Dehn surgeries and negative-definite four-manifolds
aa r X i v : . [ m a t h . G T ] A ug DEHN SURGERIES AND NEGATIVE-DEFINITEFOUR-MANIFOLDS
BRENDAN OWENS AND SAˇSO STRLE
Dedicated to Jos´e Maria Montesinos on the occasion of his 65th birthday.
Abstract.
Given a knot K in the three-sphere, we address the question: whichDehn surgeries on K bound negative-definite four-manifolds? We show that theanswer depends on a number m ( K ), which is a smooth concordance invariant. Westudy the properties of this invariant, and compute it for torus knots. Introduction
The intersection pairing of a smooth compact four-manifold, possibly with bound-ary, is an integral symmetric bilinear form Q X on H ( X ; Z ) / Tors ; it is nondegenerateif the boundary of X is a rational homology sphere. Given a rational homologythree-sphere Y there are various gauge-theoretic constraints on which bilinear formsmay be intersection pairings of manifolds bounded by Y . For example, Donaldson’scelebrated Theorem A [1] tells us that the only negative-definite pairings boundedby the three-sphere are the standard diagonal forms. Another well-known example isthe Poincar´e homology sphere P , oriented as the boundary of the positive-definite E P is +1 surgery on the right-handed trefoil knot. It is also well-knownthat +5 surgery on the same trefoil knot gives the lens space L (5 ,
1) which is theboundary of a negative-definite disk bundle over S . A natural question arises, forthe trefoil and more generally for any knot K in S : for which rational numbers r does the Dehn surgery S r ( K ) bound a smooth negative-definite four-manifold? Thisquestion is related to the computation of unknotting numbers, and also to the clas-sification of tight contact structures.An easy argument, for example using Lemma 2.5, shows that S r ( K ) bounds negative-definite whenever r is negative. Any knot can be converted to the unknot by a finitenumber of crossing changes. This enables us to show that large positive surgeries ona knot K always bound negative-definite four-manifolds, and leads to the following Date : August 25, 2011.B. Owens was supported in part by NSF grant DMS-0604876 and by the EPSRC.S.Strle was supported in part by the ARRS of the Republic of Slovenia research program No. P1-0292-0101. We also acknowledge support from the ESF through the ITGP programme. invariant:(1) m ( K ) = inf { r ∈ Q > | S r ( K ) bounds a negative-definite 4-manifold } . Some properties of m are described in the following theorem which we prove in Section3; we conjecture that the subadditivity property of the last part holds for m ( K ) aswell. Theorem 1. (a) Let K ⊂ S be a knot. If K can be unknotted by changing p positiveand n negative crossings, then m ( K ) ≤ p . In particular, if K admits a diagramwithout positive crossings then m ( K ) = 0 .(b) For all rational numbers r > m ( K ) , the r -surgery S r ( K ) on K bounds a negative-definite manifold X r with H ( X r ) → H ( S r ( K )) surjective.(c) m ( K ) is a concordance invariant of K , hence it defines a function m : C → R ≥ ,where C denotes the smooth concordance group of classical knots.(d) The integer valued invariant ⌈ m ⌉ is subadditive with respect to connected sum,i.e. ⌈ m ( K C ) ⌉ ≤ ⌈ m ( K ) ⌉ + ⌈ m ( C ) ⌉ for any knots K and C in S . In Section 4 we compute m for torus knots. The question of which nonzero Dehnsurgeries on torus knots bound negative-definite manifolds was considered previouslyin [9] and [3]; we give a complete answer here. Let p , q be coprime integers with p > q >
0. It is well known that the ( pq − T p,q boundsa negative-definite manifold (since this surgery is a lens space [7]). Let n be thenumber of steps in the standard Euclidean algorithm for p and q . Denote by q ∗ = q − (mod p ) the solution to the congruence qa ≡ p ) with 0 < a < p , and similarlylet p ∗ = p − (mod q ). Theorem 2.
Let T p,q denote the positive ( p, q ) -torus knot. Then m ( T p,q ) = pq − qp ∗ if n is even, pq − pq ∗ if n is odd.The manifold given by m ( T p,q ) surgery on T p,q bounds a negative-definite four-manifold.Moreover, for any negative-definite four-manifold this surgery bounds not all Spin c structures on the surgery manifold extend over the four-manifold.For any negative torus knot, m ( T p, − q ) = 0 . The special case of q = n = 2 follows from work of Lisca-Stipsicz [6]; they usedessentially the same obstruction to get a lower bound on m but a different constructionfor the negative-definite manifold bounded by S m ( T p,q ) ( T p,q ). Theorem 2 yields thefollowing generalisation of [6, Theorem 4.2]. EHN SURGERIES AND NEGATIVE-DEFINITE FOUR-MANIFOLDS 3
Corollary 3.
For each rational number r in the interval [ pq − p − q, m ( T p,q )) , the3-manifold given by r surgery on T p,q does not admit fillable contact structures. Continued fractions and surgery cobordisms
In this section we establish some notation and basic facts about continued fractionsand surgery cobordisms. Given a sequence of numbers c , c , . . . , c n in the extendedreal line R ∪ {∞} (typically these will be integers) one obtains two numbers[ c , c , . . . , c n ] + := c + 1 c + . . . + c n , and [ c , c , . . . , c n ] − := c − c − . . . − c n ;we refer to these as positive and negative continued fraction expansions, respectively.For a given positive rational number the standard Euclidean algorithm yields a uniqueexpansion pq = [ c , c , . . . , c n ] + with integer coefficients satisfying c ≥ c i > < i < n and c n >
1. Similarlythere is a unique expansion pq = [ a , a , . . . , a m ] − , with positive integer coefficients and a i > i >
1. (This is often referred to asthe Hirzebruch-Jung continued fraction of pq .)The following is immediate from the Euclidean algorithm for p/q . Lemma 2.1.
Let p > q > be coprime integers with pq = [ c , c , . . . , c n ] + . Writing p = c q + r we have qr = [ c , . . . , c n ] + . Lemma 2.2.
The positive and negative continued fraction expansions are related bythe formula [ c , . . . , c n ] + = [ c + 1 , , . . . , | {z } c − , c + 2 , , . . . , | {z } c − , c + 2 , . . . ] − , where the continued fraction ends with c n + 1 if n is odd and with c n − ’s if n iseven. BRENDAN OWENS AND SAˇSO STRLE
Proof.
This may be deduced easily from the equations[ c, x ] + = (cid:20) c + 1 , xx − (cid:21) − and(2) y = [ c, z ] + ⇐⇒ yy − , . . . , | {z } c − , z ] − . (Or see [11, Proposition 2.3].) (cid:3) Lemma 2.3.
Let p > q > be integers with pq = [ c , c , . . . , c n ] + . Then (3) pp − q = [2 , . . . , | {z } c − , c + 2 , , . . . , | {z } c − , c + 2 , . . . ] − , where the continued fraction ends with c n + 1 if n is even and with c n − ’s if n isodd.Proof. This follows easily from Lemma 2.2 and (2). (Or see [8, Lemma 7.2] or [11,Proposition 2.7].) (cid:3)
Given coprime natural numbers p and q , define q ∗ to be the multiplicative inverseof q modulo p , i.e. qq ∗ ≡ p ) and 1 ≤ q ∗ < p . Lemma 2.4. If p > q are coprime positive integers and pq = [ c , . . . , c n ] − with integers c i ≥ , then pq ∗ = [ c n , . . . , c ] − . Proof.
This can be seen by induction on n . For details see for example [4]. (cid:3) The importance of continued fractions in our context is due to their appearancewhen Dehn surgeries are converted to integer surgeries. If pq = [ c , . . . , c n ] − , then the 3-manifold given by pq surgery on a knot K is equivalent to that given bya framed link consisting of K with framing c and a chain of unknots with framings c , . . . , c n as in Figure 1. The equivalence of the two descriptions is established usingthe slam-dunk move (see [2, § L ( p, q ) = L ( p, q ∗ ). The next lemma is of use in computing the signatureof the resulting 4-manifold with boundary. EHN SURGERIES AND NEGATIVE-DEFINITE FOUR-MANIFOLDS 5 pq c c c c n − c n ∼ . . . Figure 1.
Converting between Dehn surgery and integral surgery.Lemma 2.5.
Let a , . . . , a n be integers with a ≥ , | a i | ≥ for < i < n and either | a n | ≥ or a n = − . Let A denote the symmetric n × n matrix whose nonzero entriesare A i,i = a i , A i,i ± = 1 . Then signature( A ) = { i | a i > } − { i | a i < } . Proof.
Write A n for A . We prove the formula by induction. Let A n − denote theminor given by deleting the last row and column of A n . We claim thatsignature( A n ) = signature( A n − ) + sign( a n ) . Let v , . . . , v n be basis vectors for a free abelian group with a bilinear pairing givenby Q ( v i , v j ) = A i,j . Extending coefficients to Q there are constants b i ∈ Q for which v ′ n = v n − n − X i =1 b i v i is Q orthogonal to v i for i < n ; in fact, b n − = 1 / [ a n − , . . . , a ] − . Inthe basis v , . . . , v n − , v ′ n the form Q has a block matrix with blocks A n − , Q ( v ′ n , v ′ n )from which it follows that the signature of A is given by the sum of the signatureof A n − and the sign of Q ( v ′ n , v ′ n ). The conditions on a , . . . , a n − ensure that − < / [ a n − , . . . , a ] − ≤ Q ( v ′ n , v ′ n )) = sign (cid:18) a n − a n − , . . . , a ] − (cid:19) = sign( a n ) . Alternatively, note that successively blowing down ± (cid:3) We start our study of negative-definite cobordisms that determine the behaviourof m ( K ) by showing that if some surgery on K bounds a negative-definite manifold,then so does any larger surgery. Lemma 2.6.
Let K be a knot in S and let r, s be rational numbers with r > s > .Then there exists a negative-definite two-handle cobordism from S s ( K ) to S r ( K ) . BRENDAN OWENS AND SAˇSO STRLE
Proof.
Suppose that the negative continued fractions of r, s agree for the first m terms, m ≥
0. In other words we have s = [ c , . . . , c m , c m +1 , . . . , c m + k ] − r = [ c , . . . , c m , c ′ m +1 , . . . , c ′ m + k ′ ] − with c , c ′ ≥ c n , c ′ n ≥ n ≥
2, and 0 < l = c ′ m +1 − c m +1 .The cobordism from S s ( K ) to S r ( K ) is then a composition of cobordisms W , . . . , W l which we proceed to describe: S s ( K ) W −→ S c ,...,c m +1 ] − ( K ) W −→ S c ,...,c m +1 +1] − ( K ) W −→ S c ,...,c m +1 +2] − ( K ) W −→· · · W l − −→ S c ,...,c ′ m +1 − − ( K ) W l −→ S r ( K );the negative-definiteness of each of W , . . . , W l follows by Lemma 2.5.To obtain the cobordism W note that S c ,...,c m +1 ] − ( K ) bounds the positive-definiteinteger surgery cobordism given by K with a chain of linked unknots in the usualway, with framings given by the continued fraction coefficients. There is an obviouspositive-definite cobordism from this to the corresponding integer surgery descriptionof S s ( K ); reversing orientation yields W .Each W i for 0 < i < l is the surgery cobordism given by attaching a ( − S c ,...,c m +1 + i − − ( K ) alongthe meridian of the last unknot in the chain.Now let r ′ = [ c ′ m +2 , . . . , c ′ m + k ′ ] − (in other words r ′ is given by the tail of the contin-ued fraction of r ) and let r ′ r ′ − a , . . . , a n ] − , with a i ≥
2. Then we have r = [ c , . . . , c m , c ′ m +1 , r ′ ] − = [ c , . . . , c m , c ′ m +1 − , − a , − a , . . . , − a n ] − , which yields the negative-definite surgery cobordism W l from S c ,...,c m ,c ′ m +1 − − ( K ) to S r ( K ). (cid:3) The next two lemmas exhibit negative-definite cobordisms from the disjoint unionof S r ( K ) and S s ( C ) to S r + s ( K C ) for certain surgery coefficients r and s . We usethem to prove subadditivity of an integer version of m ( K ) under connected sums. Wecan exhibit many other such cobordisms and in fact conjecture that such a cobordismexists for any positive rational numbers r and s . The point of the second lemma isthat we do not know if the surgery manifold S ⌈ m ( K ) ⌉ ( K ) bounds a negative-definitemanifold in general. EHN SURGERIES AND NEGATIVE-DEFINITE FOUR-MANIFOLDS 7
Lemma 2.7.
Let K and C be knots in S and let r, s be positive rational numberswhose sum is an integer. Then there exists a negative-definite cobordism from S r ( K ) ⊔ S s ( C ) to S r + s ( K C ) .Proof. We begin with the simplest case which is when r = m and s = n are bothintegers. The Kirby diagram on the right hand side of Figure 2 represents a four-manifold whose boundary is S m + n ( K C ): to see this trade the 0-framed 2-handlefor a 1-handle in dotted circle notation, slide one of the remaining 2-handles over theother and cancel the 1-handle with a 2-handle. The addition of the 0-framed 2-handlegives the required negative-definite cobordism in this case, since the four-manifold onthe left of Figure 2 is definite with b +2 = 2 and that on the right has b +2 = 2 and b − = 1. (Similar calculations show the cobordism is also negative-definite if one orboth of m, n is zero.) KK CC mm nn −→ Figure 2.
A negative-definite cobordism from S m ( K ) ⊔ S n ( C ) to S m + n ( K C ) . If r = m − q/p and s = n − ( p − q ) /p with m, n positive integers and pq =[ c , c , . . . , c n ] + we may modify the diagram in Figure 2 accordingly. See Figure 3 forthe case q/p = 1 /
3. On the left hand side, we have K with framing m and a chain ofunknots with framings c + 1 , , . . . , | {z } c − , c + 2 , , . . . , | {z } c − , c + 2 , . . . BRENDAN OWENS AND SAˇSO STRLE and also C with framing n and a chain of unknots with framings2 , . . . , | {z } c − , c + 2 , , . . . , | {z } c − , c + 2 , . . . . By Lemmas 2.2 and 2.3 this is a four-manifold bounded by S r ( K ) ⊔ S s ( C ). Thecobordism W is obtained by adding a single +1 framed unknot which links each of thetwo rightmost unknots described above once. A sequence of (+1)-blowdowns convertsthis diagram to the right hand diagram in Figure 2 (with one of m, n decreasedby 1), from which it follows that W is again a negative-definite cobordism from S r ( K ) ⊔ S s ( C ) to S r + s ( K C ). (cid:3) KK CC mm nn −→ Figure 3.
A negative-definite cobordism from S m − / ( K ) ⊔ S n − / ( C ) to S m + n − ( K C ) .Lemma 2.8. Let K and C be knots in S and let l, m, n be nonnegative integers with l > . Then there exists a negative-definite cobordism from S m +1 / l ( K ) ⊔ S n +1 / l ( C ) to S m + n +1 /l ( K C ) .Proof. The cobordism is illustrated in Figure 4 for the case l = 2. On the left hand sidewe have a Kirby diagram representing a four-manifold with boundary S m +1 / l ( K ) ⊔ S n +1 / l ( C ). The cobordism W is given by adding l ( − b +2 and increases b − by l so W is negative-definite. Blowing down the( − l -framed 2-handles over theother results in the last diagram shown in Figure 4; as in Lemma 2.7 this is seen torepresent the three-manifold S m + n +1 /l ( K C ). (cid:3) EHN SURGERIES AND NEGATIVE-DEFINITE FOUR-MANIFOLDS 9
KKK CCC mmm nnn − − − − − − − −→ ∂ ∼ Figure 4.
A negative-definite cobordism from S m +1 / ( K ) ⊔ S n +1 / ( C ) to S m + n +1 / ( K C ) . Basic properties
In this section we establish some properties of m ( K ), in particular its existence. Proof of Theorem 1. (a) Let Y be the 4 p -surgery on K . If all the crossing changes inthe unknotting of K are realized by ( − Y consists of a 0-framed unknot (corresponding to K ) along with an unlink of( − Y bounds a negative-definite 4-manifold with one 1-handle.(b) Given r = p/q > m ( K ) there exists a rational number s = p ′ /q ′ ∈ ( m ( K ) , r )with p, p ′ coprime. By the definition of m ( K ) and Lemma 2.6, S s ( K ) bounds anegative-definite four-manifold X s . Let W be the cobordism from S s ( K ) to S r ( K )given by Lemma 2.6. Since W is a two-handle cobordism, its first homology is aquotient of the first homology of either boundary component. Since the orders ofthe first homology groups of the boundary components are coprime, they have nonontrivial common quotient and so H ( W ) = 0. Taking the union X s ∪ W yields anegative-definite manifold X r bounded by S r ( K ) with the property that the inclusionof the boundary in the manifold induces trivial homomorphism on H . Using Poincar´eduality this implies that the restriction map on H is onto.(c) If K ′ is concordant to K , let A ⊂ S × I be the annulus realizing the concordance.Then r -surgeries on K and K ′ extend over A ; the resulting 4-manifold is a homologycobordism from S r ( K ) to S r ( K ′ ).(d) Let m = ⌈ m ( K ) ⌉ and n = ⌈ m ( C ) ⌉ . Suppose first that both S m ( K ) and S n ( C )bound negative-definite four-manifolds. Gluing these to the cobordism from Lemma2.7 yields a negative-definite manifold bounded by S m + n ( K C ) so that ⌈ m ( K C ) ⌉ ≤ m + n . In general S r ( K ) and S s ( C ) bound negative-definite four-manifolds for any rationalsurgery coefficients r > m and s > n . In particular we may take r = m + 1 / l and s = n + 1 / l for any positive integer l . Combining with the negative-definitecobordism from Lemma 2.8 we see that S m + n +1 /l ( K C ) bounds negative-definite.Letting l → ∞ we again see that ⌈ m ( K C ) ⌉ ≤ m + n . (cid:3) Torus knots
In this section we prove Theorem 2 and Corollary 3.Let p > q > pq = [ c , c , . . . , c n ] + , c i > c n ≥ µ ( p, q ) = pq − qp ∗ if n is even, pq − pq ∗ if n is odd. Proposition 4.1. If S r ( T p,q ) bounds a negative-definite four-manifold then r ≥ µ ( p, q ) . Proposition 4.2.
The manifold S µ ( p,q ) ( T p,q ) embeds in a connected sum of C P ’s asa separating submanifold, and hence bounds a negative-definite four-manifold. Proposition 4.3. If W is any negative-definite manifold that S µ ( p,q ) ( T p,q ) boundsthen the restriction homomorphism H ( W ; Z ) → H ( S µ ( p,q ) ( T p,q ); Z ) is not onto; con-sequently, H ( W ; Z ) contains nontrivial torsion. We use notation Y ( e ; α β , α β , α β ) to denote the 3-manifold that results by performingsurgeries with the listed fractional coefficients on disjoint fibres of the degree e S -bundle over S , as in Figure 5. If the fractional coefficients are nonzero this is aSeifert fibred space whose exceptional fibres have orders α i . We will also allow α i β i tobe zero or ∞ . eα β α β α β Figure 5. Y ( e ; α β , α β , α β ). EHN SURGERIES AND NEGATIVE-DEFINITE FOUR-MANIFOLDS 11
Lemma 4.4 (cf. Moser [7]) . For any rational number r , S r ( T p,q ) = Y (cid:18) pq ∗ , qp ∗ , pq − rpq − r − (cid:19) , Proof.
Start with the Seifert fibration of S by ( p, q ) torus knots which has twoexceptional orbits, one of order p and the other of order q . To determine the surgerycoefficient (relative to the fibration) corresponding to the r -surgery along a regularfibre note that the linking number lk( K, K ′ ) = pq , where K is the surgery curve and K ′ is a nearby regular fibre. For r = a/b the surgery curve is given by γ = aµ + bλ =( a − bpq ) µ + bK ′ , which yields r − pq for the surgery coefficient. It follows that S r ( T p,q ) = Y (cid:18) pβ , qβ , r − pq (cid:19) , for some β , β . Then the order of the first homology of the surgery is β q ( a − pqb ) + β p ( a − pqb ) + pqb = ± a = ⇒ pqb ( qβ + pβ −
1) = a ( qβ + pβ ∓ . Since the last equation holds for all r = a/b we conclude qβ + pβ − β and β by requiring 0 < β < p and | β | < q , so that β = q ∗ and β + q = p ∗ . The result now follows by applying Rolfsen twists (see for example[2, § Y (0; p/β , q/β , r − pq ) withthe framings q/β and r − pq . (cid:3) Proof of Proposition 4.1.
We may assume that r < pq −
1. Using Lemmas 4.4, 2.1,2.2 and 2.4 we find that S r ( T p,q ) is the boundary of a positive-definite plumbing P ofdisk bundles over spheres corresponding to a tree with 3 legs where the weight of thecentral vertex is 2. The weights on the three legs (listed from the central vertex) areas follows: • the weights on the first leg are the coefficients in the negative continued frac-tion for pq − rpq − r − ; • the weights on the second leg are c n + 1 , , . . . , | {z } c n − − , c n − + 2 , . . . , , . . . , | {z } c − , c + 2 , , . . . , | {z } c − , c + 1if n is odd and c n + 1 , , . . . , | {z } c n − − , c n − + 2 , . . . , , . . . , | {z } c − , c + 2 , , . . . , | {z } c − , c + 1if n > c if n = 2). • the weights on the third leg are2 , . . . , | {z } c n − , c n − + 2 , . . . , , . . . , | {z } c − , c + 2 , , . . . , | {z } c − , c + 1 if n is odd and2 , . . . , | {z } c n − , c n − + 2 , . . . , , . . . , | {z } c − , c + 2 , , . . . , | {z } c − , c + 1if n is even.(Note that if n is odd, then the second leg arises from the continued fractionexpansion of pq ∗ and the third leg corresponds to qp ∗ ; for n even qp ∗ gives the secondleg and pq ∗ gives the third.)If S r ( T p,q ) bounds a negative-definite 4-manifold X , then P ∪ ( − X ) is a closedpositive-definite manifold so by Donaldson’s Theorem the intersection form of P em-beds in some Z k (with the standard form). We seek the minimal (or a priori, infimal) r for which such an embedding is possible. We first note that minimising r is equivalentto minimising pq − rpq − r − a , . . . , a m ] − = [ a , . . . , a m , ∞ ] − , which in turn is equivalent to finding the smallest sequence a , a , . . . , a m , ∞ in lexi-cographical ordering, with integer coefficients a i ≥ E denote the central vertex of P , and let U , U , . . . denote the vertices on thefirst leg. Similarly label the vertices on the second and third legs with V i and W j respectively. Denote basis vectors of Z k by e i and f j . Suppose for some r we have anembedding of the intersection form of P in Z k . Without loss of generality, E mapsto e + f and V maps to − e + x for some x ∈ Z k .We claim there is an embedding with a = 2 so that U maps to one of − e + e or − f + f . Suppose first that U maps to − e + e , so that V maps to − e − e + x ′ . Ifwe can take a = 2 as well we must map U to − e + e and V to − e − e − e + x ′′ .Continuing in this way we find that U i maps to − e i + e i +1 for i = 1 , . . . , c n − V maps to e + e + · · · + e c n +1 and V maps to e c n +1 − e c n +2 , and so on. The requirementthat r be minimal combined with the assumption that U maps to − e + e completelydetermines the weights on the first leg and the embedding in Z k (up to automorphismof Z k ) of the first two legs. The weights on the first leg (under this assumption) are2 , . . . , | {z } c n − , c n − + 2 , . . . , , . . . , | {z } c − , c + 2 , , . . . , | {z } c − , if n is odd and 2 , . . . , | {z } c n − , c n − + 2 , . . . , , . . . , | {z } c − , c + 2 , , . . . , | {z } c − , if n is even.The reasoning from the previous paragraph may be applied to the third leg insteadof the first, showing that W maps to − f + f (the weights on the third leg representa smaller continued fraction than the minimal value found in the previous paragraph). EHN SURGERIES AND NEGATIVE-DEFINITE FOUR-MANIFOLDS 13
This enables us to eliminate the possibility that U maps to − f + y for any y ∈ Z k since orthogonality with the vertices on the third leg would then imply a > r . We have pq − rpq − r − , . . . , | {z } c n − , c n − + 2 , . . . , , . . . , | {z } c m +2 − , c m +1 + 2 , , . . . , | {z } c m − ] − , where m = 1 if n is odd, else m = 2. Applying (2) and Lemmas 2.1, 2.2 and 2.4 wefind pq − r = [ c n , c n − , . . . , c m ] + = qp ∗ if n is even, pq ∗ if n is odd. (cid:3) Proof of Proposition 4.2.
Assume for convenience that n is odd (the even case isproved in exactly the same way). From the proof of Proposition 4.1 we see that S µ ( p,q ) is the boundary of the positive-definite 4-manifold P presented by a Kirby diagramcorresponding to a three-legged tree, with framing 2 on the unknot corresponding tothe central vertex, and framings on the 3 legs given by • first leg: 2 , . . . , | {z } c n − , c n − + 2 , . . . , , . . . , | {z } c − , c + 2 , , . . . , | {z } c − , • second leg: c n + 1 , , . . . , | {z } c n − − , c n − + 2 , . . . , , . . . , | {z } c − , c + 2 , , . . . , | {z } c − , c + 1, • third leg: 2 , . . . , | {z } c n − , c n − + 2 , . . . , , . . . , | {z } c − , c + 2 , , . . . , | {z } c − , c + 1.We add further handles to this diagram to get a diagram for a manifold e X . Foreach vertex in the third leg, if the weight of the vertex is w and the valency is v (oneif the rightmost vertex, or else 2), add w − v parallel (+1)-framed meridians to thecorresponding component of the Kirby diagram. We claim the resulting e X can beobtained from S × B by a sequence of (+1)-blow-ups.Begin by blowing down (+1)-framed unknots on the third leg; this completelyeliminates the third leg and replaces the weight on the central vertex by 1. We nowhave a linear plumbing with weights c +1 , , . . . , | {z } c − , c +2 , . . . , , . . . , | {z } c n − − , c n +1 , , , . . . , | {z } c n − , c n − +2 , . . . , , . . . , | {z } c − , c +2 , , . . . , | {z } c − . A simple induction argument shows that successive blow-downs reduce this to a singlezero-framed unknot. Hence we can add a 3-handle and 4-handle to e X to get aconnected sum of C P ’s. Let X be the closure of the complement of P in the connected sum of C P ’s; then − X is a negative-definite manifold bounded by S µ ( p,q ) ( T p,q ). (cid:3) A sublattice L ⊂ Z k is called primitive if the quotient Z k /L is torsion-free. If L is not primitive in Z k , then the restriction homomorphism on the dual latticesHom( Z k , Z ) → L ′ = Hom( L, Z ) is not surjective. This observation yields the followingresult which is the main ingredient in the proof of Proposition 4.3. Lemma 4.5.
Let Y be a rational homology -sphere that bounds a positive-definite -manifold X with H ( X ; Z ) = 0 . Suppose that the intersection lattice L = ( H ( X ; Z ) , Q X ) of X does not admit a primitive embedding in any Z k . Then for any negative-definite -manifold W that Y bounds the restriction homomorphism H ( W ; Z ) → H ( Y ; Z ) is not onto; consequently, H ( W ; Z ) contains nontrivial torsion.Proof. It follows from the exact cohomology sequence of the pair (
X, Y ) (using as-sumption H ( X ; Z ) = 0) that H ( Y ; Z ) ∼ = L ′ /L , where L ′ denotes the dual latticeof L . Let W be a negative-definite manifold with boundary Y ; we may assume that b ( W ) = 0. Let Z = X ∪ Y ( − W ). It follows from the Mayer-Vietoris homologyexact sequence for Z that L embeds in H ( Z ) / Tors. Note that the intersection pair-ing of Z is isomorphic to some Z k by Donaldson’s Theorem. Since by assumption L is not primitive in Z k , the restriction H ( Z ) → L ′ is not onto. Consider now theMayer-Vietoris cohomology exact sequence for Z : · · · → H ( Z ) → H ( X ) ⊕ H ( W ) → H ( Y ) → · · · . Choose an element x ∈ L ′ = H ( X ) that is not in the image of the restrictionhomomorphism from H ( Z ). Then the image of x in H ( Y ) is not in the image ofthe restriction H ( W ) → H ( Y ); if it were the image of some y ∈ H ( W ) then x ⊕ y would be in the image of H ( Z ). (cid:3) Proof of Proposition 4.3.
We again assume for clarity of exposition that n is odd.The proof of Proposition 4.1 shows that S µ ( p,q ) ( T p,q ) bounds a plumbing P whoseintersection form (call it L ) embeds in Z k . We claim that for any such embedding L is not primitive in Z k . Proposition 4.3 is a consequence of this claim and Lemma 4.5.From the proof of Proposition 4.1 we see that there is a unique embedding in Z k (up to automorphisms) of the first two legs of the tree defining P . A simplerecursive description of all such embeddings is as follows: starting with the sequence − e − e , e , e − e one applies a finite sequence of the following modifications (blow-ups): . . . , v, e i , w, . . . . . . , v − e i +1 , e i +1 , e i − e i +1 , w, . . . or . . . , v, e i , w, . . . . . . , v, e i − e i +1 , e i +1 , w − e i +1 , . . . and then replaces the final e s with e s + f . (This becomes the central vertex of thetree, and the two chains on either side give the first two legs of the tree; comparing to EHN SURGERIES AND NEGATIVE-DEFINITE FOUR-MANIFOLDS 15 the proof of Proposition 4.1 one should reverse the order of indices of the basis vectors e , . . . , e s .) Inductively we see that the image of the sublattice L of L correspondingto the first two legs rationally spans Z s but is not equal to it as its determinant is p >
1. We claim that the image of L intersected with this Z s is equal to the imageof L and therefore L is not primitive in Z k . Indeed, the embedding of the third legcannot use any of the basis vectors e , . . . , e s , hence the only way the intersectioncould be larger is if some multiple of e s were in the image of the central vertex andthe third leg. Using that e s is rationally in the image of L , so e s = X i c i v i + X j d j w j , where v i ( w j ) are images of vectors in the first (second) leg and c i , d j ∈ Q , this wouldimply that the two sums above (representing orthogonal vectors) vanish, providing acontradiction. (cid:3) Theorem 2 follows from Propositions 4.1, 4.2, 4.3, and Theorem 1(a).
Proof of Corollary 3.
The proof is based on that of [6, Theorem 4.2]; we briefly recallthe argument here. Let Y r denote the surgery manifold S r ( T p,q ). By [6, Proposition4.1], Y r is an L -space whenever r ≥ g s ( T p,q ) − pq − p − q . By [10, Theorem 1.4],any symplectic filling of an L -space is negative-definite. By Theorem 2, Y r does notbound a negative-definite four-manifold if r ∈ [ pq − p − q, m ( T p,q )) and thus does notadmit a fillable contact structure. (cid:3) We note that the result in Corollary 3 is optimal: for any r / ∈ [ pq − p − q, m ( T p,q )), thethree-manifold obtained by r surgery on T p,q does admit a fillable contact structure.This may be deduced using the classification by Lecuona-Lisca of Seifert fibred spaceswhich admit fillable contact structures [5, Theorem 1.3]. References [1] S. K. Donaldson.
An application of gauge theory to four-dimensional topology , J. Diff. Geom. (1983), 279–315.[2] R. E. Gompf & A. I. Stipsicz. 4 -manifolds and Kirby calculus , Graduate Studies in Math. ,Amer. Math. Soc., 1999.[3] J. E. Greene. L-space surgeries, genus bounds, and the cabling conjecture , arXiv:1009.1130,2010.[4] F. Hirzebruch, W. D. Neumann & S. S. Koh.
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Ozsv´ath-Szab´o invariants and tight contact three-manifolds, I , Geom.Topol. (2004), 925–945.[7] L. Moser. Elementary surgery along a torus knot , Pacific Journal of Math. (1971), 737–745.[8] W. Neumann. A calculus for plumbing applied to the topology of complex surface singularitiesand degenerating complex curves , Trans. Amer. Math. Soc. (1981), 299344 [9] B. Owens & S. Strle.
A characterisation of the Z n ⊕ Z ( δ ) lattice and definite nonunimodularintersection forms , arXiv:0802.1495.[10] P. Ozsv´ath & Z. Szab´o. Holomorphic disks and genus bounds , Geom. Topol. (2004), 311–334.[11] P. Popescu-Pampu. The geometry of continued fractions and the topology of surface singulari-ties , Singularities in geometry and topology 2004, 119–195, Adv. Stud. Pure Math. , Math.Soc. Japan, 2007. School of Mathematics and StatisticsUniversity of GlasgowGlasgow, G12 8QW, United Kingdom
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