σ ↦ 𝔰𝔴_{σ}. Next we recall a surgery formula satisfied by them, proved in [BN10].

Let us fix one of the end–vertices j_{0} of Γ (though the statement is true for any vertex, cf. [loc.cit]). Let X̃(Γ \ j_{0}) be the tubular neighbourhood of ∪_{j≠j_{0}}E_{j} in X̃, and M_{j_{0}} its boundary. Hence, M_{j_{0}} is the plumbed 3–manifold associated with Γ \ j_{0}.

Fix any σ ∈ Spin^{c}(M), extend it to a spin^{c}–structure σ̃ ∈ Spin^{c}(X̃) of the form σ̃ = l̃′ ∗ σ̃_{can} with l̃′ ∈ L′, (l̃′, E_{j_{0}}^{∗}) ∈ [0, 1). Then consider R(l̃′) ∈ L′(Γ \ j_{0}) and the restriction σ̃_{j_{0}} and σ_{j_{0}} of σ̃ to X̃(Γ \ j_{0}) and ∂X̃(Γ \ j_{0}) respectively. Then the main result of [BN10] says the following identity. (Here we wish to draw the reader’s attention to the notational differences between the present note and [BN10]: in that article the ‘dual’ E_{j}^{∗} has opposite sign, this creates a sign difference in (l̃′, E_{j_{0}}^{∗}) ∈ [0, 1), and also in the expression of H from (2.7.1) the characters ρ ∈ (L′/L) should be replaced by their inverses. Hence, H_{σ,j_{0}} of [BN10] is our H_{[−l̃′],j_{0}}.)

Theorem 2.9.1. [BN10]

𝔰𝔴_{σ}(M) + \frac{c_{1}(σ̃)^{2} + |J |}{8} = −pc(H_{[−l̃′],j_{0}}) + 𝔰𝔴_{σ_{j_{0}}} (M_{j_{0}}) + \frac{c_{1}(σ̃_{j_{0}})^{2} + |J \ j_{0}|}{8}.

2.10. The proof of Theorem A is completed by the next result.

Theorem 2.10.1. Let 𝔰 : L′/L → ℚ be the invariant defined from the series Z(t) in (2.5.1). Then for any [l′] ∈ L′/L one has:

𝔰_{[l′]} = 𝔰𝔴_{[l′]∗σ_{can}}.

Proof. Notice that −l̄′ considered in (2.8.1) satisfies the needed assumptions for l̃′ in (2.9). Moreover, c_{1}(σ̃) = −(2l̄′ + K), and c_{1}(σ̃_{j_{0}}) = −R(2l̄′ + K), and σ = [−l̄′] ∗ σ_{can}. Hence, the surgery identities (2.9.1) and (2.8.1)(b) show that 𝔰_{[−l′]} satisfy the same surgery formula than 𝔰𝔴_{[−l′]∗σ_{can}}. One can verify that they coincide for graphs with one vertex (or one can apply [N08c, §4] which shows that they coincide for links of splice–quotient singularities, including all Seifert-manifold). Therefore, by induction on the number of vertices, we get the result. □

2.11. The proof of Theorem B is a combination of (1.3.7), (2.5.3) and (2.10.1).

3. Exact sequences of lattice cohomology.

3.1. Preliminaries and the goals. The lattice cohomology {ℍ^{q}(Γ)}_{q≥0} was introduced in [N08]. In its original version, it was associated with any connected negative definite plumbing graph Γ, or, equivalently, with any oriented 3–manifold, which might appear as the link of a local complex normal surface singularity. Lattice cohomology (together with the graded roots) plays a crucial role in the comparison of the analytic and topological invariants of surface singularities, cf. [N05, N07, N08], see also [BN10, NN02, N10] for relations with the Seiberg–Witten invariants of the link. Additionally, the lattice cohomology (conjecturally) offers a combinatorial description for the Heegaard–Floer homology of Ozsváth and Szabó (for this theory see [OSz03, OSz04, OSz04b] and the long list of articles of Ozsváth and Szabó). Indeed, in [N08] the author conjectured that ⊕_{q even/odd}ℍ^{q}(Γ) is isomorphic as a graded ℤ[U]–module with HF_{even/odd}^{+}(−M(Γ)), where M(Γ) is the plumbed 3–manifold associated with Γ. (Recall that at this moment there is no combinatorial definition/characterization of HF^{+}.)

For rational and ‘almost rational’ graphs this correspondence was established in [OSz03, N05] (see also [NR10] for a different situation and [N08] for related results). A possible machinery which might help to prove the general conjecture is based on the surgery exact sequences. They are established for the Heegaard–Floer theory in the work of Ozsváth and Szabó. Our goal is to prove the analogous exact sequences for the lattice cohomology. In fact, independently of the above conjecture and correspondence, the proof of such exact sequences is of major importance, and they are fundamental in the computation and in finding the main properties of the lattice cohomologies.

The formal, combinatorial definition of the lattice cohomology permits to extend its definition to arbitrary graphs (plumbed 3–manifolds), the connectedness and negative definiteness assumptions can be dropped. Nevertheless, in the proof of the exact sequences, we will deal only with non–degenerate graphs (they are those graphs whose associated intersection form has non–zero determinant).

More precisely, for any graph Γ and fixed vertex j_{0}, we consider the graphs Γ \ j_{0} and Γ_{j_{0}}^{+}, where the first one is obtained from Γ by deleting the vertex j_{0} and adjacent edges, while the second one is obtained from Γ by replacing the decoration e_{j_{0}} of the vertex j_{0} by e_{j_{0}} + 1. We will assume that all these graphs are non–degenerate. Then Theorem 3.6.3 establishes the following long exact sequence.

Theorem A’. Assume that the graphs Γ_{j_{0}}^{+}, Γ and Γ \ j_{0} are non–degenerate. Then

⋯ ⟶ ℍ^{q}(Γ_{j_{0}}^{+}) ⟶^{𝔸^{q}} ℍ^{q}(Γ) ⟶^{\overline{𝔹}^{q}} ℍ^{q}(Γ \ j_{0}) ⟶^{ℂ^{q}} ℍ^{q+1}(Γ_{j_{0}}^{+}) ⟶ ⋯

is an exact sequence of ℤ[U]–modules.

The first 3 terms of the exact sequence (i.e. the ℍ^{0}–part) were already used in [OSz03] (see also [N05]), and the existence of the long exact sequence was already proved over ℤ_{2}–coefficients in [Gr08]. Here we establish its validity over ℤ. In the proof we not only find the correct sign–modifications, but we also replace some key arguments. (Nevertheless, the proof follows the main steps of [Gr08].)

For negative definite graphs (i.e. when Γ_{j_{0}}^{+}, hence Γ and Γ \ j_{0} too are negative definite), the above exact sequence has some important additional properties. By general theory (cf. [N08]), if Γ is negative definite then ℍ^{0}(Γ) contains a canonical submodule 𝕋 and one has a direct sum decomposition ℍ^{0} = 𝕋 ⊕ ℍ_{red}^{0}. 𝕋 is the analogue of the image of HF^{∞} in HF^{+} in the Heegaard– Floer theory. On the other hand, in Heegaard–Floer theory by a result of Ozsváth and Szabó, the operator ℂ^{0} restricted on 𝕋(Γ \ j_{0}) is zero (this follows from the fact that the corresponding cobordism connecting Γ \ j_{0} and Γ_{j_{0}}^{+} is coming from a non–negative definite surgery). The analogue of this result is Theorem 3.6.5:

Theorem B’. Assume that Γ_{j_{0}}^{+} is negative definite. Consider the exact sequence

0 ⟶ ℍ^{0}(Γ_{j_{0}}^{+}) ⟶^{𝔸^{0}} ℍ^{0}(Γ) ⟶^{\overline{𝔹}^{0}} ℍ^{0}(Γ \ j_{0}) ⟶^{ℂ^{0}} ℍ^{1}(Γ_{j_{0}}^{+}) ⟶ ⋯

and the canonical submodule 𝕋(Γ \ j_{0}) of ℍ^{0}(Γ \ j_{0}). Then the restriction ℂ^{0}|𝕋(Γ \ j_{0}) is zero.

Using the above results one proves for a graph Γ with at most n bad vertices (for the definition, see 3.7) the vanishing ℍ_{red}^{q}(Γ) = 0 for any q ≥ n (where ℍ_{red}^{q} = ℍ^{q} for q > 0).

ℍ^{∗}(Γ) has a natural direct sum decomposition indexed by the set of the characteristic element classes (in the case of the Heegaard–Floer, or Seiberg–Witten theory, they correspond to the spin^{c}– structures of M(Γ)). Namely, ℍ^{∗}(Γ) = ⊕_{[k]}ℍ^{∗}(Γ, [k]). In the exact sequence of Theorem A’ the operators mix these classes. Theorem 4.2.2 provides an exact sequence which connects the lattice cohomologies of Γ and Γ \ j_{0} with fixed (un–mixed) characteristic element classes. More precisely, for any characteristic element k of Γ, we define a ℤ[U]–module {ℍ_{rel}^{q}(k)}_{q≥0}, the relative lattice cohomology associated with (Γ, j_{0}, k). It has finite ℤ–rank and it fits in the following exact sequence:

Theorem C’. Assume that Γ and Γ \ j_{0} are non–degenerate. One has a long exact sequence of ℤ[U]–modules:

⋯ ⟶ ℍ_{rel}^{q}(k) ⟶^{𝔸_{rel}^{q}} ℍ^{q}(Γ, [k]) ⟶^{𝔹_{rel}^{q}} ℍ^{q}(Γ \ j_{0}, [R(k)]) ⟶^{ℂ_{rel}^{q}} ℍ_{rel}^{q+1}(k) ⟶ ⋯

where R(k) is the restriction of k (and the operators also depend on the choice of the representative k).

The existence of such a long exact sequence is predicted and motivated by the surgery formula for the Seiberg–Witten invariant established in [BN10]: the results of section 4 resonate perfectly with the corresponding statements of Seiberg–Witten theory. This allows us to compute the Euler characteristic of the relative lattice cohomology in terms of the Seiberg–Witten invariants associated with M(Γ) and M(Γ \ j_{0}).

Now we start to introduce the preliminary notions necessary for the proofs of the above statements and we provide some of the key steps.

3.2. Blowing up Γ. Since in the main construction we will need some of the operators induced by blowing down, we will make explicit the involved morphisms.

Starting from now, the graph is neither necessarily connected, nor necessarily negative definite. Nevertheless, we will assume that the intersection form is non–degenerate.

We assume that Γ′ is obtained from Γ by ‘blowing up the vertex j_{0}’. More precisely, Γ′ denotes a graph with one more vertex and one more edge: we glue to the vertex j_{0} by the new edge the new vertex E_{new} with decoration −1 (and genus 0), while the decoration of E_{j_{0}} is modified from e_{j_{0}} into e_{j_{0}} − 1, and we keep all the other decorations. We will use the notations L(Γ), L(Γ′), L′(Γ), L′(Γ′). Let w′ : Char(Γ′) → ℚ be the weight function of Γ′ defined similarly as in (1.3.1). (We may use the following convention for the ordering of the indices: if j ≠ j_{0}, then j < j_{0} < j_{new}.) The following facts can be verified:

3.2.1. Consider the maps π_{∗} : L(Γ′) → L(Γ) defined by π_{∗}(∑ x_{j}E_{j} + x_{new}E_{new}) = ∑ x_{j}E_{j}, and π^{∗} : L(Γ) → L(Γ′) defined by π^{∗}(∑ x_{j}E_{j}) = ∑ x_{j}E_{j} + x_{j_{0}}E_{new}. Then (π^{∗}x, x′)_{Γ′} = (x, π_{∗}x′)_{Γ}. This shows that (π^{∗}x, π^{∗}y)_{Γ′} = (x, y)_{Γ} and (π^{∗}x, E_{new})_{Γ′} = 0 for any x, y ∈ L(Γ). Both π_{∗} and π^{∗} extend over L ⊗ ℚ and L′.

3.2.2. Set the (nonlinear) map: c : L′(Γ) → L′(Γ′), c(l′) := π^{∗}(l′) + E_{new}. Then c(Char(Γ)) ⊂ Char(Γ′) and c induces an isomorphism between the orbit spaces Char(Γ)/2L(Γ) ≃ Char(Γ′)/2L(Γ′). Moreover, for any k ∈ Char(Γ) and k′ := c(k) one has w′(k′) = w(k).

3.2.3. π_{∗}(Char(Γ′)) ⊂ Char(Γ). For any k′ ∈ Char(Γ′) write k := π_{∗}(k′). Then k′ = π^{∗}(k)+aE_{new} for some odd integer a, and w′(k′) − w(k) = \frac{a^{2}−1}{8} ∈ ℤ_{≥0}.

The maps c and π_{∗} can be extended to the level of cubes and complexes as follows.

3.2.4. For any q ≥ 0 and □ = (k′, I) ∈ Q_{q}(Γ′) one defines π_{∗}((k′, I)) := (π_{∗}(k′), I) ∈ Q_{q}(Γ), provided that j_{new} ∉ I. By (3.2.3) one has w_{Γ′}(□) − w_{Γ}(π_{∗}(□)) ≥ 0. This defines a homological morphism π_{∗}^{h} : F_{q}(Γ′) → F_{q}(Γ) by

(3.2.5)  π_{∗}^{h}(□) = \{ \begin{matrix}U^{w_{Γ′} (□)−w_{Γ}(π_{∗}(□))} π_{∗}(□) & if j_{new} ∉ I, \\ 0 & else. \end{matrix}

Using (1.3.3) one verifies that π_{∗}^{h} ∘ ∂ = ∂ ∘ π_{∗}^{h}, hence π_{∗}^{h} is morphism of homological complexes. In particular, its dual π_{∗}^{c} : F^{q}(Γ) → F^{q}(Γ′), defined by π_{∗}^{c}(ϕ) = ϕ ∘ π_{∗}^{h}, satisfies π_{∗}^{c} ∘ δ = δ ∘ π_{∗}^{c}, hence it is a morphism of complexes as well.

3.2.6. Using π_{∗} ∘ c and the definitions one gets π_{∗}^{h} ∘ c^{h} = id_{F_{∗}(Γ)}. On the other hand, c^{h} ∘ π_{∗}^{h} is not the identity, but it is homotopic to id_{F_{∗}(Γ′)}. Indeed, we define the homotopy operator K : F_{∗}(Γ′) → F_{∗+1}(Γ′) as follows. Write any k as cπ_{∗}(k) + 2aE_{new} for some a ∈ ℤ. Then define K((k, I)) as 0 if either j_{new} ∈ I or a = 0. Otherwise take for K((k, I)):

sign(a) · ∑ U^{w(k,I)−w(cπ_{∗}(k)+2lE_{new},I∪j_{new})} (cπ_{∗}(k) + 2lE_{new}, I ∪ j_{new}),

where the summation is over l ∈ {0, 1, …, a−1} if a > 0 and l ∈ {a, …, −1} if a < 0. (The exponents are non–negative because of (3.2.3).) Then, again by a computation, ∂ ∘ K − K ∘ ∂ = id − c^{h} ∘ π_{∗}^{h}.

In particular, π_{∗}^{h} and c^{h} induce (degree preserving) isomorphisms of the corresponding lattice cohomologies. In the sequel the operator π_{∗}^{h} will be crucial.

3.3. Comparing Γ and Γ \ j_{0}. We consider a non–degenerate graph Γ as in 3.2, and we fix one of its vertices j_{0} ∈ J. The new graph Γ \ j_{0} is obtained from Γ by deleting the vertex j_{0} and all its adgacent edges. We will denote by L(Γ), L′(Γ), L(Γ \ j_{0}), L′(Γ \ j_{0}) the corresponding lattices.

The operator i : L(Γ \ j_{0}) → L(Γ), i(E_{j,Γ\j_{0}}) = E_{j,Γ} identifies L(Γ \ j_{0}) with a sublattice of L(Γ). The dual operator (restriction) is R : L′(Γ) → L′(Γ \ j_{0}), R(E_{j,Γ}^{∗}) = E_{j,Γ\j_{0}}^{∗} for j ≠ j_{0} and R(E_{j_{0},Γ}^{∗}) = 0. It satisfies (i_{ℚ}(x), y)_{Γ} = (x, R(y))_{Γ\j_{0}} for any x ∈ L′(Γ \ j_{0}) and y ∈ L′(Γ). (Here, E_{j,Γ}^{∗} respectively E_{j,Γ\j_{0}}^{∗} are the usual dual generators of L′ considered in Γ, respectively in Γ \ j_{0}.)

Recall that l′ = ∑_{j} a_{j}E_{j,Γ}^{∗} is characteristic if and only if a_{j} ≡ e_{j} (mod 2). In particular, R sends characteristic elements into characteristic elements. On the other hand, i_{ℚ} not necessarily preserves characteristic elements.

Although R(Char(Γ)) ⊂ Char(Γ \ j_{0}), this operator cannot be extended to the level of cubes. Indeed, notice that

(3.3.1)  R(E_{j_{0}}) = − ∑_{(j,j_{0})∈E_{Γ}} E_{j,Γ\j_{0}}^{∗},

where the sum runs over the adjacent vertices of j_{0} in Γ. In particular, the endpoints of the 1–cube (k, j_{0}) are sent into R(k) and R(k) + 2R(E_{j_{0}}), which cannot be expressed as combination of 1–cubes in Γ \ j_{0}. In the next subsection we will consider another operator, which can be extended to the level of cubes (and which, in fact, operates in a different direction than R, cf. 1.4).

3.4. The B–operator. Consider b : L′(Γ \ j_{0}) → L′(Γ) defined by

∑_{j} a_{j}E_{j,Γ\j_{0}}^{∗} ↦ ∑_{j} a_{j}E_{j,Γ}^{∗}.

Clearly, if k ∈ Char(Γ \ j_{0}) then b(k) + a_{j_{0}}E_{j_{0},Γ}^{∗} ∈ Char(Γ) for any a_{j_{0}} with a_{j_{0}} ≡ e_{j_{0}} (mod 2). In order to see how it operates on cubes, notice that in Γ \ j_{0} for any j ∈ J (Γ \ j_{0}) one has

E_{j,Γ\j_{0}} = −e_{j} E_{j,Γ\j_{0}}^{∗} − ∑_{(i,j)∈E_{Γ\j_{0}}} E_{i,Γ\j_{0}}^{∗},

and a similar relation holds in Γ too. Therefore, one gets that

(3.4.1)  b(E_{j}) = E_{j} + (E_{j_{0}}, E_{j})_{Γ} · E_{j_{0},Γ}^{∗}.

Therefore,

(3.4.2)  b(l′) = i_{ℚ}(l′) + (E_{j_{0}}, i_{ℚ}(l′))_{Γ} · E_{j_{0},Γ}^{∗}.

In particular, b is a ‘small’ modification of i_{ℚ}, but this modification is enough to extend it to the level of cubes. Although the vertices of (k, I) are not sent to the vertices of a cube (provided that I contains elements adjacent to j_{0} in Γ), cf. (3.4.1), nevertheless if we consider all the shifts in the direction of the ‘error’ of (3.4.1) we get a well–defined operator

B_{q} : F_{q}(Γ \ j_{0}) → F_{q}(Γ)

given by

(3.4.3)  B_{q}((k, I)) := ∑_{a_{j_{0}} ≡e_{j_{0}} (mod 2)} (b(k) + a_{j_{0}}E_{j_{0},Γ}^{∗}, I).

(Here we keep the notation ‘B’, since this notation was used in similar contexts, as in [OSz03, N05, Gr08].)

B_{q} induces a morphism B^{q} : F^{q}(Γ) → F^{q}(Γ \ j_{0}) via (B^{q}ϕ)(□) = ϕ(B_{q}□). A straightforward (slightly long) computation shows that B_{q} commutes with the boundary operator ∂, hence B^{∗} ∘ δ = δ∘B^{∗} too. In particular, one gets a well–defined morphism of ℤ[U]–modules 𝔹^{∗} : ℍ^{∗}(Γ) → ℍ^{∗}(Γ\j_{0}).

3.5. The sign–modified B–operator. In the exact sequences considered in the next section we will need to modify the B–operator by a sign (compare with the end of §2 of [OSz03], where the case q = 0 is discussed). The definition depends on some choices.

For each L–orbit [k] := k +2L(Γ\j_{0}) ⊂ Char(Γ\j_{0}) we will fix a representative r_{[k]} ∈ [k]. Hence, for any characteristic element k, with orbit [k], k − r_{[k]} ∈ 2L(Γ \ j_{0}), hence (k − r_{[k]}, E_{j,Γ\j_{0}}^{∗}) ∈ 2ℤ for any j ≠ j_{0}. In particular, (k − r_{[k]}, R(E_{j_{0}})) ∈ 2ℤ for R(E_{j_{0}}) defined in (3.3.1). Then the modified operator

\overline{B}_{q} : F_{q}(Γ \ j_{0}) → F_{q}(Γ)

is given by

(3.5.1)  \overline{B}_{q}((k, I)) := ∑_{a_{j_{0}} ≡e_{j_{0}} (mod 2)} (−1)^{n(k,a_{j_{0}} )/2} · (b(k) + a_{j_{0}}E_{j_{0},Γ}^{∗}, I),

where

n(k, a_{j_{0}}) := (k − r_{[k]}, R(E_{j_{0}})) + a_{j_{0}} + e_{j_{0}}.

Then, again, \overline{B}^{∗} commutes with the boundary operator, hence \overline{B}^{∗} : F^{q}(Γ) → F^{q}(Γ \ j_{0}) defined by (\overline{B}^{q}ϕ)(□) = ϕ(\overline{B}_{q}□) commutes with δ, hence it also defines a ℤ[U]–modules morphism \overline{𝔹}^{∗} : ℍ^{∗}(Γ) → ℍ^{∗}(Γ \ j_{0}).

Remark 3.5.2. The morphisms 𝔹^{∗} and \overline{𝔹}^{∗} do not preserve the gradings neither the orbits Char/2L (i.e. they do not split in direct sum with respect to these orbits).

3.6. The surgery exact sequence and applications. Now we are ready to sketch the proofs and give some applications.

Let Γ be a non–degenerate graph as above with s = |J | ≥ 2. We fix a vertex j_{0} ∈ J. Associated with j_{0} we consider two other graphs, namely Γ \ j_{0} and Γ_{j_{0}}^{+}. The second one is obtained from Γ by modifying the decoration e_{j_{0}} of the vertex j_{0} into e_{j_{0}} + 1. In order to stay in our category of objects, we will assume that both graphs Γ \ j_{0} and Γ_{j_{0}}^{+} are non–degenerate.

Our goal is to establish a long exact sequence connecting the lattice cohomologies of these three graphs, similar which is valid for the Heegaard–Floer cohomologies provided by surgeries. As usual, in order to define a long exact sequence, we need first to determine a short exact sequence of complexes.

The graphs Γ and Γ \ j_{0} will be connected by the \overline{B}–operator, cf. 3.5. We define the ‘A’–operator connecting Γ and Γ_{j_{0}}^{+} as follows. Let Γ^{b} be the graph obtained from Γ by attaching a new vertex j_{new} (via a single new edge) to j_{0}, such that the decoration of the new vertex is −1. Then, for the pair Γ, Γ^{b} (since Γ = Γ^{b} \ j_{new}), we can apply the construction and results of §3.3. In particular, we get morphisms of complexes: B_{∗} : F_{∗}(Γ) → F_{∗}(Γ^{b}) and B^{∗} : F^{∗}(Γ^{b}) → F^{∗}(Γ). Next, since Γ_{j_{0}}^{+} obtained from Γ^{b} by blowing down the new vertex j_{new}, by § 3.2 we get morphisms of complexes: π_{∗}^{h} : F_{∗}(Γ^{b}) → F_{∗}(Γ_{j_{0}}^{+}) and π_{∗}^{c} : F^{∗}(Γ_{j_{0}}^{+}) → F^{∗}(Γ^{b}). By composition we get the A–operators:

A_{∗} := π_{∗}^{h} ∘ B_{∗} : F_{∗}(Γ) → F_{∗}(Γ_{j_{0}}^{+}), and A^{∗} := B^{∗} ∘ π_{∗}^{c} : F^{∗}(Γ_{j_{0}}^{+}) → F^{∗}(Γ).

In particular, we can consider the short sequence of complexes

(3.6.1)  0 → F^{∗}(Γ_{j_{0}}^{+}) ⟶^{A^{∗}} F^{∗}(Γ) ⟶^{\overline{B}^{∗}} F^{∗}(Γ \ j_{0}) → 0.

Theorem 3.6.2. The short sequence of complexes (3.6.1) is exact.

The proof is omitted here, for details see [N10b].

As a corollary we get:

Theorem 3.6.3. Assume that the graphs Γ_{j_{0}}^{+}, Γ and Γ \ j_{0} are non–degenerate. Then

⋯ ⟶ ℍ^{q}(Γ_{j_{0}}^{+}) ⟶^{𝔸^{q}} ℍ^{q}(Γ) ⟶^{\overline{𝔹}^{q}} ℍ^{q}(Γ \ j_{0}) ⟶^{ℂ^{q}} ℍ^{q+1}(Γ_{j_{0}}^{+}) ⟶ ⋯

is an exact sequence of ℤ[U]–modules.

Next, we analyze negative definite graphs.

For any graph Γ we write det(Γ) for the determinant of the negative of the intersection form associated with Γ. If Γ is negative definite then det(Γ) is obviously positive.

If Γ is negative definite then Γ \ j_{0} is automatically so for any j_{0}. Nevertheless, this is not the case for Γ_{j_{0}}^{+} (although, if Γ_{j_{0}}^{+} is negative definite then Γ is so too).

Lemma 3.6.4. Assume that Γ is negative definite. Then Γ_{j_{0}}^{+} is negative definite if and only if det(Γ) > det(Γ \ j_{0}). If these conditions are satisfied then E_{j_{0},Γ}^{∗} ∉ L(Γ) and (E_{j_{0},Γ}^{∗})^{2} ∉ ℤ.

Proof. Γ_{j_{0}}^{+} is negative definite if and only if det(Γ_{j_{0}}^{+}) is positive (provided that Γ \ j_{0} is negative definite). But det(Γ_{j_{0}}^{+}) = det(Γ) − det(Γ \ j_{0}). The last statement follows from −(E_{j_{0},Γ}^{∗})^{2} = det(Γ \ j_{0})/ det(Γ), which cannot be an integer. □

In the sequel we assume that all the graphs are negative definite, but not necessarily connected. The next theorem is an addendum of Theorem 3.6.3.

Theorem 3.6.5. Assume that Γ_{j_{0}}^{+} is negative definite. Consider the exact sequence

0 ⟶ ℍ^{0}(Γ_{j_{0}}^{+}) ⟶^{𝔸^{0}} ℍ^{0}(Γ) ⟶^{\overline{𝔹}^{0}} ℍ^{0}(Γ \ j_{0}) ⟶^{ℂ^{0}} ℍ^{1}(Γ_{j_{0}}^{+}) ⟶ ⋯

and the canonical submodule 𝕋(Γ \ j_{0}) of ℍ^{0}(Γ \ j_{0}). Then 𝕋(Γ \ j_{0}) ⊂ im \overline{𝔹}^{0}; or, equivalently, the restriction ℂ^{0}|𝕋(Γ \ j_{0}) is zero.

3.7. Application. Graphs with n bad vertices. We say that a negative definite connected graph is ‘rational’ if it is the plumbing representation of a the link of a rational singularity (or, the resolution graph of a rational singularity). They were characterized combinatorially by Artin, for more details see [N99, N05].

We fix an integer n ≥ 0. We say that a negative definite graph has at most n ‘bad’ vertices if we can find n vertices {j_{k}}_{1≤k≤n}, such that replacing their decorations e_{j_{k}} by some more negative integers e′_{j_{k}} ≤ e_{j_{0}} we get a graph whose all connected components are rational. (Notice that this is a generalization of the notion of ‘bad’ vertices of [OSz03]. A graph with at most one bad vertex is called ‘almost rational’ in [N05, N08]. Any ‘star–shaped’ graph, i.e. normal form of a Seifert manifold, has at most one bed vertex, namely the ‘central’ vertex.)

Theorem 3.7.1. If Γ has at most n bad vertices then ℍ_{red}^{q}(Γ) = 0 for q ≥ n.

This is a generalization of [N08, (4.3.3)], where it is proved for n = 1 (compare also with the vanishing theorems of [OSz03, N05]).

Proof. We run induction over n. If n = 0, then all the components of Γ are rational. By [N05], their reduced lattice cohomology is vanishing. This fact remains true for more components too, since the cohomology of a tensor product of two acyclic complexes is acyclic.

Assume now that the statement is true for n − 1 and take Γ with n bad vertices. Let j be one of them. Let Γ_{j}(−ℓ) be the graph obtained from Γ by replacing the decoration e_{j} by e_{j} − ℓ (ℓ ≥ 0). Then consider the long exact sequence (3.6.3) associated with Γ_{j}(−ℓ), Γ_{j}(−ℓ − 1) and Γ \ j_{0}, for all ℓ ≥ 0. Then, by the inductive step, we get that ℍ^{q}(Γ) = ℍ^{q}(Γ_{j}(−ℓ)) for all ℓ and q ≥ n. (Here, in the case n = 1, Theorem 3.6.5 is also used.) Since for ℓ ≫ 0 the graph Γ_{j}(−ℓ) has only n − 1 bad vertices, all these modules vanish. □

In fact, the above statement can be improved as follows.

Theorem 3.7.2. Assume that Γ has at most n ≥ 2 bad vertices {j_{k}}_{1≤k≤n} such that Γ \ j_{1} has at most (n − 2) bad vertices. Then ℍ_{red}^{q}(Γ) = 0 for q ≥ n − 1.

Proof. The proof is same as above, if one eliminates first the vertex j_{1}. □

See [N05, (8.2)(5.b)] for a graph Γ with 2 bad vertices {j_{1}, j_{2}} such that Γ \ j_{1} has only rational components.

4. The ‘relative lattice cohomology’ and its surgery exact sequence

4.1. Preliminaries. The motivation for the next exact sequence is two–folded. First, the exact sequence (3.6.3) mixes the classes of the characteristic elements. (Note that the surgery exact sequence valid for the Heegaard–Floer theory — which is one of out models for the theory — does the same.) These classes, in topological language, correspond to the spin^{c}–structures of the corresponding plumbed 3–manifolds. It would be desirable to have a surgery exact sequence which do not mix them, and allows inductively the computation of each ℍ^{∗}(Γ, [k]) for each [k] independently.

The second motivation is the main result of [BN10]. This is a surgery formula for the Seiberg– Witten invariant of negative definite plumbed 3–manifolds, it compares these invariants for Γ and Γ \ j_{0} for fixed (non–mixed) spin^{c}–structures. The third term in the main formula of [BN10] comes from a ‘topological’ Poincaré series associated with the plumbing graph, and its nature is rather different than the other two terms.

Here our goal is to determine an exact sequence connecting ℍ^{∗}(Γ, [k̄]) and ℍ^{∗}(Γ \ j_{0}, [R(k̄)]) (where R(k̄) is the restriction of k̄) with the newly defined third term, playing the role of the relative cohomology. Its relationship with the Poincaré series used in [BN10] will also be treated.

4.2. The ‘relative’ complex and cohomology. We consider a non–degenerate graph Γ and j_{0} one of its vertices.

We fix [k] ∈ Char(Γ \ j_{0})/2L(Γ \ j_{0}) and a characteristic element k_{m} ∈ [k] with w(k_{m}) = min_{k∈[k]} w(k). Furthermore, we fix a_{0} satisfying a_{0} ≡ e_{j_{0}} (mod 2). Then k_{a_{0}} := i(k_{m})+( (i(k_{m}), E_{j_{0}})+ a_{0} )E_{j_{0}}^{∗} ∈ Char(Γ). In fact, for any k′ ∈ [k] one gets that i(k′) + ( (i(k_{m}), E_{j_{0}}) + a_{0} )E_{j_{0}}^{∗} ∈ Char(Γ) and it is an element of [k_{a_{0}}]. For simplicity we write r_{0} for (i(k_{m}), E_{j_{0}}) + a_{0}.

We define

B_{0,rel} : F_{0}(Γ \ j_{0}, [k]) → F_{0}(Γ, [k_{a_{0}}])

by

(4.2.1)  B_{0,rel}(k′) = i(k′) + ( (i(k_{m}), E_{j_{0}}) + a_{0} )E_{j_{0}}^{∗} = i(k′) + r_{0}E_{j_{0}}^{∗}.

This extends to the level of complexes B_{∗,rel} : (F_{∗}(Γ\j_{0}, [k]), ∂) → (F_{∗}(Γ, [k_{a_{0}}]), ∂) by B_{∗,rel}((k, I)) = (B_{0,rel}(k), I). Its dual B_{rel}^{∗} : (F^{∗}(Γ, [k_{a_{0}}]), δ) → (F^{∗}(Γ \ j_{0}, [k]), δ) is defined by B_{rel}^{∗}(ϕ) = ϕ ∘ B_{∗,rel}.

By a similar argument as for the abstract version of the theory, we get that

B_{rel}^{∗} : F^{∗}(Γ, [k_{a_{0}}]) → F^{∗}(Γ \ j_{0}, [k]) is surjective.

We define the ‘relative’ complex F_{rel}^{∗} = F_{rel}^{∗}(Γ, j_{0}, [k], a_{0}]) via ker(B_{rel}^{∗}). Let the cohomology of the complex (F_{rel}^{∗}, δ) be ℍ_{rel}^{∗} = ℍ_{rel}^{∗}(Γ, j_{0}, [k], a_{0}). It is a graded ℤ[U]–module. We refer to it as the relative lattice cohomology.

Note that both F_{rel}^{∗} and B_{rel}^{∗} depend on the choice of the representative k_{a_{0}} of [k_{a_{0}}] and are not invariants merely of the classes [k_{a_{0}}] and [k].

Theorem 4.2.2. One has the short exact sequence of complexes:

0 ⟶ F_{rel}^{∗} ⟶^{A_{rel}^{∗}} F^{∗}(Γ, [k_{a_{0}}]) ⟶^{B_{rel}^{∗}} F^{∗}(Γ \ j_{0}, [k]) ⟶ 0,

which provides a long exact sequence of ℤ[U]–modules:

⋯ ⟶ ℍ_{rel}^{q} ⟶^{𝔸_{rel}^{q}} ℍ^{q}(Γ, [k_{a_{0}}]) ⟶^{𝔹_{rel}^{q}} ℍ^{q}(Γ \ j_{0}, [k]) ⟶^{ℂ_{rel}^{q}} ℍ_{rel}^{q+1} ⟶ ⋯

(Again, the relative cohomology modules and the operators in the above exact sequence depend on the choice of k_{a_{0}}, and not only on its class [k_{a_{0}}].)

In the case when Γ is negative definite, the restriction of 𝔹_{rel}^{0} to T_{d[k_{a_{0}} ]}^{+} has its image in T_{d[k]}^{+}.

Proposition 4.2.3. Assume that Γ (but not necessarily Γ_{j_{0}}^{+}) is negative definite. Then

𝔹_{rel}^{0} : T_{d[k_{a_{0}} ]}^{+} → T_{d[k]}^{+} is onto.

In particular, ℍ_{rel}^{∗} has finite rank over ℤ. Moreover, one has an exact sequence of finite ℤ–modules:

0 ⟶ ℍ_{rel}^{0} ⟶ ℍ_{red}^{0}(Γ, [k_{a_{0}}])⊕ℤ^{n} ⟶ ℍ_{red}^{0}(Γ\j_{0}, [k]) ⟶ ℍ_{rel}^{1} ⟶ ℍ_{red}^{1}(Γ, [k_{a_{0}}]) ⟶ ℍ_{red}^{1}(Γ\j_{0}, [k]) ⋯ where n := w_{Γ}(i(k_{m}) + r_{0}E_{j_{0}}^{∗}) − min{ w_{Γ} | [k_{a_{0}}]} ∈ ℤ_{≥0}.

Proof. First note that

w_{Γ}(i(k′) + r_{0}E_{j_{0}}^{∗}) = w_{Γ\j_{0}}(k′) − \frac{1 + r_{0}^{2}(E_{j_{0}}^{∗})^{2}}{8} .

This applied for k′ = k_{m} provides

(4.2.4)  n + \frac{d[k_{a_{0}}]}{2} = \frac{d[k]}{2} − \frac{1 + r_{0}^{2}(E_{j_{0}}^{∗})^{2}}{8} .

For any l ≥ 0 and k̄ ∈ [k_{a_{0}}] set ϕ̄_{l} ∈ F^{0}(Γ, [k_{a_{0}}]) defined by

ϕ̄_{l}(k̄) = U^{−l−d[k_{a_{0}} ]/2+w_{L}(k̄)} (k̄ ∈ [k_{a_{0}}]).

For different l ≥ 0 they generate T_{d[k_{a_{0}} ]}^{+} ⊂ ℍ^{0}(Γ, [k_{a_{0}}]). Similarly, set {ϕ_{l}}_{l≥0} in F^{0}(Γ\j_{0}, [k]), where

ϕ_{l}(k′) = U^{−l−d[k]/2+w(k′)} (k′ ∈ [k]).

They generate T_{d[k]}^{+}. From the above identities and from the definition of B_{rel}^{0} one gets for any l ≥ 0:

𝔹_{rel}^{0}(ϕ̄_{n+l}) = ϕ_{l}.

Hence the restriction 𝔹_{rel}^{0} : T_{d[k_{a_{0}} ]}^{+} → T_{d[k]}^{+} is onto and the ℤ–rank of its kernel is n. □

The reader is invited to recall the definition of the Euler characteristic of the lattice cohomology from (1.3.5). We define the Euler characteristic of the relative lattice cohomology by

eu(ℍ_{rel}^{∗}) := ∑_{q≥0}(−1)^{q} rank_{ℤ} ℍ_{rel}^{q}.

Then the exact sequence of Proposition 4.2.3 and equation (4.2.4) provide

Corollary 4.2.5. With the notation r_{0} := (i(k_{m}), E_{j_{0}}) + a_{0} one has

eu(ℍ_{rel}^{∗}(Γ, j_{0}, [k], a_{0})) = eu(ℍ^{∗}(Γ, [k_{a_{0}}])) − eu(ℍ^{∗}(Γ \ j_{0}, [k])) − \frac{1 + r_{0}^{2}(E_{j_{0}}^{∗})^{2}}{8} .

Fix any l ∈ L(Γ \ j_{0}), then k_{a_{0}} + 2l = i(k_{m} + 2l) + r_{0}E_{j_{0}}^{∗}, hence we also get

Corollary 4.2.6. For any l ∈ L(Γ \ j_{0}) one has

\begin{matrix}eu(ℍ_{rel}^{∗}(Γ, j_{0}, [k_{m}], a_{0})) = eu(ℍ^{∗}(Γ, [k_{a_{0}}])) − \frac{(k_{a_{0}} + 2l)_{Γ}^{2} + |J |}{8} \\ −eu(ℍ^{∗}(Γ \ j_{0}, [k_{m}])) + \frac{(k_{m} + 2l)_{Γ\j_{0}}^{2} + |J \ j_{0}|}{8}. \end{matrix} ; (4.2.7)

4.3. Reinterpretation. Above, we started with an element k_{m} of the class [k], and we constructed one of its extensions k_{a_{0}}. Since k_{a_{0}} + i(2l) = i(k_{m} + 2l) + r_{0}E_{j_{0}}^{∗}, we have k_{m} + 2l = R(k_{a_{0}} + 2l) for any l ∈ L(Γ \ j_{0}).

This procedure can be inverted. Indeed, let us fix any k̄ ∈ Char(Γ) (which plays the role of k_{a_{0}} + i(2l)). Then set R(k̄) and define r_{0} by the identity r_{0}E_{j_{0}}^{∗} = k̄ − iR(k̄). Finally, define

B_{0,rel} : F_{0}(Γ \ j_{0}, [R(k̄)]) → F_{0}(Γ, [k̄]) by B_{0,rel}(k′) = i(k′) + r_{0}E_{j_{0}}^{∗},

whose kernel is the relative complex F_{rel}^{∗} with cohomology ℍ_{rel}^{∗}(Γ, j_{0}, k̄). Then (4.2.7) reads as

\begin{aligned}eu(ℍ_{rel}^{∗}(Γ, j_{0}, k̄)) & = eu(ℍ^{∗}(Γ, [k̄])) − \frac{(k̄)_{Γ}^{2} + |J |}{8} \\ & − eu(ℍ^{∗}(Γ \ j_{0}, [R(k̄)])) + \frac{(R(k̄))_{Γ\j_{0}}^{2} + |J \ j_{0}|}{8}. \end{aligned} ; (4.3.1)

The identity (4.3.1) depends essentially on the choice of the choice of k̄ ∈ Char(Γ). In fact, even if we fix the class [k̄], the choice of the representative k̄ from the class [k̄] provides essentially different identities of type (4.3.1): not only the terms eu(ℍ_{rel}^{∗}(Γ, j_{0}, k̄)), (k̄)_{Γ}^{2} and (R(k̄))_{Γ\j_{0}}^{2} depend on the choice of k̄, but even the class [R(k̄)].

4.4. The connection with the topological Poincaré series. Let K_{Γ} ∈ L′ denote the canonical characteristic element of Γ defined by the adjunction formulae (K_{Γ} + E_{j}, E_{j}) + 2 = 0 for all j ∈ J. Similarly, one defines K_{Γ\j_{0}} ∈ Char(Γ \ j_{0}). Note that Char(Γ) = K + 2L′(Γ) and K_{Γ\j_{0}} = R(K_{Γ}). The next result computes eu(ℍ_{rel}^{∗}(Γ, j_{0}, K+2l′)) in terms of l′ via the coefficients of a series associated with Γ.

Consider the multi-variable Taylor expansion Z(t) = ∑ p_{l′}t^{l′} as in (2.1.1) and define H_{[l′],j_{0}}(t) as in (2.7).

Proposition 4.4.1. Fix the vertex j_{0} of Γ and write H_{[l′],j_{0}}(t) as ∑_{i≥0} c_{i}t^{i}.

(a) If l′ = ∑_{j} a_{j}E_{j}^{∗} = ∑_{j} l′_{j}E_{j} ∈ L′(Γ) with all a_{j} sufficiently large then

∑_{i<dl′_{j_{0}}} c_{i} = eu(ℍ_{rel}^{∗}(Γ, j_{0}, K + 2l′)).

(b) Take l̄′ = ∑_{j} l̄′_{j}E_{j} ∈ L′(Γ) with l̄′_{j_{0}} ∈ [0, 1). Then

pc(H_{[l̄′],j_{0}}) = eu(ℍ_{rel}^{∗}(Γ, j_{0}, K + 2l̄′)).

Proof. Use (4.3.1), (2.6.4) and (2.8.1). □

Example 4.4.2. Let Γ be the next graph, where all the vertices have decoration −2 except the j_{0} vertex which has −3.

[figure: j_{0}]

det(Γ) = det(Γ \ j_{0}) = 1, hence for both graphs we have only one class.

Moreover, Γ \ j_{0} is rational (an E_{8}–graph), hence min w_{Γ\j_{0}} = 0, ℍ_{red}^{∗}(Γ \ j_{0}) = 0 and eu(ℍ^{∗}(Γ \ j_{0})) = 0.

On the other hand, Γ is minimally elliptic, min w_{Γ} = 0, ℍ_{red}^{∗}(Γ) = ℍ_{red}^{0}(Γ) = ℤ_{(0)}, the rank one ℤ–module concentrated at degree zero. Hence eu(ℍ^{∗}(Γ)) = 1.

By the long exact sequence, we get ℍ_{rel}^{q}(Γ, j_{0}, k̄) = 0 for any k̄ and q > 0.

It is easy to see that K_{Γ} = −E_{j_{0}}^{∗}, and (E_{j_{0}}^{∗})^{2} = −1. Therefore, if k̄ = K +2l′, and r_{0} := −(k̄, E_{j_{0}}^{∗}) and l′_{j_{0}} := −(l′, E_{j_{0}}^{∗}), then r_{0} = 2l′_{j_{0}} − 1. By (4.2.5) or (4.3.1)

rank ℍ_{rel}^{0}(Γ, j_{0}, k̄) = eu(ℍ^{∗}(Γ, j_{0}, k̄)) = \frac{r_{0}^{2} + 7}{8} = 1 + \frac{l′_{j_{0}}(l′_{j_{0}} − 1)}{2}.

By a computation one obtains

H_{[0],j_{0}}(t) = \frac{1 − t^{6}}{(1 − t^{3})(1 − t^{2})(1 − t)} = \frac{1 − t + t^{2}}{(1 − t)^{2}} = 1 + t + 2t^{2} + 3t^{3} + 4t^{4} + · · · .

Then

∑_{i<l′_{j_{0}}} c_{i} = 1 + 1 + 2 + 3 + ⋯ + (l′_{j_{0}} − 1) = 1 + \frac{l′_{j_{0}}(l′_{j_{0}} − 1)}{2},

hence (4.4.1)(a) follows. In order to exemplify part (b), we have to take l̄′ with l̄′_{j_{0}} = 0, hence in this case eu(ℍ^{∗}(Γ, j_{0}, k̄)) = 1. But one also has

pc \frac{1 − t + t^{2}}{(1 − t)^{2}} = 1.

Example 4.4.3. Assume that Γ is a star–shaped graph with central vertex j_{0} and we fix k̄ = K_{Γ}. Since all the connected components of Γ\j_{0} are strings (i.e. rational graphs), for them (cf. [N08])

eu(ℍ^{∗}(Γ \ j_{0}, [K_{Γ\j_{0}}])) − \frac{(K_{Γ\j_{0}})^{2} + |J \ j_{0}|}{8} = 0.

Some properties of the lattice cohomology 19

Moreover, ℍ^{q}(Γ, [K]) = ℍ_{rel}^{q}(Γ, j_{0}, [K]) = 0 for q > 0, and

rank ℍ_{rel}^{0}(Γ, j_{0}, [K]) = eu(ℍ^{∗}(Γ, [K]) − \frac{K^{2} + |J |}{8} = rank ℍ^{0}(Γ, [K]) − min w_{Γ} − \frac{K^{2} + |J |}{8}
= rank ℍ^{0}(Γ, [K]) − min χ_{K}.

(4.4.4)

This equals the periodic constant of H_{[0],j_{0}}(t) by [N08, BN10]. Moreover, if (X, o) is a weighted homogeneous normal surface singularity with minimal good resolution graph Γ, then its geometric genus p_{g} equals the last term of (4.4.4), cf. e.g. [NN04, N05]. Hence p_{g}(X, o) = rank ℍ_{rel}^{0}(Γ, j_{0}, [K]).

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

   E-mail address: nemethi@renyi.hu