On the mu-bar invariant of rational surface singularities
aa r X i v : . [ m a t h . G T ] M a y On the µ invariant of rational surface singularities Andr´as I. Stipsicz
R´enyi Institute of MathematicsH-1053 Budapest, Re´altanoda utca 13–15, Hungary andDepartment of Mathematics, Columbia University, NY, New York, 10027
Email: [email protected], [email protected]
Abstract
We show that for rational surface singularities with odd deter-minant the µ invariant defined by W. Neumann is an obstruction for thelink of the singularity to bound a rational homology 4–ball. We identifythe µ invariant with the corresponding correction term in Heegaard Floertheory. AMS ClassificationKeywords
Smoothings of surface singularities play a prominent role in constructing newand interesting smooth (and symplectic) 4–manifolds. It is of particular inter-est when the singularity at hand admits a rational homology 4–ball smooth-ing. 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. Re-strictions for a singularity to admit rational homology 4–ball smoothing havebeen found recently in [22].A topological obstruction for a Z –homology 3–sphere (that is, a 3–manifold Y with H ∗ ( Y ; Z ) = H ∗ ( S ; Z )) to bound a spin rational homology 4–ball isits µ –invariant, defined modulo 16. An integral lift µ of µ has been definedby Neumann in [14] (cf. also [21]) for plumbed Z –homology 3–spheres, butit was unclear whether this integer valued invariant obstructs the 3–manifoldto bound a spin rational homology 4–ball. Special cases, like Seifert fibered 3–manifolds, have been considered by Saveliev [21]. More recently, based on workof Ozsv´ath and Szab´o [16, 17, 18] the correction term of spin c µ –invariant is defined for any spin rational homology 3–sphere which can be given by plumbing spheres along a tree (i.e., the assumption1n the parity of the determinant of the plumbing graph can be relaxed). Byidentifying µ of a spin 3–manifold ( Y, s ) which is a link of a rational surfacesingularity with the appropriate correction term, we show Theorem 1.1
Suppose that Y Γ is given as a plumbing of spheres along anegative 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 ob-struction for the existence of a spin c rational homology 4–ball ( X, t ) withboundary ( Y Γ , s ) . • If Π s ∈ Spin ( Y Γ ) µ ( Y Γ , s ) = 0 then the rational singularity does not bound aspin rational homology 4–ball. • Specifically, if det Γ is odd and µ ( Y Γ ) = 0 then Y Γ is not the boundary ofa rational homology 4–ball. Consequently the corresponding singularitydoes 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 ho-mology 4–ball smoothing. If det Γ is odd, then for rational surface singularitiesTheorem 1.1 concludes the proof.We hope that this obstruction will be useful in completing the characteriztionof surface singularities with rational homology 4–ball smoothing, along the lineinitiated in [22]. Remark 1.3
The assumption on the parity of det Γ cannot be relaxed ingeneral, since for example the singularity with resolution graph having a singlevertex of weight ( −
4) has two spin structures with µ invariants − d ( Y, s ) of a spin c Y, s ) is the correction term in Heegaard Floer theory. (For more on HeegaardFloer theory see Section 4.) 2 heorem 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 onthe associated 3–manifold Y Γ . Then µ ( Y Γ , s ) = − d ( Y Γ , s ) . µ and µ invariants Suppose that Y is a rational homology 3–sphere, and the rank | H | of its firsthomology is odd. Then H ( Y ; Z ) = H ( Y ; Z ) = 0, hence Y admits a uniquespin structure. Consider a spin 4–manifold X with ∂X = Y . The classicaldefinition of Rokhlin’s µ –invariant is µ ( Y ) ≡ σ ( X ) mod 16 , where σ ( X ) is the signature of the 4–manifold X . The invariance of thisquantity is a simple consequence of Rokhlin’s famous result on the divisibilityof the signature of a closed spin 4–manifold by 16. (If Y is an integral homologysphere, that is, H ( Y ; Z ) = 0 also holds, then the signature σ ( X ) of a spin 4–manifold X with ∂X = Y is divisible by 8, and in this case sometimes Rokhlin’sinvariant is defined as σ ( X )8 ∈ Z .)It is not hard to see that if X is a spin rational homology 4–ball (i.e., H ∗ ( X ; Q ) = H ∗ ( D ; Q )) with ∂X = Y and H ( Y ; Z ) = 0 then µ ( Y ) = 0. Consequently,the µ –invariant of a Z –homology sphere Y provides an obstruction for Y tobound a spin rational homology 4–ball. (The spin assumption on X is impor-tant, since for example the Brieskorn sphere Σ(2 , ,
7) has µ = 1 and boundsa nonspin rational homology 4–ball, cf. [3].) Since µ is defined only mod 16, itis typically less effective than an integer valued invariant. Interest in integrallifts (or related obstructions) was motivated also by a result of Galewski andStern [4] about higher dimensional (simplicial) triangulation theory.In [14] Walter Neumann defined a lift µ ∈ Z of µ for spin 3–manifolds given bythe plumbing construction along a weighted tree. Before giving the definitionof this invariant we shortly review a few basic facts about plumbing trees. Fora general reference see [14].Suppose that Γ is a weighted tree with nonzero determinant. Let X Γ denotethe 4–manifold defined by plumbing disk bundles over spheres according to theweighted tree Γ, and define Y Γ as ∂X Γ . As it is described in [14], the mod2 homology H ( Y Γ ; Z ) can be determined by a simple algorithm, which weoutline 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.3 Move 2 : If the weight of v is odd, then erase v and change the parityof the weight on w .This procedure stops once we reach a graph Γ ′ with no edges. Suppose that Γ ′ contains p vertices, q of them with even weights. Lemma 2.1
The dimension of the vector space H ( Y Γ ; Z ) over Z is equalto q . Proof
Denote the set of vertices of the given weighted plumbing tree Γ withnonzero determinant by V = { v , . . . , v n } . It is known (cf. [5, Proposi-tion 5.3.11]) that the homology group H ( Y Γ ; Z ) admits a presentation by tak-ing elements of V as generators, and equations n i · v i + X j = i h v j , v i i · v j = 0as relations ( i = 1 , . . . , n ), with the convention that n i is the weight on v i ,and h v j , v i i is one or zero depending on whether v j and v i (as vertices of thetree Γ) are connected or not. These relations follow easily from the existence ofSeifert surfaces for the components of the surgery link. The mod 2 reduction ofthe relations (with the same generators) provide a presentation for H ( Y Γ ; Z ).Now the moves for simplifying the graph (until it becomes a disjoint union ofsome vertices) obviously correspond to base changes and expressions of gen-erators in terms of others. Indeed, when Move 1 applies to v and w thenthe relation for v shows w = 0, while the relation for w expresses v in termsof 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 easilyfollows: a single point with odd weight gives rise to a 3–manifold with vanish-ing first mod 2 homology, while with even weight the first mod 2 homology is1-dimensional.Recall that an oriented 3–manifold Y always admits a spin structure, andthe space of spin structures is parametrized by the first mod 2 cohomology H ( Y ; Z )( ∼ = H ( Y ; Z )) of Y . A convenient parametrization of the set of spinstructures on the rational homology 3–sphere Y Γ is given as follows. First wedefine a set of subsets of the vertex set for every plumbing graph Γ. We startwith a graph Γ ′ having no edges: in that case consider the subsets of the verticeswhich contain all vertices with odd weights. Every such subset will give rise toa unique subset S ⊂ V for the original graph Γ as follows. We describe thechange of S under one step in the process giving Γ ′ from Γ. Suppose that Γ ′
4s 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 (viaerasing v i ) then let S be equal to S ′ or S ′ ∪ { v i } depending on whether v j wasin 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 submanifoldΣ 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, theabove surface is a disjoint union of embedded spheres. Let c S ∈ H ( X Γ ; Z )denote the Poincar´e dual of Σ S . The inductive definition (and the startingcondition) shows that c S is a characteristic element , that is, for every surfaceΣ v ⊂ X Γ defined by a vertex v we have c S (Σ v ) ≡ n v mod 2 . On the simply connected 4–manifold X Γ a characteristic cohomology classuniquely specifies a spin c structure t S , which restricts to a spin c structure s S on the boundary Y Γ . Since P D ( c S ) = S v Σ v = Σ S is in H ( X Γ ; Z ), onthe 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 abovedefines a spin structure s S on Y Γ ; the set S is called the Wu set of the cor-responding spin structure. Since this construction provides a spin structure onthe complement X − Σ S , it is obvious that two different sets induce differentspin structures: if S and S differ on the vertex v of even weight (in thedisconnected graph our construction started with) then only the spin structurecorresponding to the Wu set not containing v will extend to the cobordismwe get by the appropriate handle attachment along v . In conclusion, we getan identification of H ( Y Γ ; Z )( ∼ = H ( 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 correspondingfirst 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 Wuset S and embedded Wu surface Σ S ⊂ X Γ . Define µ ( Y Γ , s ) ∈ Z as the differ-ence µ ( Y Γ , s ) = σ ( X Γ ) − [Σ S ] .
5y applying the handle calculus developed in [15] together with the Wu set S ,the proof of the following statement easily follows. Proposition 2.3 ( [14, Theorem 4.1] ) The quantity µ ( Y Γ , s ) is an invariantof the spin 3–manifold ( Y Γ , s ) and is independent of the choices made in thedefinition. Consider the plumbing tree Γ and suppose that Γ is negative definite. Accord-ing to a classical result of Grauert [6], for any negative definite plumbing graphthere exists a normal surface singularity such that the plumbing along the givengraph is diffeomorphic to a resolution of the singularity.
Definition 3.1
A normal surface singularity S Γ is called rational if its geo-metric genus p g = 0. A negative definite plumbing graph Γ is rational if thereis a rational singularity S Γ with resolution diffeomorphic to X Γ .Although the singularity corresponding to a plumbing graph might not beunique, it is known that rationality is a topological property and can be fairlyeasily read off from the plumbing graph through Laufer’s algorithm. Namely,consider the homology class K = X v ∈ Γ [Σ v ] ∈ H ( X Γ ; Z ) . In the i th step, consider the product K i · Σ v j = h P D ( K i ) , [Σ v j ] i . If it is at least2 then the algorithm stops and the singularity is not rational. If the productis 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 valueof the product for all vertices of Γ again. If all products are nonpositive, thealgorithm stops and the graph gives rise to a rational singularity. Lemma 3.2
A rational plumbing graph is always a (negative definite) treeof spheres, and the link is a rational homology 3–sphere. In addition, for anyvertex v i ∈ Γ the sum of its weight n i and the number d i of its neighbours isat most 1. Notice that in a rational graph a vertex with weight ( −
1) has degree d ≤ n i ≤ − v i ∈ Γ.6
Heegaard Floer groups
In [17, 18] a set of very powerful invariants, the Ozsv´ath–Szab´o homology groups d HF ( Y, s ) , HF ± ( Y, s ) and HF ∞ ( Y, s ) of a spin c Y, s ) were intro-duced. 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-mology 3–sphere Y is an L –space if d HF ( Y, s ) = Z for every spin c structure s ∈ Spin c ( Y ). (In the version of the theory we are about to apply, we use Z –coefficients.) In this case we can label the unique nonzero element of d HF ( Y, s )by the corresponding spin c structure s . Recall also that for a rational homology3–sphere Y the groups are equipped with a natural Q –grading. The gradingof the unique nontrivial element of d HF ( Y, s ) for an L –space Y is called the correction term d ( Y, s ) of the spin c Y, s ). For the proof of thenext proposition, see for example [8, Theorem 2.3]. Proposition 4.1
Suppose that d ( Y, s ) = 0 . Then there is no spin c rationalhomology 4–ball ( X, t ) with ∂ ( X, t ) = ( Y, s ) . Proposition 4.2
Suppose that det Γ is odd. If d ( Y Γ , s ) = 0 for the uniquespin structure s then Y Γ does not bound any rational homology 4–ball. 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 ( Y Γ ; Z ) | is odd, the size of the subgroup Im ϕ ∗ is also odd. Thisimplies 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 concludethat the spin structure s = s of Y Γ extends to X as a spin c structure, thereforeProposition 4.1 concludes the proof.A relation between the singularity’s holomorphic structure and its HeegaardFloer theoretic behaviour was found by A. N´emethi: Theorem 4.3 (N´emethi, [13])
Suppose that the negative definite plumbingtree Γ gives rise to a rational singularity. Then Y Γ is an L –space. µ ( Y Γ , s ) and d ( Y Γ , s ) The proof of our main result about the µ –invariant relies on the identificationof it with the appropriate multiple of the d –invariant of the spin 3–manifold athand. 7 roof of Theorem 1.4 Let Γ be a given negative definite rational plumbingtree 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 theneighbours 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 arein S .)The proof of the theorem will proceed by induction on m Γ ,S . Let us start withthe easy case when m Γ ,S = 0, that is, for any vertex v i in Γ we have c S (Σ v i ) < − n i . (5.1)For v i ∈ S we have c S (Σ v i ) = n i , while if v i is not in S then c S (Σ v i ) is thenumber of neighbours of v i which are in S . In particular, 0 ≤ c S (Σ v i ) ≤ d i holds for all v i not in S . Since c S is characteristic, Inequality (5.1) actuallymeans that c S (Σ v i ) ≤ − n i −
2. In conclusion, c S satisfies n i ≤ c S (Σ v i ) ≤− n i − c S is a terminal vector in the sense of [19]. Bysubtracting twice the Poincar´e duals of the homology classes represented bysurfaces corresponding to vertices in S , eventually we get a path back to avector K ∈ H ( X Γ ; Z ) which satisfies K (Σ v i ) = − n i for v i ∈ S and K (Σ v i ) ≥− d i ≥ n i + 2 if v i is not in S . 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 d HF ( Y, s ) of degree ( c S − σ ( X Γ ) − χ ( X Γ )). (Here, as costumary in HeegaardFloer theory, χ ( X Γ ) is understood as the Euler characteristic of the cobordismwe get from S 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 negativedefinite, χ ( X Γ ) = − σ ( X Γ ), hence the above formula for the degree shows that − µ ( Y Γ , s ) = c S − σ ( X Γ ) is equal to 4 d ( 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 HeegaardFloer homologies, proved for a surgery triple, see [18, 9]. To this end, fixa graph Γ with Wu set S and corresponding spin structure s ∈ Spin ( Y Γ )having m Γ ,S = m > v denote a vertex with − n i neighbours in S .(Consequently v is not in S .) Consider the following plumbing graphs (withspin structures specified by the various Wu sets): • Let Γ ′ , Γ ′′ denote the same graphs as Γ with the alteration of the framingon v from n i to n i − n i −
4, resp. It is easy to see that S stillprovides 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 notin the Wu set S , we see directly that µ ( Y Γ , s ) = µ ( Y Γ ′ , s ′ ) = µ ( Y Γ ′′ , s ′′ ).8aufer’s algorithm shows that Γ ′ , Γ ′′ are also rational. • Let Γ be the disjoint union of Γ ′ and the graph on a single vertex w withframing ( − S is chosen as S ∪ { w } . Simple computationshows that µ ( Y Γ , s ) = µ ( Y Γ , s ) + 1. In the surgery picture for Y Γ resulting from the plumbing let K denote the unknot linking the unknotcorresponding to v ∈ Γ chosen above and the new ( − w ) once.Attach a 4–dimensional 2–handle to the 3–manifold Y Γ along K with framing( − X . Lemma 5.1
The result of the above surgery is Y Γ , and the spin structure s on Y Γ defined by S extends as a spin structure to provide a spin cobordism ( X, t ) from ( Y Γ , s ) to ( Y Γ , s ) . Proof
By sliding K and the handle corresponding to w down, the first state-ment is obvious. The extension follows from the fact that for the graph con-taining Γ together with K , the vertex corresponding to K is not in S .Notice that by induction on m Γ ,S the statement of the theorem holds for Γ and Γ ′ , hence we have that − d ( Y Γ , s ) = µ ( Y Γ , s ) = µ ( Y Γ , s ) + 1 and − d ( Y Γ ′ , s ′ ) = µ ( Y Γ ′ , s ′ ) = µ ( Y Γ , s ).If the spin cobordism ( X, t ) of Lemma 5.1 between ( Y Γ , s ) and ( Y Γ , s ) in-duces a nontrivial map on the Ozsv´ath–Szab´o homology groups, we can easilyconclude the argument: since a negative definite spin cobordism with χ = 1 and σ = − , the unique nontrivialelement of d HF ( Y Γ , s ) maps to the unique nontrivial element of d HF ( Y Γ , s ) ofdegree d ( Y Γ , s ) + = 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 upthe exact triangle defined by the surgery triple ( Y Γ , Y Γ , Y Γ ′′ ) along the knot K ⊂ Y Γ : d HF ( Y Γ ) d HF ( Y Γ ) d HF ( Y Γ ′′ ) F X F U F V for the identification of the two manifolds Y Γ , Y Γ ′′ simple Kirby calculus ar-guments are needed. Recall that the map F X is the sum of all F X, u for u ∈ Spin c ( X ). 9e claim first that F X ( s ) has nonzero s –component. Since U is not negativedefinite, the map F ∞ U vanishes, and since Y Γ is an L –space, this implies thesame 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 ) has zero s –component.Then F X ( s ) = a + a for some a ∈ d HF ( Y Γ ), where a is a formal sum ofsome 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 ∈ d HF ( Y Γ ) with F X ( x ) = a ,hence s + x + x is in the kernel of F X , so in the image of F V . If F V ( y ) = s + 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 spincomponent, 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 . By the uniqueness of extensions this z must be s ′′ , and the spin c cobordism ( V, t ′ ) connecting z = s ′′ and s mustbe spin. Therefore the grading shift between the elements s ′′ and s is . Thisimplies d ( Y Γ ′′ , s ′′ ) + 14 = d ( Y Γ , s ) . (5.2)Recall that µ ( Y Γ ′′ , s ′′ ) = µ ( Y Γ , s ) = µ ( Y Γ , s ) − . (5.3)Since by induction for the spin 3–manifolds ( Y Γ , s ) and ( Y Γ ′′ , s ′′ ) the invariant µ actually computes the correction term, that is, − d ( Y Γ , s ) = µ ( Y Γ , s )and − d ( Y Γ ′ , s ′ ) = µ ( Y Γ ′ , s ′ ), Equations (5.2) and (5.3) contradict each other.Therefore the element F X ( s ) has nontrivial s –component, verifying our claim.The nontriviality of F X between s and s , however, implies that there is aconnecting spin structure t with F X, t ( s ) = s , cf. [10, Lemma 3.3]. Conse-quently the degree shift given by F X, t is , hence the inductive step concludesthe proof of Theorem 1.4. Proof of Theorem 1.1
Combining Propositions 4.1 and 4.2 with the identi-fication of Theorem 1.4 the proof follows at once. A CKNOWLEDGEMENTSWe would like to thank Stefan Friedl, Josh Greene and an anonymous refereefor helpful comments and corrections. The author was partially supported byOTKA 49449, by EU Marie Curie TOK program BudAlgGeo and by the ClayMathematics Institute. 10 eferences [1] A. Casson and J. Harer,
Some homology lens spaces which bound rational ho-mology balls , Pacific J. Math. (1981) 23–36.[2] R. Fintushel and R. Stern, Rational blowdowns of smooth 4–manifolds , J. Diff.Geom. (1997) 181–235.[3] R. Fintushel and R. Stern, A µ –invariant one homology 3–sphere that boundsan orientable rational ball , Contemporary Math. (1984) 265–268.[4] D. Galewski and R. Stern, Classification of simplicial triangulations of topolog-ical manifolds , Ann. Math. (1981) 1–34.[5] R. Gompf and A. Stipsicz, , Graduate Studiesin Mathematics AMS, 1999.[6] H. Grauert, ¨Uber Modifikationen und exzeptionelle analytische Mengen , Math.Ann. (1962) 498–507.[7] J. Greene and S. Jabuka,
The slice-ribbon conjecture for 3–stranded pretzel knots ,arXiv:0706.3398.[8] S. Jabuka and S. Naik,
Order in the concordance group and Heegaard Floerhomology , arXiv:math.GT/0611023.[9] P. Lisca and A. Stipsicz,
Ozsv´ath–Szab´o invariants and tight contact 3–manifolds, I , Geom. Topol. (2004) 925–945.[10] P. Lisca and A. Stipsicz, Ozsv´ath–Szab´o invariants and tight contact 3–manifolds, II , J. Differential Geom. (2007) 109–141.[11] P. Lisca and A. Stipsicz, Ozsv´ath–Szab´o invariants and tight contact 3–manifolds, III , J. Symplectic Geometry, to appear, arXiv:math.SG/0505493.[12] P. Lisca and A. Stipsicz,
On the existence of tight contact structures on Seifertfibered 3–manifolds , arXiv:0709.0737[13] A. N´emethi,
On the Ozsv´ath-Szab´o invariant of negative definite plumbed 3-manifolds , Geom. Topol. (2005) 991–1042.[14] W. Neumann, An invariant of plumbed homology spheres , Topology Symposium,Siegen 1979, 125–144. Lect. Notes in Math. , Springer, Berlin, 1980.[15] W. Neumann,
A calculus for plumbing applied to the topology of complex surfacesingularities and degenerating complex curves , Trans. Amer. Math. Soc. (1981) 299–344.[16] P. Ozsv´ath and Z. Szab´o,
Absolutely graded Floer homologies and intersectionsforms for four–manifolds with boundary , Adv. Math. (2003) 179–261.[17] P. Ozsv´ath and Z. Szab´o,
Holomorphic disks and topological invariants for closedthree-manifolds , Ann. of Math. (2004) 1027–1158.[18] P. Ozsv´ath and Z. Szab´o,
Holomorphic disks and three-manifold invariants:properties and applications , Ann. of Math. (2004) 1159–1245.
19] P. Ozsv´ath and Z. Szab´o,
On the Floer homology of plumbed three-manifolds ,Geom. Topol. (2003) 185–224.[20] J. Park, Seiberg–Witten invariants of generalized rational blow–downs , Bull.Austral. Math. Soc. (1997) 363–384.[21] N. Saveliev, Fukumoto–Furuta invariants of plumbed homology 3–spheres , PacificJ. Math. (2002) 465–490.[22] A. Stipsicz, Z. Szab´o and J. Wahl,
Rational blow–downs and smoothings ofsurface singularities , J. Topology (2008) 477–517.(2008) 477–517.