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On the mu-bar invariant of rational surface singularities

We show that for rational surface singularities with odd determinant the mu-bar invariant defined by W. Neumann is an obstruction for the link of the singularity to bound a rational homology 4-ball. We identify the mu-bar invariant with the corresponding correction term in Heegaard Floer theory.


Introduction
Smoothings of surface singularities play a prominent role in constructing new and interesting smooth (and symplectic) 4-manifolds. It is of particular interest when the singularity at hand admits a rational homology 4-ball smoothing. Such smoothings led to the discovery of the rational blow-down procedure [2,20], which in turn provided a major tool for finding exotic 4-manifolds. Restrictions for a singularity to admit rational homology 4-ball smoothing have been found recently in [22].
A topological obstruction for a Z 2 -homology 3-sphere (that is, a 3-manifold Y with H * (Y ; Z 2 ) = H * (S 3 ; Z 2 )) to bound a spin rational homology 4-ball is its µ-invariant, defined modulo 16. An integral lift µ of µ has been defined by Neumann in [14] (cf. also [21]) for plumbed Z 2 -homology 3-spheres, but it was unclear whether this integer valued invariant obstructs the 3-manifold to bound a spin rational homology 4-ball. Special cases, like Seifert fibered 3manifolds, have been considered by Saveliev [21]. More recently, based on work of Ozsváth and Szabó [16,17,18] the correction term of spin c 3-manifolds (stemming from gradings on the Ozsváth-Szabó homology groups) provided further obstructions. For applications of these invariants along similar lines see [7].
In fact, in [14] the µ-invariant is defined for any spin rational homology 3sphere which can be given by plumbing spheres along a tree (i.e., the assumption on the parity of the determinant of the plumbing graph can be relaxed). By identifying µ of a spin 3-manifold (Y, s) which is a link of a rational surface singularity with the appropriate correction term, we show Theorem 1.1 Suppose that Y Γ is given as a plumbing of spheres along a negative definite tree Γ, defining a rational surface singularity.
• For a spin structure s ∈ Spin(Y Γ ) the invariant µ(Y Γ , s) ∈ Z is an obstruction for the existence of a spin c rational homology 4-ball (X, t) with boundary (Y Γ , s).
• Specifically, if det Γ is odd and µ(Y Γ ) = 0 then Y Γ is not the boundary of a rational homology 4-ball. Consequently the corresponding singularity does not admit rational homology 4-ball smoothing.
Corollary 1.2 Suppose that S Γ is a normal surface singularity with det Γ odd. If µ(Y Γ ) = 0 then S Γ does not admit a rational homology 4-ball smoothing.
Proof If S Γ is not a rational singularity then it does not admit rational homology 4-ball smoothing. If det Γ is odd, then for rational surface singularities Theorem 1.1 concludes the proof.
We hope that this obstruction will be useful in completing the characteriztion of surface singularities with rational homology 4-ball smoothing, along the line initiated in [22].

Remark 1.3
The assumption on the parity of det Γ cannot be relaxed in general, since for example the singularity with resolution graph having a single vertex of weight (−4) has two spin structures with µ invariants −3 and +1, but the link of the singularity is the boundary of a rational homology 4-ball: the complement of a quadric in the complex projective plane. In fact, this rational homology 4-ball can be given as a smoothing of the singularity. In accordance with Theorem 1.1 the spin structures on the link of the singularity do not extend to the rational homology 4-ball.
As it was indicated earlier, the proof of Theorem 1.1 above rests on the following, more technical statement. Here the invariant d(Y, s) of a spin c 3-manifold (Y, s) is the correction term in Heegaard Floer theory. (For more on Heegaard Floer theory see Section 4.) Theorem 1.4 Suppose that Γ is a negative definite plumbing tree of spheres, giving rise to a rational surface singularity. Let s be a given spin structure on the associated 3-manifold Y Γ . Then µ(Y Γ , s) = −4d(Y Γ , s).

The µ and µ invariants
Suppose that Y is a rational homology 3-sphere, and the rank |H 1 | of its first homology is odd.
where σ(X) is the signature of the 4-manifold X . The invariance of this quantity is a simple consequence of Rokhlin's famous result on the divisibility of the signature of a closed spin 4-manifold by 16. (If Y is an integral homology sphere, that is, H 1 (Y ; Z) = 0 also holds, then the signature σ(X) of a spin 4manifold X with ∂X = Y is divisible by 8, and in this case sometimes Rokhlin's invariant is defined as σ(X) 8 ∈ Z 2 .) It is not hard to see that if X is a spin rational homology 4-ball (i.e., H * (X; Q) = H * (D 4 ; Q)) with ∂X = Y and H 1 (Y ; Z 2 ) = 0 then µ(Y ) = 0. Consequently, the µ-invariant of a Z 2 -homology sphere Y provides an obstruction for Y to bound a spin rational homology 4-ball. (The spin assumption on X is important, since for example the Brieskorn sphere Σ(2, 3, 7) has µ = 1 and bounds a nonspin rational homology 4-ball, cf. [3].) Since µ is defined only mod 16, it is typically less effective than an integer valued invariant. Interest in integral lifts (or related obstructions) was motivated also by a result of Galewski and Stern [4] about higher dimensional (simplicial) triangulation theory.
In [14] Walter Neumann defined a lift µ ∈ Z of µ for spin 3-manifolds given by the plumbing construction along a weighted tree. Before giving the definition of this invariant we shortly review a few basic facts about plumbing trees. For a general reference see [14].
Suppose that Γ is a weighted tree with nonzero determinant. Let X Γ denote the 4-manifold defined by plumbing disk bundles over spheres according to the weighted tree Γ, and define Y Γ as ∂X Γ . As it is described in [14], the mod 2 homology H 1 (Y Γ ; Z 2 ) can be determined by a simple algorithm, which we outline below. Consider a leaf v of Γ, connected to the vertex w .
• Move 1: If the weight on v is even, then erase v and w from Γ. This procedure stops once we reach a graph Γ ′ with no edges. Suppose that Γ ′ contains p vertices, q of them with even weights.
Proof Denote the set of vertices of the given weighted plumbing tree Γ with nonzero determinant by V = {v 1 , . . . , v n }. It is known (cf. [5, Proposition 5.3.11]) that the homology group H 1 (Y Γ ; Z) admits a presentation by taking elements of V as generators, and equations as relations (i = 1, . . . , n), with the convention that n i is the weight on v i , and v j , v i is one or zero depending on whether v j and v i (as vertices of the tree Γ) are connected or not. These relations follow easily from the existence of Seifert surfaces for the components of the surgery link. The mod 2 reduction of the relations (with the same generators) provide a presentation for H 1 (Y Γ ; Z 2 ). Now the moves for simplifying the graph (until it becomes a disjoint union of some vertices) obviously correspond to base changes and expressions of generators in terms of others. Indeed, when Move 1 applies to v and w then the relation for v shows w = 0, while the relation for w expresses v in terms of the other neighbours of w . In the situation of Move 2 the relation for v simply asserts that v = w (mod 2). From this observation the statement easily follows: a single point with odd weight gives rise to a 3-manifold with vanishing first mod 2 homology, while with even weight the first mod 2 homology is 1-dimensional.
Recall that an oriented 3-manifold Y always admits a spin structure, and the space of spin structures is parametrized by the first mod 2 cohomology A convenient parametrization of the set of spin structures on the rational homology 3-sphere Y Γ is given as follows. First we define a set of subsets of the vertex set for every plumbing graph Γ. We start with a graph Γ ′ having no edges: in that case consider the subsets of the vertices which contain all vertices with odd weights. Every such subset will give rise to a unique subset S ⊂ V for the original graph Γ as follows. We describe the change of S under one step in the process giving Γ ′ from Γ. Suppose that Γ ′ is given by Move 1 from Γ (via erasing v = v i and w = v j ), and a set S ′ ⊂ V ′ is specified for Γ ′ . Now we define the set S ⊂ V by taking it to be equal to S ′ or S ′ ∪ {v i } according as the number of indices in S ′ adjacent to w = v j have the same parity as n j or n j − 1. If Γ ′ is derived from Γ by Move 2 (via erasing v i ) then let S be equal to S ′ or S ′ ∪ {v i } depending on whether v j was in S ′ or not. It is not hard to see from this algorithm that if v i , v j ∈ S then v i and v j are not connected by an egde in Γ.
Suppose now that S ⊂ V is a subset defined as above. Consider the submanifold Σ S ⊂ X Γ defined as the union of the spheres corresponding to the vertices in S . Notice that since by construction S does not contain adjacent vertices, the above surface is a disjoint union of embedded spheres. Let c S ∈ H 2 (X Γ ; Z) denote the Poincaré dual of Σ S . The inductive definition (and the starting condition) shows that c S is a characteristic element, that is, for every surface On the simply connected 4-manifold X Γ a characteristic cohomology class uniquely specifies a spin c structure t S , which restricts to a spin c structure , on the boundary the spin c structure s S = t S | ∂X Γ has vanishing first Chern class, therefore it is a spin structure on Y Γ . Hence every subset S constructed above defines a spin structure s S on Y Γ ; the set S is called the Wu set of the corresponding spin structure. Since this construction provides a spin structure on the complement X − Σ S , it is obvious that two different sets induce different spin structures: if S 1 and S 2 differ on the vertex v of even weight (in the disconnected graph our construction started with) then only the spin structure corresponding to the Wu set not containing v will extend to the cobordism we get by the appropriate handle attachment along v . In conclusion, we get an identification of H 1 (Y Γ ; Z)( ∼ = H 1 (Y Γ ; Z)) with the set of spin structures on Y Γ : take the characteristic function of S on the starting disconnected graph Γ ′ (which by the above said determines S ), and associate to it the corresponding first mod 2 cohomology class. Now the definition of the µ invariant of Neumann (cf. also [14]) is as follows.
Definition 2.2 For a spin structure s on Y Γ consider the corresponding Wu set S and embedded Wu surface Σ S ⊂ X Γ . Define µ(Y Γ , s) ∈ Z as the difference By applying the handle calculus developed in [15] together with the Wu set S , the proof of the following statement easily follows.

Rational singularities
Consider the plumbing tree Γ and suppose that Γ is negative definite. According to a classical result of Grauert [6], for any negative definite plumbing graph there exists a normal surface singularity such that the plumbing along the given graph is diffeomorphic to a resolution of the singularity.
Definition 3.1 A normal surface singularity S Γ is called rational if its geometric genus p g = 0. A negative definite plumbing graph Γ is rational if there is a rational singularity S Γ with resolution diffeomorphic to X Γ .
Although the singularity corresponding to a plumbing graph might not be unique, it is known that rationality is a topological property and can be fairly easily read off from the plumbing graph through Laufer's algorithm. Namely, consider the homology class In the i th step, consider the product If it is at least 2 then the algorithm stops and the singularity is not rational. If the product is nonpositive, move to the next vertex. Finally, if the product is 1 for some v ∈ Γ, then replace K i with K i+1 = K i + [Σ v ] and start checking the value of the product for all vertices of Γ again. If all products are nonpositive, the algorithm stops and the graph gives rise to a rational singularity.
Lemma 3.2 A rational plumbing graph is always a (negative definite) tree of spheres, and the link is a rational homology 3-sphere. In addition, for any vertex v i ∈ Γ the sum of its weight n i and the number d i of its neighbours is at most 1.
Notice that in a rational graph a vertex with weight (−1) has degree d ≤ 2, hence can be blown down by keeping Γ a plumbing tree. For this reason, we might assume that n i ≤ −2 for all vertices v i ∈ Γ.

Heegaard Floer groups
In [17,18] a set of very powerful invariants, the Ozsváth-Szabó homology groups HF (Y, s), HF ± (Y, s) and HF ∞ (Y, s) of a spin c 3-manifold (Y, s) were introduced. In the following we will use these groups and relations among them; for a more thorough introduction see [17,18,10]. Recall that a rational ho- Proof Suppose that Y Γ = ∂X for a rational homology 4-ball X . Let ϕ : Y Γ → X denote the embedding of the boundary, inducing the map ϕ * on homology.
Since |H 1 (Y Γ ; Z)| is odd, the size of the subgroup Im ϕ * is also odd. This implies that an odd number of spin c structures in Spin c (Y Γ ) extend to X . Since s ∈ Spin c (Y Γ ) and its conjugate s extend at the same time, we conclude that the spin structure s = s of Y Γ extends to X as a spin c structure, therefore Proposition 4.1 concludes the proof.
A relation between the singularity's holomorphic structure and its Heegaard Floer theoretic behaviour was found by A. Némethi: [13]) Suppose that the negative definite plumbing tree Γ gives rise to a rational singularity. Then Y Γ is an L-space.

A relation between µ(Y Γ , s) and d(Y Γ , s)
The proof of our main result about the µ-invariant relies on the identification of it with the appropriate multiple of the d-invariant of the spin 3-manifold at hand.
Proof of Theorem 1.4 Let Γ be a given negative definite rational plumbing tree with a spin structure s (represented by its Wu set S ⊂ V ). Let m Γ,S denote the number of those vertices v i ∈ Γ which are not in S but −n i of the neighbours of v i are in S . (Notice that by the rationality of Γ this means that v i has −n i or −n i + 1 neighbours and either all or all but one neighbours are in S .) The proof of the theorem will proceed by induction on m Γ,S . Let us start with the easy case when m Γ,S = 0, that is, for any vertex v i in Γ we have In conclusion, c S satisfies n i ≤ c S (Σ v i ) ≤ −n i − 2 for all vertices, hence c S is a terminal vector in the sense of [19]. By subtracting twice the Poincaré duals of the homology classes represented by surfaces corresponding to vertices in S , eventually we get a path back to a vector K ∈ H 2 (X Γ ; Z) which satisfies K( This means that K is an initial vector, hence c S is in a full path (again, in the terminology of [19]). By the identification of [13] this implies that c S gives rise to a Heegaard Floer homology element in HF (Y, s) of degree 1 4 (c 2 S −3σ(X Γ )−2χ(X Γ )). (Here, as costumary in Heegaard Floer theory, χ(X Γ ) is understood as the Euler characteristic of the cobordism we get from S 3 to Y Γ by deleting a point from X Γ .) Since Y Γ is an L-space, this degree must be equal to d(Y, s). On the other hand, since Γ is negative definite, χ(X Γ ) = −σ(X Γ ), hence the above formula for the degree shows that −µ(Y Γ , s) = c 2 S − σ(X Γ ) is equal to 4d(Y Γ , s). Next we assume that the statement is proved for graphs (Γ, S) with m Γ,S ≤ m − 1. In the inductive step we will utilize the exact triangle for Heegaard Floer homologies, proved for a surgery triple, see [18,9]. To this end, fix a graph Γ with Wu set S and corresponding spin structure s ∈ Spin(Y Γ ) having m Γ,S = m > 0 and let v denote a vertex with −n i neighbours in S . (Consequently v is not in S .) Consider the following plumbing graphs (with spin structures specified by the various Wu sets): • Let Γ ′ , Γ ′′ denote the same graphs as Γ with the alteration of the framing on v from n i to n i − 2 and n i − 4, resp. It is easy to see that S still provides Wu sets S ′ , S ′′ (and hence spin structures s ′ , s ′′ ) for Γ ′ and Γ ′′ . Notice that m Γ ′ ,S ′ = m Γ ′′ ,S ′′ = m Γ,S − 1. In addition, since v was not in the Wu set S , we see directly that µ(Y Γ , s) = µ(Y Γ ′ , s ′ ) = µ(Y Γ ′′ , s ′′ ).
• Let Γ 1 be the disjoint union of Γ ′ and the graph on a single vertex w with framing (−2). The set S 1 is chosen as S ∪ {w}. Simple computation shows that µ(Y Γ 1 , s 1 ) = µ(Y Γ , s) + 1. In the surgery picture for Y Γ 1 resulting from the plumbing let K denote the unknot linking the unknot corresponding to v ∈ Γ chosen above and the new (−2)-framed circle (corresponding to w ) once.
Attach a 4-dimensional 2-handle to the 3-manifold Y Γ 1 along K with framing (−1). The resulting coboridsm will be denoted by X .
The result of the above surgery is Y Γ , and the spin structure s 1 on Y Γ 1 defined by S 1 extends as a spin structure to provide a spin cobordism (X, t) from (Y Γ 1 , s 1 ) to (Y Γ , s).
Proof By sliding K and the handle corresponding to w down, the first statement is obvious. The extension follows from the fact that for the graph containing Γ 1 together with K , the vertex corresponding to K is not in S 1 .
Notice that by induction on m Γ,S the statement of the theorem holds for Γ 1 and Γ ′ , hence we have that −4d(Y Γ 1 , If the spin cobordism (X, t) of Lemma 5.1 between (Y Γ 1 , s 1 ) and (Y Γ , s) induces a nontrivial map on the Ozsváth-Szabó homology groups, we can easily conclude the argument: since a negative definite spin cobordism with χ = 1 and σ = −1 shifts degree for Ozsváth-Szabó homologies by 1 4 , the unique nontrivial element of HF (Y Γ 1 , s 1 ) maps to the unique nontrivial element of HF (Y Γ , s) of degree d(Y Γ 1 , s 1 ) + 1 4 = d(Y Γ , s), reducing the proof to elementary arithmetics. The nontriviality of the map F X,t is, however, not so obvious. Let us set up the exact triangle defined by the surgery triple (Y Γ 1 , Y Γ , Y Γ ′′ ) along the knot K ⊂ Y Γ 1 : for the identification of the two manifolds Y Γ , Y Γ ′′ simple Kirby calculus arguments are needed. Recall that the map F X is the sum of all F X,u for u ∈ Spin c (X).
We claim first that F X (s 1 ) has nonzero s-component. Since U is not negative definite, the map F ∞ U vanishes, and since Y Γ is an L-space, this implies the same for the maps F + U and F U . In particular, by exactness we get that F V is injective and F X is surjective. Suppose that F X (s 1 ) has zero s-component. Then F X (s 1 ) = a + a for some a ∈ HF (Y Γ ), where a is a formal sum of some spin c structures on Y Γ and a denotes the sum of the conjugate spin c structures, cf. [10]. By surjectivity now there is x ∈ HF (Y Γ 1 ) with F X (x) = a, hence s 1 + x + x is in the kernel of F X , so in the image of F V . If F V (y) = s 1 + x + x then the same holds for y , hence by the injectivity of F V the element y satisfies y = y . In order F V (y) to have spin component, y must have a spin component, hence we have found some spin and spin c structures z ∈ Spin(Y Γ ′′ ) and t ′ ∈ Spin c (V ) with F V,t ′ (z) = s 1 . By the uniqueness of extensions this z must be s ′′ , and the spin c cobordism (V, t ′ ) connecting z = s ′′ and s 1 must be spin. Therefore the grading shift between the elements s ′′ and s 1 is 1 4 . This implies Since by induction for the spin 3-manifolds (Y Γ 1 , s 1 ) and (Y Γ ′′ , s ′′ ) the invariant µ actually computes the correction term, that is, −4d(Y Γ 1 , s 1 ) = µ(Y Γ 1 , s 1 ) and −4d(Y Γ ′ , s ′ ) = µ(Y Γ ′ , s ′ ), Equations (5.2) and (5.3) contradict each other. Therefore the element F X (s 1 ) has nontrivial s-component, verifying our claim.
The nontriviality of F X between s 1 and s, however, implies that there is a connecting spin structure t with F X,t (s 1 ) = s, cf. [10,Lemma 3.3]. Consequently the degree shift given by F X,t is 1 4 , hence the inductive step concludes the proof of Theorem 1.4.