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Dehn Surgery Along a Torus T2-Knot II by IWASE, Zjunici

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Dehn Surgery Along a Torus T2-Knot II by IWASE, Zjunici Japan. J. Math. Vol. 16, No. 2, 1990 Dehn surgery along a torus T2-knot II By IWASE, Zjunici (Received November 2, 1988) (Revised April 17, 1989) This paper is a continuation of "Dehn-surgery along a torus T2-knot" , in which the author defined and classified torus T2-knots in S4 and studied some properties of the 4-manifolds obtained by Dehn surgeries along them . In this paper we investigate the diffeomorphism types of such manifolds , which has been determined in some cases. •˜ 1. Introduction An embedded 2-torus in S4 is called a torus T2-knot if K is incompres sibly embedded in aN(U), where U is an embedded 2-torus which bounds a solid torus S1 X D2 in S4. This is an analogy of classical torus knots in S3 , Put (S4, K(p, q, 0))=((S3, k(p, q))\Int B3)XS1Uid 52XD2, (S4, K(p, q, q))=((S3, k(p, q))\Int B3)X53Ut52XD2, where k(p, q) means the classical torus knot of type (p , q) and r is an automorphism of 52 X S1(i.e. a diffeomorphism from 52 X 51 to itself) which is not isotopic to identity. In [3], we proved the following. PROPOSITION 1.1. Any torus T2-knot is equivalent to one and only one of the following: (i) K(p, q, 0), 1<p<q, gcd (p, q)=1; (ii) K(p, q, q), 1<p<q, gcd (p, q)=1; (iii) unknotted T2-knot(=K(1, 0, 0)). 0 A Dehn surgery is an operation of cutting off and gluing back the tubular neighborhood of a submanifold. In [3], we studied some properties of the 4-manifolds obtained by Dehn surgeries along torus T2-knots. The main theorem of [3] is the following. THEOREM 1.2. ([3], Theorem 1.3 (i)) Assume that a Dehn surgery of type 172 IWASE, ZJUNICI (a, j3, r) (for definition, see [3]) is performed along K(p, q, 0) or K(p, q,. q). If a;ipgj3-a0, then the manifold obtained is the total space of a good torus fibration over S2 with one twin singular fiber of multiplicity p and two multi ple tori of multiplicity q and a. 0 In this paper, we investigate the diffeomorphism types of the manifolds obtained by Dehn surgeries along torus T2-knots. A Dehn surgery of type (a, p, y) is called special if gcd (a, j3)=1. Our main theorem is the following. THEOREM 1.3 (Theorem 9.3). The difeomorphism types of the manifolds obtained by special Dehn surgeries along torus T2-knots with 1/p+1/q+1/a<_1 or ar=0 are classified by the fundamental groups, spinners, and the equi variant intersection forms on the universal covering spaces. In the appendix, we prove the following theorem. THEOREM A.2. Any orientable closed 4-manifold is obtained from a connected sum of some 51XS3's, CP2's and -CP2's by a Dehn surgery along some T2-link in it. D The author would like to express his thanks to Prof. Yukio Matsu moto for uncountable suggestions and encouragements. He also thanks Prof. M. Ue and Prof. K. Kuga. •˜ 2. Preliminaries For notations, see [3]. Let K be K(p, q, 0) (resp. K(p, q, q)). The manifold obtained by Dehn surgery of type (a, J9, r) along K is denoted by M(p, q, 0; a, p, r) (resp. M(p, q, q; a, /3, Y)) or M(K; a, j9, r). We assume that a is non-negative. For a 3-manifold Mo and a 3-ball D in M0, put and call them the untwisted spin, the twisted spin of M4 respectively, where r is an automorphism of S2XS1 which is not isotopic to identity. •˜ 3. Automorphism of the exterior of a torus T2-knot PROPOSITION 3.1. For K=K(p, q, 0) (resp. K(p, q, q)), an automorphism of 6N(K) can be extended to an automorphism of the exterior if and only if Dehn surgery alontg a toru,s T2-knot II 173 PROOF. Let U be an unknotted T2-knot . We may assume that K(p, q, 0) and K(p, q, q) are contained in N(U) . (l, r, s> is the basis of H1(aN(U); Z) as is in •˜2 in [3]. By Theorem 5 .3 in [4] and Lemma 2.6 in [2], an automor phism of aN(U) can be extended to an automorphism of S4 if and only if Ah is of the form (a+b+c+d is even). Therefore h fixes the homology class [p(rxs)+q(sxl)]EH2(aN(U); Z) up to sign and can be extended to an automorphism of S4 if and only if (a+b+c+d is even). This implies We may assume that the extension of h to S4 maps K(p , q, 0) and N(K(p, q, 0)) to themselves. Since 12=s and h*(m)=•}m , we have h(pgm +l1)=h(-pl+qr)=+pl+q(•}r+2c's)=•}(pqm+lj+2c'gl2 and Since K(p, q, 0)=K(q, p, 0), the automorphism h'4, with also maps K (p, q, o) to itself and cann be extended to S4. Since gcd (p , q)=1, an automorphism c of the exterior of K(p, q, 0) with 174 IWASE, ZJUNICI is constructed. On the other hand, h fixes the homology class [p(rxs)+q(sxl)+q(lXr)J eH2(aN(U); Z) up to sign and can be extended to an automorphism of 54 if and only if (a+b+c+d is even). Therefore d-c=1 and -b+a=1. Substitute them into ad-bc=e_±1 and we have (b+1)(c+1)-be=e_±1 and we obtain b+c=0 if e=1 and b+c+2=0 if e=-1. Therefore we have Since 12=s-r, if=1, we have h,~(pgm+l1)=h*(-pl+qr)=-pl+q((b+ 1)r-bs)=-pl+qr+bq(r-s)=(pqm+l1)-bql2 and h*(l2)=12. If e=-1, we have h*(pqm+l1)=h*(-pl+qr)=pl+q(ar+(-a-1)s)=pl-qr+(a+1)q(r s)=-(pqm+l1)-(a+1)q12 and h*(l2)=12. By an argument similar to the case of K(p, q, 0), we can construct an automorphism ~ of the exterior of K(p, q, q) with Let be a Seifert surface of k(p, q). For K=K(p, q, 0) or K(p, q, q), X 51 is properly embedded in the exterior of K. Identify the tubular neighborhood of. ~'XS1 with xS1x[-1, 1] and let (x, 0, t) be its coordinates. Define an automorphism fr of the exterior of K by (x, 0, t)(x, 0+t, t) in ~' X six [-1, 1] and id otherwise. Since a~=h, we have We have proved the "if" part. Denote the exterior of K by E. 12 generates Ker (r1(dN(K))ir1E) and <l1, l2> generates Ker (H1(dN(K); Z)H1(E; Z)). This implies that 12-, 13-, and 23-components must be zero. Assume that an h with h*(m)=•}m+al1+b12 is extended to the exterior. Then, there is a diffeomorphism between S4 and M=M(K; •}1, a, b). The fundamental group of M is isomorphic to the one of the 3-manifold obtained Dehn surgery along a toms T2-knot TI 175 by a Dehn surgery of type (•}1, a) along k(p, q). Since the torus knot k(p, q) has the Property P, a must be zero. The 32-component is even in the case of K(p, q, 0) by Lemma 3.4 in [3]. (Note that Y(11)=Y(12)=0, Y(4+12)=1.) a COROLLARY 3.2. If rr' (mod gcd (a, 2j9)), then M(p, q, 0; a, j3, r) and M(p, q, 0; a, j3, r') are difeomorphic. If r-r' (mod gcd (a, j3)), then M(p, q, q; a, j9, r) and M(p, q, q; a, ,9, r') are difeomorphic. PROOF. Assume that K=K(p, q, 0). There are integers x and y with r'r ax+2 jay. A diffeomorphism h: aN(K)dN(K) with extends to an automorphism h of the exterior. Since h*(am+p11+r12)= M am+P11+r'l2, h can be extended to a diffeomorphism required. The case K=K(p, q, q) can be treated similarly. 0 •˜ 4. Good torus fibration structures In [3], we described good torus fibration structures of the manifolds obtained by Dehn surgeries along torus T2-knots. In this section, we inves tigate them more closely. The singular fibers we encounter are twin singular fibers and multiple tori, which have trivial monodromies. Therefore our manifolds are obtained from (3 punctured sphere) X T2 and one Tw and two sT4's glued along the boundaries. We want to know the gluing maps. Recall that a torus T2-knot K=K(p, q, 0) can be regarded to be con tained in aN(U), where. U is an unknotted T2-knot. Assume that a tubular neighborhood aN(U)XI of aN(U) contains N(K). The closure of S4\aN(U) XI consists of Tw and sT4. Assume that aN(U)x{0}=a(Tw) and 6N(U)X {1}=a(sT4). We have already chosen a basis <1, r, s> of H1(aN(U); Z). We choose another basis as follows.
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