A NOTE ON E 8 -INTERSECTION FORMS AND CORRECTION TERMS
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1 A NOTE ON E 8 -INTERSECTION FORMS AND CORRECTION TERMS MOTOO TANGE Abstract. In this paper we construct families of homology spheres which bound 4-manifolds with intersection forms isomorphic to E 8. We show that these families have arbitrary large correction terms. 1. Introduction 1.1. Definite bounds and homology cobordism invariants. Any oriented homology sphere Y bounds a spin 4-manifold W. We call the 4- manifold W a spin bound of Y. The restriction for spin bound is classically well-known through the Rohlin invariant µ(y ). It is defined by σ(w )/8 Z/Z for a spin bound W of Y. If the intersection form of a spin bound is (positive/negative) definite, then we call it (positive/negative) definite spin bound. In this paper we would like to consider definite spin bounds for a homology sphere. If the homology of a 4-manifold has no - torsions, the definite even bound implies spin and definite bound. In this article we assume homologically 1-connected i.e., H 1 (W ) = {0} for bounding 4-manifold W. Ozsváth and Szabó defined a homology coboridism invariant d in [5]. If a 3-manifold bounds a negative-definite 4-manifold, then the d-invariant has the following restriction. Theorem 1.1 ([5]). Let Y be an integral homology three-sphere, then for each negative-definite four-manifold X which bounds Y, we have the inequality ξ + rk(h (X, Z)) 4d(Y ) for each characteristic vector ξ. Thus, if a homology sphere Y has an even negative-definite bound W, then the second Betti number satisfies b (W ) 4d(Y ). For example Σ(, 3, 5) is the boundary of the E 8 -plumbing. Here E 8 is the unimodular, even, negative-definite, rank 8 quadratic form. The computation d(σ(, 3, 5)) = means that any negative-definite even bound satisfies b 8. On the other Date: November 6, Mathematics Subject Classification. 57R55, 57R65. Key words and phrases. 4-manifold, E 8 -bounds, signature zero 4-manifolds, logtransformation. The author was partially supported by JSPS KAKENHI Grant Number
2 MOTOO TANGE hand, the d-invariant of Σ(, 3, 7) is 0. Thus, if there exists a negativedefinite even bound, the second Betti number has to be b = 0. Since µ = 1 of Σ(, 3, 7), it has no homology 1-connected definite spin bound. The plumbing coming from the Seifert structure of Σ(, 3, 7) gives ( E 8 ) H, where H is the hyperbolic intersection form. Hence, the H-component cannot be removed by some surgery keeping the bound spin. In [6] the author defined invariants measuring the Betti number of the bound of direct sums of E 8 -intersection form. If a homology sphere Y has a 4-dimensional bound which is the ne 8 -intersection form, then we say that Y has an ne 8 -bound. If Y has such a bound, then we define g 8 (or g 8 ) to be g 8 (Y ) = max{ n Y = X, H 1 (X) = {0}, Q X = ne8 } g 8 (Y ) = min{ n Y = X, H 1 (X) = {0}, Q X = ne8 }. If Y does not have any ne 8 -bound at all, then g 8 (Y ) =. We call the invariant g 8 E 8 -genus. If g 8 (Y ) <, then immediately, we can see the following bound (1) g 8 (Y ) 4 d(y ). For example, for any integer n, d(σ(, 3, 1n + 5)) = holds. The author in [6] showed g 8 (Σ(, 3, 1n + 5)) = 1 when 0 n 13 or n = 15. In [6] we gave the examples with 4 d(y ) g 8 (Y ) = 0. The simple question is the following: Question 1.1. We assume that a homology sphere Y has an even definite bound W. Then, is 4 d(y ) g 8 (Y ) bounded? We give families of Brieskorn homology spheres to obtain negative answers for this question. 1.. Main results. Here we give the main result: Theorem 1.. For any integer n, the following homology spheres have E 8 - bound, i.e., satisfy g 8 = 1. (i) Σ(, 8n 3, 14n 5), (ii) Σ(, 14n + 3, 4n + 5) (iii) Σ(, 16n + 3, 6n + 5), (iv) Σ(, 10n 3, 16n 5) (v) Σ(5, 35n, 50n 3), (vi) Σ(5, 5n, 40n 3) (vii) Σ(3, 15n, 36n 5), (viii) Σ(3, 9n, 4n 5) (ix) Σ(3, 1n 4, 36n 7), (x) Σ(3, 7n 4, 48n 7) (xi) Σ(4, 8n 3, 64n 7), (xii) Σ(4, 3n 3, 76n 7) These constructions will be useful for realization of E 8 intersection form of 4-manifold bounding 3-manifolds restricted by gauge theory. For example, see recent Scaduto s study [10]. Theorem 1.3. For positive integer n the correction terms of Brieskorn homology spheres (i), (ii) (iii) and (iv) in Theorem 1. have the following inequalities:
3 A NOTE ON E 8 -INTERSECTION FORMS AND CORRECTION TERMS 3 n d(σ(, 8n 3, 14n 5)), n+1 d(σ(, 14n+3, 4n+5)), n+1 d(σ(, 16n+3, 6n+5)), n Σ(, 10n 3, 16n 5). These theorems say that for any positive integer n, the Brieskorn homology spheres (i), (ii), (iii), and (iv) have E 8 -bounds and 4d(Y ) g 8 are arbitrarily large. Remark 1.1. As a conjecture, the inequalities in Theorem 1.3 would become the equalities actually. The evidence is due to Karakurt s program []. Similarly, we predict the following equalities for other Brieskorn homology spheres in Theorem 1.. d(σ(5, 35n, 50n 3)) = d(σ(5, 5n, 40n 3)) = 6n d(σ(3, 15n, 36n 5)) = d(σ(3, 9n, 4n 5)) = n d(σ(3, 1n 4, 36n 7)) = d(σ(3, 7n 4, 48n 7)) = n d(σ(4, 8n 3, 64n 7)) = d(σ(4, 3n 3, 76n 7)) = 4( n + n ) For negative n we can conjecture that the d-invariants are all. Acknowledgements This study was started by Christopher Scaduto s question in the Gauge Theory in Fukuoka in 018 February: Does Σ(, 5, 9) bound any 4-manifold with E 8 intersection form? The author is grateful for motivating to the calculating. His question is also answered by Scaduto and Golla in [1]. The author is grateful to Macro Golla for giving me many useful comments and advice for writing this article.. Notations and preliminaries.1. Plumbing diagram. We define a plumbing diagram (or graph) as explained in [9]. Let V be the set of vertices with weight function m : V Z and E the set of edges. Let Γ be a Z-weighted tree graph (V, E, m). If a Z-weighted graph (V, E, m) is a tree graph, then it is called plumbing graph. Let (V, E, m) be a plumbing diagram. We assign for v V the D -bundle over S with the Euler number m(v). Each edge {v, w} E means the plumbing process below between the two D -bundles over S. Plumbing is a surgery obtained by identifying two D -bundles over D along the inverse images over each disk embedded in S in such a way that one exchanges the roles of their sections and fibers. As a result the plumbing for any plumbing graph Γ gives a 4-manifold P (Γ) and we call P (Γ) a plumbed 4-manifold. The boundary P (Γ) of P (Γ) is called a plumbed 3-manifold. Here [v] is the class represented by the core sphere of the D -bundle corresponding to the vertex v. The intersection form (, ) : H (P (Γ)) H (P (Γ)) Z of P (Γ) is computed from the linear extension of the following definition. m(v) v = w ([v], [w]) = 1 v w and {v, w} E, 0 v w and {v, w} E.
4 4 MOTOO TANGE A graph with at most one vertex of degree larger than two is called a star-shaped graph. A plumbed 3-manifold with a star-shaped graph is called a Seifert manifold. For i = 1,,, n, let (α (1) i, α () i,, a (r i) i ) be a sequence of weights of vertices of the i-th branches of a star-shaped graph. We compute the continued fraction for the sequence as follows: () a i /b i = [α (i) 1, α(i),, α(i) r i ], where α (i) j is some integer and (a i, b i ) are coprime integers. Here the continued fraction is defined to be 1 [c 1, c,, c m ] = c 1 c 1. c m The Seifert manifold with the rational numbers a i /b i (i = 1,,, n) and the central weight e. We present such a Seifert manifold as S(e; (a 1, b 1 ), (a, b ),, (a n, b n )). We call these integers the Seifert invariant. Instead of (a i, b i ) we also present it as α (i) 1 α (i) α(i) r i. Here we present several consecutive integers by the power as follows: = [m]. Brieskorn homology sphere Σ(p, q, r) is defined to be {(z 1, z, z 3 ) C 3 z p 1 + zq + zr 3 = 0} S 5, where p, q, r are pairwise coprime positive integers. The manifold is a plumbed 3-manifold with a branch number three star-shaped graph. The Seifert invariant is S(e; (p, p ), (q, q ), (r, r )), where 0 < p < p, 0 < q < q, 0 < r < r and e (p /p+q /q+r /r) = ±1/pqr. In [6], the author showed that the intersection matrix of the 4-manifold of (minimal) plumbing diagram of Σ(, 3, 5) and Σ(3, 4, 7) are isomorphic to E 8. For example, the plumbing diagram of Σ(, 3, 5) is described as follows: S(, ( ) [4], ( ) [], )... Linear diagram. To construct a 4-manifold bounding a Brieskorn homology sphere, we introduce two new notations as below. Let us consider the following notation (3) L(a 1 a a n (k) b m b m 1 b 1 ), where the integers a i, b j and k are integers. The notation presents a 3- manifold designated by the following convention. The dot means the transversal intersection for two -spheres and (k) means the intersection with geometrically linking number k between the two components. The surgery description diagram of the plumbing is Figure 1. Here, the box with the k means the full k-twist. As an example, we consider a linear notation of Σ(, 3, 5). Sliding the first branch of S( ; ( ) [4], ( ) [], ) to
5 A NOTE ON E 8 -INTERSECTION FORMS AND CORRECTION TERMS 5 Figure 1. A surgery description of (3). the third branch, we have the following linear notation: S( ; ( ) [4], ( ) [], ) = L(( ) [] ( ) ( ) ( ) ( 4) ( ) [3] ) = L(( ) [4] ( ) ( 4) ( ) [3] ). The 4-manifold having the framed link as in Figure 1 is called 4-manifold having linear diagram (3)..3. Torus knot component linear diagram and n-twisting. Next, we introduce a torus knot component linear diagram. We introduce the following notation. Consider a linear diagram that the framing of the nearest component to (k) is zero. We call the 3-manifold Y. Then, we deform the linear diagram as follows: Y = L( q n 0 (k) p ) = L( q (k) p + nk (k,nk+1) ). The right equation presents the move of the diagram as in Figure. The component with the underline having index (k, nk+1) stands for the (k, nk+ 1)-torus knot with framing p + nk. The last surgery diagram gives a 4- manifold X bounding Y. The intersection form of X is isomorphic to the intersection form of the 4-manifold having the linear diagram as follows: L( q (k) (p + nk ) ). Thus, if the intersection form of the linear notation is isomorphic to E 8, then Y is a 3-manifold which bounds a 4-manifold with intersection form E An estimate of µ-invariant. Neumann-Siebenmann s µ-invariant in [3] is defined for any plumbed 3-manifold M = P (Γ). Here we set Γ = (V, E, m). In the case where P (Γ) is a homology sphere, we define w(γ) H (P (Γ), Z) as follows: (1) The class w(γ) is written by w(γ) = v V ϵ v[v] for ϵ v = 0 or 1. () For any v V we have (w(γ), [v]) ([v], [v]) mod. Let σ(γ) be the signature of the intersection form (, ) associated with Γ. Then we define the µ-invariant of M to be σ(γ) w(γ) µ(m) =. 8 The invariant µ can be extended to any rational plumbed spin 3-manifold (M, c) naturally as in [3].
6 6 MOTOO TANGE Theorem.1 ([11]). Suppose that a Seifert rational homology 3-sphere M with spin structure c bounds a negative-definite spin 4-manifold Y with spin structure c Y. Then b (Y ) 8 µ(m, c) mod 16 σ(γ) w(γ, c) µ(m, c) = 8 8 µ(m, c) b (Y ) 8 µ(m, c). 9 In particular, if a Seifert homology sphere M bounds a spin negativedefinite bound Y, then b (Y ) 8 µ(m) holds. Figure. A formula of n-twisting 3. The families of Brieskorn homology spheres in Theorem Proof of Theorem 1.. We prove Theorem1.. Proof. The Seifert invariants of Σ(, 14n 5, 8n 3), Σ(, 4n + 5, 14n + 3) Σ(, 6n + 5, 16n + 3), Σ(, 10n 3, 16n 5) Σ(5, 35n, 50n 3), Σ(5, 5n, 40n 3) Σ(3, 15n, 36n 5), Σ(3, 9n, 4n 5) Σ(, 14n 5, 8n 3), Σ(, 16n 5, 10n 3) Σ(3, 1n 4, 36n 7), Σ(3, 7n 4, 48n 7) Σ(4, 8n 3, 64n 7), Σ(4, 3n 3, 76n 7)
7 A NOTE ON E 8 -INTERSECTION FORMS AND CORRECTION TERMS 7 with reverse orientation are as follows: S(1; (p 1, 1), (p, q ), (p 3, q 3 ). p 1 (p, q ) (p 3, q 3 ) p 1 (p, q ) (p 3, q 3 ) (14n 5, 7n + 1) (8n 3, ) (14n + 3, 7n ) (4n + 5, 6) (6n + 5, 13n 4) (16n + 3, 4) (10n 3, 5n 4) (16n 5, 4) 5 (35n, 8n 3) (50n 3, ) 5 (40n 3, 3n 4) (5n, 1) 3 (15n, 10n 3) (36n 5, 4) 3 (4n 5, 16n 6) (9n, 1) 3 (1n 4, 14n 5) (36n 7, 4) 3 (48n 7, 3n 10) (7n 4, 3) 4 (8n 3, 1n 4) (64n 7, 4) 4 (76n 7, 57n 10) (3n 3, ) We deform the presentation as follows: S(1;, (n + 1) [6], ( 4n) ) = S(0,, ( ) n [6], ( 4n) ) = L( [6] n 0 () (4 4n) ) = L( [6] () 4 (,n+1) ) L( [6] () 4 ) S(1;, (n + 1) 4, ( 4n) [5] ) = S(0;, ( ) n 4, ( 4n) [5] ) = L( [5] ( 4n) () 0 n 4 ) = L( [5] (,n+1) () 4 ) L( [6] () 4 ) S(1;, (n + 1) 4 [3], ( 4n) [3] ) = S(0;, ( ) n 4 [3], ( 4n) [3] ) = L( [3] 4 n 0 () ( 4n) [3] ) = L( [3] 4 () (,n+1) [3] ) L( [3] 4 () [4] ) S(1;, (n + 1) [4], ( 4n) [3] ) = S(0;, ( ) n [4], ( 4n) [3] ) = L( [4] n 0 () (4 4n) [3] ) = L( [4] () 4 (,n+1) [3] ) L( [4] () 4 [3] ) S(0; 5, 5 n ( 7), ( 5n) ) = L(( 7) n 0 ( 5) ( 3 5n) ) = L(( 7) ( 5) 3 ( 5, 5n+1) ) = L( 1 ( 5) ( 5) 3 ( 5, 5n+1) ) Here we slide the 3-framed ( 5, 5n + 1)-torus knot component to the component with ( 5)-framed component. Then we have a 4-manifold with intersection form of the plumbed 4-manifold for S(1;, [], 5). By doing four blow-ups and one blow-down, we have the intersection E 8. In the same way, the 4-manifolds that the last diagrams in the following equalities present can be deformed into 4-manifolds with intersection form E 8. The results of the latter four equalities are a plumbed 4-manifold of type S(; [], [4], 4) which is the Seifert structure for Σ(3, 4, 7). S(0; 5, 5 n ( 8), ( 5n)) = L(( 3 5n) ( 5) 0 n ( 5) 1 [] ) = L( 3 ( 5, 5n+1) ( 5) ( 5) 1 [] ) S(0; 3, 3 n ( 5), ( 9n) [3] ) = L(( 5) n 0 ( 3) ( 1 9n) [3] ) = L( 1 ( 3) ( 3) ( 1) ( 3, 3n+1) [3] )
8 8 MOTOO TANGE S(0; 3, 3 n ( 8), ( 9n)) = L( [4] 1 ( 3) n 0 ( 3) ( 1 9n)) = L( [4] 1 ( 3) ( 3) ( 1) ( 3, 3n+1) ) S(1;, n ( 7), ( 4n) [] ) = S(0;, n ( 7), ( 4n) [] ) = L( [4] 1 ( ) n 0 ( ) ( 4n) [] ) = L( [4] 1 ( ) ( ) (, n+1) [] ) S(1;, n ( 6), ( 4n) [3] ) = S(0;, n ( 6), ( 4n) [3] ) = L( [3] 1 ( ) n 0 ( ) ( 4n) [3] ) = L( [3] 1 ( ) ( ) 0 (, n+1) [3] ) S(0; 3, 3 n ( 7), ( 9n) 4) = L( [3] 1 ( 3) n 0 ( 3) ( 1 9n) 4) = L( [3] 1 ( 3) ( 3) ( 1) ( 3, 3n+1) 4) S(0; 3, 3 n ( 5) 3, ( 9n) [] ) = L( [] ( 1 9n) ( 3) 0 n ( 3) 1 4) = L( [] ( 1) ( 3, 3n+1) ( 3) ( 3) 1 4) S(0; 4, 4 n ( 7), ( 16n) 4) = L(4 ( 16n) ( 4) 0 n ( 4) 1 [] ) = L(4 ( ) ( 4, 4n+1) ( 4) ( 4) 1 [] ) S(0; 4, 4 n ( 6) 3, ( 16n) ) = L( ( 16n) ( 4) 0 n ( 4) 1 4) = L( ( ) ( 4, 4n+1) ( 4) ( 4) 1 4) According to the definition of µ as above, computing the µ-invariants for these Brieskorn homology spheres, we obtain µ = 1 easily. From the description under Theorem.1 we obtain g 8 1. Namely, the homology spheres have all g 8 = Brieskorn homology spheres with E 8 -bound and arbitrarily large correction terms Heegaard Floer homology and one preparation. In [5] for any spin c rational homology sphere (Y, s) the Heegaard Floer homology HF + (Y, s) has the following exact sequence: 0 T + d(y,s) HF + (Y, s) HF red (Y, s) 0. T + s is isomorphic to T + := F[U, U 1 ]/U F[U] with the minimal degree s. HF red (Y, s) is a finite dimensional torsion F[U]-module. d(y, s) is called the
9 A NOTE ON E 8 -INTERSECTION FORMS AND CORRECTION TERMS 9 correction term of (Y, s). We call the submodule T + d(y,s) in HF + (Y, s) the T + -part of HF + (Y, s). Here we prepare a lemma to prove Theorem 1.3. We denote d(l(p, q), i) by d(p, q, i). Here we define L(p, q) to be the p/q-surgery of the unknot. The identification of spin c structures with Z/pZ is due to Fig. in [5]. Here the p-dehn surgery of a knot K in a homology sphere Y is the surgery (Y \ S 1 D ) V, where V = S 1 D. Here the attaching meridian of the new solid torus V is mapped to p [m] + [l] H 1 ( (S 1 D )) where m is the meridian of K and l is the homologically trivial longitude of K. We denote the p-dehn surgery of a knot in Σ by Σ(p). Lemma 4.1. Let Σ be a homology sphere and K Σ a knot. For some positive integer p, if Σ(p 1) is an L-space and Σ(p) is a lens space L(p, q), then the correction term d(σ) is computed as follows. (4) d(σ) = max {d (p, q, ki + c) d(p, 1, i) 0 i < p}, where k is the dual class of [ K] H 1 (L(p, q), Z) and c = (k +1+p)(k 1)/, where K is the surgery dual of the lens space surgery. Here the dual class, which is used on the same situation in [8], is defined to be an element representing the 1st homology class of the dual knot for the Dehn surgery Σ(p). In the case of lens space surgery, by using the genus one Heegaard decomposition, the class is identified with an integer modulo p. Proof. We use the following surgery exact sequence (Corollary 9.13 in [5]): HF + (Σ) HF + (Σ(p 1)) G+ HF + (Σ(p)) F + HF + (Σ). Since Σ(p 1) and Σ(p) are L-spaces and the corresponding map F on HF is surjective, F + is also surjective onto the T + -part in HF + (Σ). The map F + is induced from the cobordism Σ(p) to Σ obtained by attaching a 0-framed -handle along the meridian of K. The spin c structures on Σ(p) are identified with Z/pZ due to the description in p.13 in [5]. For any integer j with 0 j < p consider the surgery exact sequence in Theorem 9.19 in [4]: HF + (Σ) HF + (Σ(0), [j]) HF + (Σ(p), j) F + j HF + (Σ). F j + is a component of F + restricted to the spin c structure j. It is also a sum of homogeneous maps f i + with respect to the spin c cobordism from (Σ(p), j) to the unique spin c manifold on Σ. Namely, F j + is described by the sum F j + = j i mod p f i +. The degree shift of f i + is (4p (i p) )/(4p) due to [5]. The maximal degree shift among {f i + j i mod p} is (4p (j p) )/(4p) = d(p, 1, j). Since F j + is a surjective U-equivariant map, for 0 j < p we have d(σ) d(p, q, kj + c) d(p, 1, j).
10 10 MOTOO TANGE The 1 to 1 correspondence Z/pZ Spin c (L(p, q)) in Corollary 7.5 in [5] is described by ki + c. See [7]. Suppose that d(σ) > d(p, q, kj + c) d(p, 1, j) for any integer j with 0 j < p. Then any element with the minimal degree in HF + (Σ(p)) is included in the kernel of F + = 0 j<p F + j. Thus the kernel of F + includes at least p components. On the other hand, for a sufficient large number N, ker(f + )/(U N = 0) is (p 1)-times direct sum of T + from the exact sequence of the version of HF. Hence, this implies that in the image of G + there is a torsion F[U]-module by at least one component. However, since Σ(p 1) is an L-space, the image of G + does not have any torsion F[U]-module. This is a contradiction. Therefore for some j, d(σ) = d(p, q, kj + c) d(p, 1, j) holds. 4.. The d-invariants for the four families of Brieskorn homology spheres. We prove Theorem 1.3. Proof. The Seifert presentations of Brieskorn homology spheres from (i) to (iv) in Theorem 1. are the below: (i) S(1;, ( n + 1) 7, (4n 1) ) (ii) S(1;, ( n) ( 3), (4n + 1) 6) (iii) S(1;, ( n) ( 3) 4, (4n + 1) 4) (iv) S(1;, ( n + 1) 5, (4n 1) 4) Let Σ n be one of Brieskorn homology spheres parametrized by n in the list above. We do 0-surgery and +1-surgery of the homology sphere along the meridian of the singular fiber of multiplicity. We call the meridian K n. Note the coefficients 0 and 1 are the framing of the unknot K n in the diagram. The 0-surgery and 1-surgery give lens spaces L(r n, s n ) and L(p n, q n ). The results are the lens spaces in the list below. 0-surgery (r n, s n ) 1-surgery (p n, q n ) (i) (56n 41n + 7, 8n 7n + ) (56n 41n + 8, 8n 7n + 1) (ii) (168n + 71n + 7, 7n + 7n + 4) (168n + 71n + 8, 7n + 7n + 1) (iii) (08n + 79n + 7, 48n + 17n + ) (08n + 79n + 8, 48n + 17n + 1) (iv) (80n 49n + 7, 16n 13n + 4) (80n 49n + 8, 16n 13n + 1) These examples satisfy p n = r n + 1. As a result, the 0-surgery means a positive r n -Dehn surgery along K n. We set d n := d(σ n ). Here using Lemma 4.1, we compute the lower bound of d n. We argue the case of (i) only. Other cases are able to be proven by similar arguments. Let Σ n be the Brieskorn homology sphere in the type (i). We set p n = 56n 41n + 8, q n = 8n 7n + 1, k n = 14n 5 and c n 4n 9n + 4 mod p n. The k n is the dual class in the lens space L(p n, q n ) which presents K n. Here we set i = qn+1 n. Then modulo p n we have k n i + c n { 7n 5 n: odd n: even. 7n
11 A NOTE ON E 8 -INTERSECTION FORMS AND CORRECTION TERMS 11 If n is an odd number, then by using the reciprocity formula in [5], we have ( d L(p n, q n ), 7n 5 ) = 4n3 + 8n 95n + 5 4p n and d(l(p n, 1), i) = 5n 37n + 7 4p n Thus we have d(l(p n, q n ), k n i + c n ) d(l(p n, 1), i) = n + 1. If n is an even number, then we have ( d L(p n, q n ), 7n ) = 4n3 16n + 73n 8 4p n Thus we have d(l(p n, 1), i) = 5n 41n + 8 4p n. d(l(p n, q n ), k n i + c n ) d(l(p n, 1), i) = n. Therefore we have d n n. References [1] M. Golla and C. Scaduto, On definite lattices bounded by integer surgeries along knots with slice genus at most. arxiv: [] C. Karakurt, HFNem, (Computer program). [3] W. Neumann, An invariant of plumbed homology spheres, Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen, Siegen, 1979), pp , Lecture Notes in Math., 788, Springer, Berlin, [4] P. Ozsváth and Z. Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. () 159 (004), no. 3, [5] P. Ozsváth and Z. Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (003), no., [6] M. Tange, The E 8 -boundings of homology spheres and negative sphere classes in E(1), Topology Appl. 0 (016), [7] M. Tange, Ozsváth-Szabó s correction term and lens surgery, Math. Proc. Cambridge Philos. Soc. 146 (009), no. 1, [8] M. Tange, Homology spheres yielding lens lens spaces, Proceedings of 4th Gökova Geometry-Topology Conference pp [9] N. Saveliev, Invariants for homology 3-spheres, Encyclopaedia of Mathematical Sciences, 140. Low-Dimensional Topology, I. Springer-Verlag, Berlin, 00. xii+3 pp. [10] C. Scaduto, On the determination of definite lattices bounding 3-manifolds using Yang-Mills instanton Floer homology arxiv: [11] M. Ue. The Fukumoto-Furuta and the Ozsváth-Szabó invariants for spherical 3- manifolds, Algebraic topology old and new, , Banach Center Publ., 85, Polish Acad. Sci. Inst. Math., Warsaw, 009. Motoo Tange University of Tsukuba,
12 1 MOTOO TANGE Ibaraki , Japan. Institute of Mathematics, University of Tsukuba, Tennodai, Tsukuba, Ibaraki , Japan address:
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