TRANSACTIONSOFTHE AMERICANMATHEMATICALSOCIETY Volume00,Number0,Pages000–000 S0002-9947(XX)0000-0 ON SYMPLECTIC FILLINGS OF LENS SPACES 6 0 PAOLOLISCA 0 2 Abstract. Let ξ be the contact structure naturally induced on the lens n st a space L(p,q)=S3/Z/pZ bythe standard contact structure ξst onthe three– sphere S3. We obtain a complete classification of the symplectic fillings of J (L(p,q),ξst)uptoorientation–preservingdiffeomorphisms. Inviewofourre- 6 sults,weformulateaconjectureonthediffeomorphismtypesofthesmoothings 2 ofcomplextwo–dimensional cyclicquotientsingularities. ] G S 1. Introduction and statement of results . h t A (coorientable) contact three–manifold is a pair (Y,ξ), where Y is a closed a three–manifold and ξ ⊂ TY a two–dimensional distribution given as the kernel of m a one–form α ∈ Ω1(Y) such that α∧dα is a volume form. The orientation on Y [ determined by α∧dα only depends on the distribution ξ. We shall always assume 4 that the underlying manifold of a contact three–manifold is oriented, and that the v orientation is the one induced by the contact structure. 4 A symplectic filling of a closed contact three–manifold (Y,ξ) is pair (X,ω) con- 5 sisting of a smooth, compact, connected four–manifold X and a symplectic form 3 2 ω on X such that, if X is oriented by ω ∧ω and ∂X is given the boundary ori- 1 entation, there exists an orientation–preserving diffeomorphism ϕ: Y → ∂X such 3 that ω| 6= 0 at every point of ∂X. For example, the unit four–ball B4 ⊂ C2 dϕ(ξ) 0 endowed with the restriction of the standard Kähler form on C2 is a symplectic / h filling of (S3,ξ ), where the standard contact structure ξ on S3 is, by definition, st st at the distribution of complex lines tangent to S3 ⊂C2. m The first classificationresult for symplectic fillings is due to Eliashberg [6], who proved that if (X,ω) is a symplectic filling of (S3,ξ ), then X is diffeomorphic : st v to a blowup of B4. McDuff [19] extended Eliashberg’s result to the lens spaces i X L(p,1) endowed with the contact structure ξ defined in the following paragraph. st r Ohta and Ono [23] determined the diffeomorphism types of the strong symplectic a fillings of links of simple elliptic singularities endowed with their natural contact structures. In[15,24]resultsontheintersectionformsofsymplecticfillingsoffinite quotients of S3 are proved. The distribution ξ on S3, being invariant under the natural action of U(2), is st a fortiori invariant under the induced action of the subgroup (1.1) G ={ ζ 0 | ζp =1}⊂U(2), p,q 0ζq where p,q ∈ Z. It follows that w(cid:16)hen p(cid:17)> q ≥ 1 and p,q are coprime, ξ descends st to a contact structure ξ on the lens space L(p,q)=S3/G . st p,q 2000 Mathematics Subject Classification. Primary57R17;Secondary53D35. (cid:13)c1997 American Mathematical Society 1 2 PAOLOLISCA Let D denote the disk bundle over the two–sphere with Euler class e(D )=p. p p The contact three–manifold (L(p,1),ξ ) admits a symplectic filling of the form st (D ,ω) for every p > 1. On the other hand, (L(4,1),ξ ) also admits a symplec- −p st tic filling of the form CP2\ν(C),ω , where ν(C) is a strictly pseudo–concave 0 neighborhoodof a smooth conic C ⊂CP2 and ω is the restriction of the standard (cid:0) (cid:1) 0 Kähler form on CP2. McDuff proved [19] that if (X,ω) is a symplectic filling of (L(p,1),ξ ), then X st is orientation preserving diffeomorphic to a smooth blowup of: (a) D if p6=4, −p (b) D or CP2\ν(C) if p=4. −4 In this paper we obtain a complete classification up to diffeomorphism for the symplectic fillings of the contact three–manifolds L(p,q),ξ . In order to state st our result we need to introduce some notation. (cid:0) (cid:1) Let p and q be coprime integers such that p 1 (1.2) p>q ≥1, =b − , b ,...,b ≥2. 1 1 k p−q 1 b − 2 ...− 1 b k The integers b are uniquely determined by the rationalnumber p . Without the i p−q assumptionb ≥2,this uniquenesspropertyfails. Thestandardsymbol[b ,...,b ] i 1 k willbe usedthroughoutthe papertodenoteacontinedfractionasthe onein(1.2). Definition. Ak–tupleofnon–negativeintegers(n ,...,n )is admissible ifeach 1 k of the denominators appearing in the continued fraction [n ,...,n ] is positive. 1 k Letn=(n ,...,n )beanadmissiblek–tupleofnon–negativeintegerssuchthat 1 k (1.3) [n ,...,n ]=0. 1 k Let N(n) be the closed, oriented three–manifold given by surgery on S3 along the framed link of Figure 1. It is easy to check that Assumption (1.3) ensures the existence of an orientation preserving diffeomorphism (1.4) ϕ: N(n)→S1×S2. n1 n2 nk−1 nk . . . . Figure 1. The manifold N(n) Definition. Fix a diffeomorphism ϕ as in (1.4), and let L⊂N(n) be the thick framed link drawn in Figure 2. Define W (n) to be the smooth four–manifold p,q with boundary obtained by attaching two–handles to S1 ×D3 along the framed link ϕ(L)⊂S1×S2. ON SYMPLECTIC FILLINGS OF LENS SPACES 3 Remark. The diffeomorphism type of W (n) is independent of the choice of p,q the diffeomorphism ϕ, because every self–diffeomorphism of S1 ×S2 extends to S1×D3 [10]. n1 n2 nk−1 nk . . . . ... ... ... ... L −1 −1 −1 −1 −1 −1 −1−1 −1 −1−1 −1 b1−n1 b2−n2 bk−1−nk−1 bk −nk Figure 2. The framed link L⊂N(n) Definition. Let Z ⊂ Zk be the set of admissibible k–tuples of non–negative p,q integers (n ,...,n ) such that 1 k [n ,...,n ]=0 and 0≤n ≤b for i=1,...,k. 1 k i i It is easy to check (see Section 2) that the set Z admits the involution: p,q n=(n ,...,n )7→n=(n ,...,n ) 1 k k 1 Given coprime integers p>q ≥1, we denote by q the only integer satifsying p>q ≥1, qq ≡1modp. Theorem 1.1. Let p>q ≥1 be coprime integers. Then, (1) Let(W,ω)beasymplectic fillingofthecontactthree–manifold (L(p,q),ξ ). st Then, there exists n ∈ Z such that W is orientation preserving diffeo- p,q morphic to a smooth blowup of W (n). p,q (2) For every n∈Z , the four–manifold W (n) carries a symplectic form ω p,q p,q such that (W (n),ω) is a symplectic filling of the contact three–manifold p,q L(p,q),ξ . Moreover, there are no classes in H (W (n);Z) with self– st 2 p,q intersection equal to −1. (3) L(cid:0)et n∈Zp,q(cid:1) and n′ ∈Zp′,q′. Then, Wp,q(n)#rCP2 is orientation preserving diffeomorphic to Wp′,q′(n′)#sCP2 if and only if: (a) (p′,s)=(p,r) and (q′,n′)=(q,n), or (b) (p′,s)=(p,r) and (q′,n′)=(q,n). Theorem 1.1 gives a complete diffeomorphism classification of the symplectic fillingsof(L(p,q),ξ ),extendingtheresultofMcDuffquotedabove1. Forinstance, st as explained in [16], by Theorem 1.1 there are exactly two symplectic fillings of (L(p2,p−1),ξ )uptoblowupsanddiffeomorphisms. Onecomesfromthecanonical st resolutionoftheassociatedsingularity,whiletheotheristherationalhomologyball used in the symplectic rational blowdown construction [8, 27]. The following corollary gives three new results as applications of Theorem 1.1: Corollary 1.2. Let p>q ≥1 be coprime integers. 1Portions(1)and(2)ofTheorem1.1wereannounced in[16,17]. 4 PAOLOLISCA (a) Given a positive integer n, there exist infinitely many lens spaces L(p,q) such that thecontact three–manifold (L(p,q),ξ ) has morethan n symplec- st tic fillings pairwise distinct up to homotopy equivalence and whose underlying four–manifolds are smoothly minimal 2. (b) There exist infinitely many lens spaces L(p,q) such that q > 1 and the contact three–manifold (L(p,q),ξ ) has only one symplectic filling up to st blowups and diffeomorphisms. (c) Thecontactthree–manifold (L(p,q),ξ )has asymplectic filling(W,ω)with st b (W)=0 if and only if (p,q)=(m2,mh−1), for some m and h coprime 2 natural numbers. The statement of Theorem 1.1 – not the proof given here – is related to the deformation theory of complex two–dimensional cyclic quotient singularities. In fact, the contact three–manifold L(p,q),ξ can be viewed as the link of the st singularity (C2/G ,0) together with the natural contact structure given by the p,q (cid:0) (cid:1) complex tangents. Every smoothing of (C2/G ,0) determines Stein fillings F of p,q (L(p,q),ξ ), and Theorem 1.1 implies that any such F must be diffeomorphic to 0 W (n) for some n ∈ Z . It seems likely that a converse to this fact should p,q p,q also hold, because every irreducible component of the reduced base space S of red the versal deformation of (C2/G ,0) gives a smoothing of the singularity, and p.q Stevens[26]provedtheexistenceofaone–to–onecorrespondencebetweenZ and p,q the set S of irreducible components of S (see also [3]). In light of the results p,q red obtained in this paper, we propose the following: Conjecture. Let F(n) be a Stein filling of L(p,q),ξ determined by the st smoothing of (C2/G ,0) corresponding to n∈Z under the Stevens correspon- p.q p(cid:0),q (cid:1) dence [26]. Then, F(n) is diffeomorphic to W (n). p,q Thereisevidencesupportingthisconjecture. Infact,inthispaperweprovethat eachW (n)carriesSteinstructures(Corollary5.2). Moreover,thesmoothingcor- p,q respondingto (1,2,...,2,1)∈Z is knownto be isomorphic to the canonicalres- p,q olutionX ofthesingularity,anditisnothardtoverifythatifn=(1,2,...,2,1) p,q then W (n) is indeed diffeomorphic to a regular neighborhood of the exceptional p,q divisor in X . Finally, by [28, 5.9.1], a singularity (C2/G ,0) has a smoothing p,q p.q withb =0ifandonlyif(p,q)=(m2,mh−1),withmandhcoprime,inagreement 2 with Corollary 1.2(c). The paper is organized as follows. In Section 2 we prove Corollary 1.2 assum- ing Theorem 1.1. In Section 3 we show that every symplectic filling (W,ω) of (L(p,q),ξ ) can be compactified to a rational symplectic four–manifold X so that st X\W isaneighborhoodofanimmersedsymplecticsurfaceΓ⊂X ofaspecialkind. In Section 4 we prove Theorem 1.1(1). In Section 5 we construct Stein structures on the smooth four–manifolds W (n). In Section 6 we prove Theorem 1.1(2). In p,q Section 7 we complete the proof of Theorem 1.1. Acknowledgements. The author gratefully acknowledges support and hospi- tality from the Department of Mathematics of the University of Georgia during the preparationof part of this paper. Warm thanks go to PatrickPopescu–Pampu for his interest in my work and for several remarks and suggestions which allowed 2A smooth, oriented four–manifold X is smoothly minimal if the interior of X contains no smoothlyembeddedtwo–sphereofself–intersection −1. ON SYMPLECTIC FILLINGS OF LENS SPACES 5 me to improve this paper in a few points where it lacked precision, and to Andra´s Stipsicz for useful comments. 2. The proof of Corollary 1.2 Definition 2.1. Given a k–tuple of positive integers (n ,...,n ,n ,n ,...,n ) 1 s−1 s s−1 k with n =1, we say that the (k−1)–tuple s (n ,...,n −1,n −1,...,n ) 1 s−1 s+1 k is obtained by a blowdown at n (with the obvious meaning of the notation when s s = 1 or s = k). The reverse process is a blowup. A blowdown at n is strict if s s>1. The reverse process is a strict blowup. It is showed in [22, Appendix] that a k–tuple of positive integers (n ,...,n ) is 1 k admissible 3 if and only if the symmetric matrix n −1 1 −1 n −1 2   −1 n 3  ... −1    −1 −n   k   is positive semi–definite ofrank≥k−1. It immediately follows fromthis factthat if (n ,...,n ) is admissible then (n ,...,n ) is admissible, 1 k k 1 (n ,n ,...,n ,n ) i i+1 j−1 j is admissible for every 1 ≤ i ≤ j ≤ k, and blowing up and blowing down preserve admissibility. Lemma 2.2. Let (n ,...,n )be a k–tuple of positive integers. Then, the following 1 k conditions are equivalent: • (n ,...,n ) is admissible and satisfies [n ,...,n ]=0; 1 k 1 k • (n ,...,n ) is obtained from (0) by a sequence of strict blowups. 1 k Proof. Clearly,a k–tuple ofpositive integers is obtainedfrom(0) via a sequence of strictblowupsifandonlyifitisobtainedfrom(1,1)viasuchasequence. Moreover, both the property of being admissible and that of having vanishing associated continuedfractionarepreservedunderblowup. Therefore,since(1,1)isadmissible and [1,1]= 0, any k–tuple of positive integers obtained from (0) by a sequence of strict blowups is admissible and has vanishing associated continued fraction. Conversely, let (n ,...,n ) be an admissible k–tuple of positive integers with 1 k [n ,...,n ] = 0. Then, we must have k ≥ 2. We shall argue by induction that 1 k (n ,...,n ) is obtained from (0) by a sequence of strict blowups. For k = 2 the 1 k statement is obvious, so suppose that k > 2 and the statement corresponding to k − 1 holds true. An easy induction argument shows that if n ≥ 2 for every i i = 1,...,k, then [n ,...,n ] > 1. Thus, if [n ,...,n ] = 0 then necessarily 1 k 1 k 3The definition of admissibilitygiven in [22, Appendix] is easily seen to be equivalent to the onegiveninSection1. 6 PAOLOLISCA n =1 for some i∈{1,...,k}. We conclude that (n ,...,n ) is obtained from the i 1 k admissible (k−1)–tuple (n ,...,n −1,n −1,...,n ) 1 i−1 i+1 k by a blowup (which is strict if and only if i>1). By the induction hypothesis, (n ,...,n −1,n −1,...,n ) 1 i−1 i+1 k is obtained from (0) by a sequence of strict blowups, hence n = 1 for some j > j 1. Therefore, the k–tuple (n ,...,n ) is obtained by a strict blowup from the 1 k admissible (k−1)–tuple (n ,...,n −1,n −1,...,n ), 1 j−1 j+1 k and the conclusion follows by induction. (cid:3) Lemma 2.3. Let p>q ≥1 be coprime integers, and suppose that p =[a ,...,a ], with a ,...,a ≥5. 1 h 1 h q Then, Z ={(1,2,...,2,1)}. p,q Proof. Let p =[b ,...,b ], b ,...,b ≥2. 1 k 1 k p−q UsingRiemenschneider’spointdiagram[25],one caneasilycheckthatthe assumption on the a ’s implies the following three conditions: i • k ≥4, • b ,...,b ≤3, 1 k • if b =b =3 then either 3 4. By Lemma 2.2, n > 0 for every i = 1,...,k and n = 1 i j for some index j > 1. We claim that j = k. In fact, suppose j < k. If j = 2 or j = k−1, then Assumption (3) implies n = n = 2. By Lemma 2.2 this j−1 j+1 is impossible, because two blowdowns would give a string of length bigger than 1 containing 0. Therefore we have 2 < j < k −1. Blowing down once yields the sequence (n ,...,n −1,n −1,...,n ), 1 j−1 j+1 k which still satisfies the three assumptions. Therefore, by induction (n ,...,n −1,n −1,...,n )=(1,2,...,2,1) 1 j−1 j+1 k ON SYMPLECTIC FILLINGS OF LENS SPACES 7 But then n =n =3, which goes against Assumption (3). We conclude that j−1 j+1 j =k. Blowing down once yields the sequence (n ,...,n −1), 1 k−1 which satisfies the assumptions if and only if n < 3. Notice that if n < 3, k−3 k−3 we can apply induction and reach the conclusion. Blowing down two more times yields the sequence (n ,...,n −1), 1 k−3 which satisfies the assumptions, so by induction we get n =2. (cid:3) k−3 We need the following elementary facts about continued fractions (see e.g. [22, Appendix] for the proofs): • Let p>q ≥1 be coprime integers, and suppose that p =[a ,a ,...,a ], a ,...,a ≥2. 1 2 h 1 h q Then, p [a ,a ,...,a ]= , h h−1 1 q where p>q ≥1 and qq ≡1modp. • If (n ,...,n ) is an admissible k–tuple of positive integers, then 1 k [n ,n ,...,n ]=0 1 2 k if and only if [n ,n ...,n ]=0. k k−1 1 Lemma 2.4. Let (n ,...,n ) be an admissible k–tuple of positive integers such 1 k that [n ,n ,...,n ]=0, k ≥3. 1 2 k Supposethat there is exactly oneindex j ∈{1,...,k}such that n =1. Then, there j are coprime integers m and n such that m2 [n ,n ,...,n ,2,n ,...,n ]= . 1 2 j−1 j+1 k mn+1 Proof. Wearguebyinductiononk ≥3. Fork =3,theassumptionsandLemma2.2 imply (n ,n ,n )=(2,1,2), 1 2 3 and 4 [2,2,2]= 3 is of the stated form. Now suppose k >3. Since, by Lemma 2.2, (n ,...,n ) must 1 k be obtained from (2,1,2) by a sequence of strict blowups, we have n =2, n >2, or n >2, n =2. 1 k 1 k Observe that m2 [n ,n ,...,n ,2,n ,...,n ]= 1 2 j−1 j+1 k mn+1 if and only if m2 [n ,n ,...,n ,2,n ,...,n ]= . k k−1 j+1 j−1 1 m(m−n)+1 8 PAOLOLISCA Therefore, without loss of generality we may assume n =2 and n >2. 1 k Under these assumptions we claim that (2.1) [n ,...,n −1]=0. 2 k Inordertoprovethe claim,wearguebyinductiononk ≥4. Ifk =4wemusthave (n ,n ,n ,n )=(2,2,1,3), 1 2 3 4 and the claim is clear. If k > 4, observe that, since by Lemma 2.2, (n ,...,n ) is 1 k a blowup of (2,2,1,3), we must have j > 2. Then, blowing down once yields the string (n =2,...,n −1,n −1,...,n ), 1 j−1 j+1 k to which we may apply induction to conclude [n ,...,n −1,n −1,...,n −1]=0 2 j−1 j+1 k if j q ≥1 be coprime and such that p =[b ,...,b ], b ,...,b ≥2, 1 k 1 k p−q with k ≥4, b ,...,b ≥3 and b ≥k−2. Let r ∈Z, 0≤r ≤k−4. Then, 2 k−2 k r k−4−r n =(1,2,...,2,3,2,...,2,1,k−2−r)∈Z r p,q z }| { z }| { ON SYMPLECTIC FILLINGS OF LENS SPACES 9 One can easily check that k (2.2) χ(W (n ))=5+ (b −3)+r. p,q r i i=1 X Then,Equation(2.2)andTheorem1.1implythat(L(p,q),ξ )admitsatleastk−3 st smoothly minimal symplectic fillings up to homotopy equivalence. (b) If p =[a ,...,a ], a ,...,a ≥5, 1 h 1 h q then by Lemma 2.3 Z ={(1,2,...,2,1)}. p,q The conclusion follows by Theorem 1.1. (c) Suppose that p =[b ,...,b ]. 1 k p−q Itiseasytochecktheattachingcircleofeachtwo–handleofW (n)ishomologically p,q non–trivial in S1×S2. Therefore, b (W (n)) = 0 if and only if there is exactly 2 p,q one index j ∈{1,...,k} such that n =1, and j (b ,...,b )=(n ,n ,...,n ,2,n ,...,n ). 1 k 1 2 j−1 j+1 k Then, by Lemma 2.4, p m2 = , p−q mn+1 with m and n coprime. Therefore q =mh−1, with h=m−n. (cid:3) 3. Compactifications of symplectic fillings The purpose ofthis sectionis to proveTheorem3.2 below. In orderto state the theorem, we need a definition. Definition 3.1. Let (X,ω) be a symplectic four–manifold. A symplectic string in X is an immersed symplectic surface Γ=C ∪C ∪···∪C ⊂X, 0 1 k where: • C is a connected, embedded symplectic sphere for i=0,...,k; i • C and C intersect transversely and positively at a single point, for i i+1 i=0,...,k−1; • C ∩C =∅ if |i−j|>1, for i,j =0,...,k. i j We say that Γ as above is of type (m ,...,m ) if, furthermore, 0 k • C ·C =m for i=0,...,k. i i i Theorem 3.2. Let p>q ≥1 be coprime integers, and suppose that p =[b ,...,b ], b ,...,b ≥2. 1 k 1 k p−q Let(W,ω)beasymplecticfillingof(L(p,q),ξ ). Then, forsomeintegerM ≥0,W st is orientation preserving diffeomorphic tothecomplement of aregularneighborhood of a symplectic string Γ=C ∪C ∪···∪C ⊂CP2#MCP2 0 1 k 10 PAOLOLISCA of type (1,1−b ,−b ,...,−b ), where CP2#MCP2 is endowed with a symplectic 1 2 k blowup ω of the standard Ka¨hler form on CP2, and C is a complex line in CP2. M 0 Theorem 3.2 will be used in Section 4. We start with an auxiliary construction of a suitable symplectic form on (0,∞)×L(p,q). The function ρ: C2 −→(0,+∞) (z ,z ) 7→|z |2+|z |2 1 2 1 2 is a Kähler potential for the standard symplectic form ω , i.e. 0 2 i i ω = dz ∧dz = ∂∂ρ. 0 k k 2 2 k=1 X Let J be the standard complex structure on C2, and consider the one–form 0 2 1 i i σ =− dρ◦J =− (∂ρ−∂ρ)=− (z dz −z dz ). 0 0 k k k k 4 4 4 k=1 X Let i: S3 = ρ−1(1) ֒→ C2 be the inclusion map, and define α = i∗σ ∈ Ω1(S3). 0 0 Since the standard contact structure ξ on S3 is given by complex tangent lines, st we have ξ ={α =0}⊂TS3. Define π: C2\{(0,0)}→S3 by setting: st 0 z π(z)= . ρ(z)12 The pair (ρ,π) induces an orientation preserving diffeomorphism: (ρ,π): C2\{(0,0)}→(0,+∞)×S3. The diffeomorphism (ρ,π) sends the standard symplectic form ω to the form 0 d(tα ), where t is the coordinate on the first factor. To see this, notice that 0 2 i z z 1 (i◦π)∗σ =− z d k −z d k = σ , 0 4k=1(cid:18) k (cid:18)ρ(z)12(cid:19) k (cid:18)ρ(z)12(cid:19)(cid:19) ρ 0 X and therefore: i i (3.1) (ρ,π)∗d(tα )=d(ρπ∗i∗σ )=dσ =− ∂∂ρ−∂∂ρ = ∂∂ρ. 0 0 0 4 2 Since U(2)acts onC2 vianorm–preservingcomple(cid:0)xlineartran(cid:1)sformations,σ and 0 ω are clearly U(2)–invariant while α is invariant under the naturally induced 0 0 actionon S3. Moreover,(ρ,π) is U(2)–equivariantinanobvious sense,so we have: • asymplecticformω inducedbyω onC2\{(0,0)}/G ,whereG isthe 0 0 p,q p,q subgroup of U(2) defined by (1.1), • a contact form α induced by α on L(p,q)=S3/G , and 0 0 p,q • a symplectomorphism (3.2) C2\{(0,0)}/Gp,q,ω0 ∼=((0,+∞)×L(p,q),d(tα0)). With the nota(cid:0)tion of Section 1, we h(cid:1)ave ξ ={α =0}. The action st 0 (3.3) S1×S3 −→ S3 (3.4) (eiθ,z) 7−→ eiθz

On Symplectic Fillings of Lens Spaces

Similar On Symplectic Fillings of Lens Spaces