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 (ii) K(p, q, 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 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. Put v=-pl+qr, w=s, then a general fiber is vX w. Choose two integers p and q with pp+qq=1 and put u=ql+ pr. Since u•~v•~w=l•~r•~s, (u, v, w) is a basis which determines the same orientation as <1, r, s>. aN(U)•~{1} in the natural way. Taking the orientation into consideration, put ua=-u in aN(U)•~{0}, u1=u in aN(U)•~{1}. We shall mainly use the new bases I\Int N(K), because u0+u1 bounds an annulus in aN(U)•~I which meets each fiber exactly in one point transversely and u2 can be taken on the annulus. Note that u2=-m. We have chosen bases of the three connected components of aN(U)•~ I\Int N(K), which is diffeomorphic to Q•~T2, where Q means a 3-punctured sphere. The boundary of Q is u0, u1 and u2 and The projection map of the good torus fibration structure is the projection to the first factor Q. First, consider the gluing map between Q•~T2 and Tw. Recall that we have chosen a basis <1, r, s> for a(Tw). By definition, holds. Therefore we have Let sT42 denote the sT4 which is cut off and pasted back in the Dehn surgery, and sTl denote another sT4. Next, we consider the gluing map between Q•~T2 and s T. Recall that we have chosen a basis <1, r, s> for a(sT4) and S4 is obtained from Tw and sT4 by a gluing map with l=1, r=r, s=s. Here we assume that the gluing map is 1=-l, r=r, s=s so that it is orientation reversing. Then we have Finally, we consider the gluing map between Q•~T2 and sT2. Recall that the meridian of sT2 is mapped to am+/3ll+112. From m=-u2, pqm+h =-pl+gr=v and 12=8=w, we have am+j9h+r12=-au2+~(v+pqu2)+rw =(pq~-a)u2+Pv+Yw. In the case of K=K(p, q, q), we modify as follows. Recall that a general fiber represents (-pl+qr)•~(s-r). Put u=ql+pr, v=-pl+qr and Dehn surgery along a torus Ta-knot II 177 w=s-r. Then we have The meridian of sT2 is mapped to am+ph+r12=-au2+j3(v+pqu2)+rw= (pgQ-a)u2+j9v+rw. Hence we have the following. PROPOSITION4.1. M(p, q, 0; a, j9,r) and M(p, q, q; a, 9, r) are obtained as follows. Let Q be a 3 -punctured sphere and its boundaries be u0, u1, u2. Denote a basis of H,(T2; Z) by For K(p, q, q), glue them by 178 IWASE, ZJUNICI 0 Note that the proposition above still holds in the case of a=(pgj9-a =0. LEMMA 4.2. Let f: TwD2 be a good torus fibration which has only one multiple twin of multiplicity m as singular fibers. Then, we can modify f into f' without changing the fibration of a(Tw) so that f' has one multiple torus of multiplicity m and one simple twin singular fibers as singular fibers. PROOF. Consider S1•~D2•~S1. Define three loops 1, r, s in the bound ary by S'•~{*}•~{*}, {*}•~6D2•~{*}, {*}•~{*}•~S1. We can construct a good torus fibration cp: S1•~D2•~S'D2 with lkr, rmr, s0 where k is an integer with gcd (k, m)=1 and r is a generator of H,(aD2; Z) (see Definition 3.16, Remark 3.17 in [2]). We may assume that ~o(x, y, z)_~o(x, y, z') for any x, y, z, z' and that the singular fiber is S'•~{0}•~S1. A surgery on {*}•~{0}•~S1 yields Tw. If the framing is trivial, then 1, r, s taken above are 1, r, s for Tw. If it is non-trivial, then 1, r, r+s are 1, r, s for Tw. In any case, by Proposition 3.12 in [2], we can construct a good torus fibration TwD2 which has one multiple twin singular fiber of multiplicity m as singular fibers without changing the fibration of the boundary. Fact in the proof of Proposition 3.12 in [2] shows that a given f: TwD2 is of this form by composing some automorphism of Tw. Here we return to the good torus fibration S1•~D2•~S1D2 above. We have obtained Tw by a surgery, on {*}•~{0}•~S1. We can move this loop by an isotopy into a general fiber. The fibered neighborhood of the general fiber is diffeomorphic to S1•~D2•~S1 and we may assume that the loop is {*}•~{0}•~S1 in this neighborhood. After surgery, this part is a twin. Since we can apply the argument above with m=1, this twin has a good torus fibration structure whose singular fiber is a simple twin singular fiber. Therefore the whole manifold has the good torus fibration structure of one simple twin singular fiber and one multiple torus of multiplicity m. Since Dehn surgery along a torus T'-knot II 179 the loop is moved by an isotopy, the diffeomorphism type of the whole mani fold is not changed. The gluing map between the tubular neighborhood of the multiple twin singular fiber (i.e. a twin) and Q•~T2 is given by Proposition 4.1. By Lem ma 4.2, decompose the twin into (3-punctured sphere)•~T2 and Tw and sT4, whose core is the multiple torus and which is denoted by sTo. Let us investigate the gluing map between them. The union of (3 punctured sphere)•~T2 and Q•~T2 is Q'•~T2 where Q' means a 4-punctured sphere. Denote the boundary components of Q' which are not contained in Q by uoA, uoB and assume that uoA•~T2 is glued to Tw, uoB•~T2 is glued to sTo. The automorphism of Tw in the proof of Lemma 4.2 is id in our case. Therefore we may assume that we have Tw by a surgery along a loop parallel to s in sT4 whose meridian is r. Therefore the meridian to of sTo is glued to r=-puoB+qv. In the case of K(p, q, 0), r, s of the new twin are glued to a cross section and s. Therefore we have r=uoA+*v+*'w, s=w. In the case of K(p, q, q), we have r=uoA+*v+*'w, s=(uoA+*v+*'w)+w. Therefore we have the following. PROPOSITION 4.3. M(p, q, 0; a, j3, r) and M(p, q, q; a, ,9, r) are obtained as follows. Let Q' be a 4-punctured sphere and its boundaries be uoA, uoB, u1, u2. Denote a basis of Hl(T2; Z) by r=uoA+*v+*'w, s=w for K(p, q, 0), (uoA+*v+*'w)+w for K(p, q, q), l0=-pu0B+qv, l1=-qua+pv, where *, *' are integers. The structure of the base orbifold V of the good torus fibration is as follows. PROPOSITION4.4. If a=JpqRa~0, the orbi f olds o f the good torus 180 IWASE, ZJUNICI fibration structure of M(p, q, 0; a, j3, Y) and M(p, q, q; a, j3, 1) are the 2-sphere with three cone points, whose cone angles are 2ir/p, 2ir/q, and 27r/6. •˜ 5. Covering spaces In this section, we investigate some covering spaces of our manifolds. If c=ppqj3-a~~0, then M=M(p, q, 0; a, /3, Y) or M(p, q, q; a, j3, r) has the structure of a good torus fibration M-~(9. Let C~--~( be the universal orbifold of (9. PROPOSITION5.1. Let M-~(9 be tht pull-back of M-~(9 by £-&. (i) MM is a covering space. (ii) MC9 is a good torus fibration. If C9is elliptic, Euclidean, hyper bolic, then C9is S2 (2-dimensional spherical space), E2 (2-dimensional Euclid ean space), H2 (2-dimensional hyperbolic space) respectively. Singular fibers are simple twin singular fibers. (iii) Let Q' denote (9 with the interior of the tubular neighborhoods of the singular points of (9 (including the image of twin singular fibers) removed. N Let Q' be the inverse image of Q' by U &. The components of dQ' corres ponding to ui are denoted by ui,1 (AE A). Denote a basis of H1(T2; Z) by (v, w). l 1,2=-u1,+pv, yields M. PROOF. (i) It is clear that MM is a covering space if the neighbor hood of the multiple fibers of M is removed. In a neighborhood of each multiple fiber, the fibration D2•~S1•~S1D2 decomposes as p' 7r by composing an automorphism of D2•~S1•~S1, where p': D2•~S1-D2 is a Seifert fibration and 7r: (D2•~S1)•~S1D2•~S1 is the projection to the first coordinate. The problem is reduced to the case of the Seifert spaces, which can be proved by drawing pictures. (ii), (iii) Note that u,, 2-~puo, u,,1-~qu1, u2,1-J pqjSa~u2. Propositions 4.3, 4.4 completes the proof. 0 Dehn surgery along a torus T2-knot II 181 PROPOSITION 5.2. Let K be K(p, q, 0) or K(p, q, q). If a'=epq/3-af0 and the base orbifold is Euclidean or hyperbolic, then the universal covering spaces of M=M(K; a, p, r) and M'=M(K; a, j3, r') are di f eomorphic and their equivariant intersection forms are equivalent. PROOF. The universal covering space M of M in Proposition 5.1 is the universal covering space of M. Similarly we construct M' and M'. In Proposition 5.1, we can choose another u , 's (i=OA, OE, 1, 2) so that r2= uOA,2, lo,z=-uOB,2, ll, _-u1,2, l2 ,=sgn (pqp-a)u2,a holds. Therefore M, M' are obtained by surgeries along {*2}•~f*}•~S1 in R2•~T2=R2•~S1•~S1. Therefore M and M' are obtained by surgeries along {*2}•~{n}•~S1 (n e Z) in R2•~R1•~S1 with the same framing. Therefore they are diffeomorphic to each other. The difference between M and M' is the gluing of sT2, the inverse image of sT2. Decompose M as (M\Int sT2) U sT2 and calculate H2(M; Z) by the Mayer-Vietoris exact sequence. No elements are added to the second homology by adding sT2 to M\Int sT2. And the elements killed are all the elements of H2(a(M\Int sT2); Z), because sT2 is diffeomorphic to D2•~ S1•~R. Therefore we have H2(M; Z)=H2((1V1\Int sT42); Z)/H2(a(M\Int sT2); Z). The same holds for M'. Therefore the lift M\Int sT2M'\Int sT2 of the identity map between the exteriors of the knots induces the required map H2(M; Z) H2(M'; Z). o Next, consider the universal abelian covering spaces M of M, where M is M(K; a, j9, r) and K is K(p, q, 0) or K(p, q, q). M is the covering space corresponding to the subgroup [ir1M, ,r1M] of ic1M. Therefore the covering transformation group is isomorphic to rc1M/[2r1M, 7r1M]Z/a. Assume that a>1. Then, MM is a finite covering. Since H1(M; Z) is generated by the meridian m, M consists of X and sT2 where X is the a-fold branched covering space of S4 branched along K with the inverse image of Int sT2 removed and sT2 is the inverse image of sT2. Since aX is connected, sT2 is diffeomorphic to sT4. Let' be a Seifert surface of k(p, q). Then ~'•~S1 is contained in S4 with a(~ •~S1)=K. X is obtained from a copies of X cut along X S1 and pasted cyclically. LEMMA 5.3. The N -fold cyclic branched covering space of S4 branched along K(p, q, 0) (resp. K(p, q, q)) is diffeomorphic to the connected sum of the untwisted (resp. twisted) spin of the N-fold cyclic branched covering space Y of S3 branched along k(p, q) and (N-1)S2•~S2, PROOF. The argument above shows that the manifold obtained is Y•~S1 with N loops {*4}•~S1 surgered. All the framings of the surgeries are 182 IWASE, ZJUNICI trivial (resp. non-trivial). {*i}•~S1 (i=2, 3,•c, N) are contractible in x(Y•~S1, {*1}•~S1). Since all the framings are the same, surgeries yield S2•~S2's, not S2•~S2's. PROPOSITION5.4. Let K be K(p, q, 0) or K(p, q, q). If a*0, then the universal abelian covering spaces of M=M(K; a, j9,r) and M'=M(K; a, j9, r') are difeomorphic. Moreover,their equivariant intersection forms are iso morphic. PROOF. Lift m, 11,12 in aX to ax. m is lifted to a loop m. Since l~ (i=1, 2) is lifted to a loops, denote one of them by lz. Cm,11, l2> is a basis of H1(aX;Z). The meridian of sT2 is glued to am+j3ll+rl2. Since the merid ian of sTti is the inverse image of the meridian of sT2, it is glued to m+/3l, +rl2. Recall that we constructed a diffeomorphism fr: XX in the proof of Proposition 3.1. Similarly, we can construct a diffeomorphism w: XX with We have proved the first half. Apply the Mayer-Vietoris exact sequence to M=XUsT and calculate H2(II; Z). Then we have H2(M; Z) w H2(X; Z)/H2(dX; Z). The same argu ment holds for M'. An argument similar to the one in the proof of Proposi tion 5.2 completes the proof. E The d-fold cyclic covering space Zd of the exterior of k(p, q) has the relation Zd+pq=Zd (see for example p. 85 in [1]). Therefore the d-fold cyclic covering space Xd of the exterior of K(p, q, 0) or K(p, q, q) has the relation Xd+pq=Xd#pq(S2•~S2). •˜ 6. Intersection forms and linking forms Let M be M(p, q, 0; a, j3, r) or M(p, q, q; a, j3, r) and consider the inter section forms on their homology groups. If a•‚0, then H0(M; Z) H4(M; Z)~Z, H,(M; Z)~H2(M; Z) Z/a, H3(M; Z)=0. Consider the intersection forms on H*(M; Z/a). Note that H0(M; Z/a) ~'Hi(M; Z/a) H3(M; Z/a)=H4(M; Z/a)Z/a and H2(M; Z/a)=Z/a®Z/a. Let ,8, r be elements in Z/a with p~+rY=1 and put=7(mXlJ+/3(l2Xm). And let , be the union of a mod a 2-chain in X bounded by pl1+rl2 and the Dehn surgery along a torus T2-knot II 183 meridian disk DMd of sT4. Then, <, ~) is a generator of H2(M; Z/a). The boundary of 2•~S1 is 11•~12, which bounds a mod a 3-chain in sT4. the union of 2 X S1 and the 3-chain, is a generator of H3(M; Z/a), The intersection matrix on H2(M; Z/a)•~H2(M; Z/a) is of the form 0 c c since .=0. If a is even, it is equivalent to 0 1 or 0 1. Since CC,r~) is a generator of H2(M; Z/2) too, it contains no more information than the spinness.Ifa is odd, itis equivalent to[ 0 1 j. Next, we compute 8.8. If a=0, recall that 8 (1X=2•~S1. We can move this slightly into 2'•~51 so that 2•~51 n2'•~S1=0. Since a(2•~S1) and a(2'•~S1) are parallel 2-tori in aX, they bound disjoint parallel solid tori in sT4. This implies 8.8=0. Assume that a~0. Then there exists a basis such that Md is the meridian and a(2•~S1)=a(2'•~S1)=(aL,+xMd)•~L2 where x is an integer. Figure 1 shows T2=Md•~L1. The solid line shows a(.'•~51) and the dotted line shows a(2'•~S1). Figure 2 shows aL, and xMd. Let a(sT4)•~I be the collar of sT4. We may assume a(sT4)•~{0} is (Figure 1)•~ S1 and a(sT4)•~{1} is (Figure 2)•~S1. Then 8.8 is the intersection of the solid and dotted lines during the deformation between them. Since the intersection is 2-dimensional, it can be measured by the intersection with the meridian disk. Then, the problem is to see how many times the solid and dotted lines meet in the loop Md during the deformation. Let us call a connected component of T2\a(2•~51) U a(2'•~S1) a band. In Figure 3, the band intersects Md a-times. Watch a neighborhood of Md during the defor mation, then a pieces of the band move to overlap each other. While the second piece from right moves to overlap the first one, the solid and dotted lines intersect once in Md. While the third one moves to overlap them, they intersect twice. While the n-th one moves they intersect (n-1) times. Therefore 8.8 is 1+2+...+(a-1)=a(a-1)/2 times the generator. If a is Figure 1 Figure 2 184 IWASE, ZJUNICI Figure 3 even and non-zero, then a(a-1)/2-a/2 (mod a), therefore S• S=(a/2)(L1•~L2). Since =L1•~L2, we have v• v=(a/2)C. If a is odd, then since a(a-1)/2.0 (mod a), 8.8=0. Since S f is contained in X and the map Hl(Sn~; Z/a)-+H1(X; Z/a) induced by the inclusion map is the 0-map, S• C=0. To calculate S• i, note that (S• S=• (5.5). Therefore if a=0 then S•=0 and if a is even and non-zero, then S• 1,1 0 and 2(5• )=0. Note that if Cc','2;> is a basis such that the intersection matrix with this basis is 0 1 or 0 .1, then (a/2)C=(a/2)C' holds. If a is odd then S• r is 0. 1 0 1 1 We have proved the following. THEOREM 6.1. The intersection forms on the integral homology groups of M=M(p, q, 0; a, /9, r) or M(p, q, q; a, /3, r) are trivial if a•‚0. If a=0, they are trivial except for H2(M; Z) X H2(M; Z)H0(M; Z), which is equivalent to 0 1 0 if M is spin, O l if M is not spin. Q 1 1 1 THEOREM 6.2. Let M be M(p, q, 0; a, /3, r) or M(p, q, q; a, j9, r). Then for some bases u'i=(a/2)e if a is even, 0 if a is odd, ~.'`1, 1•'2=0 if M is spin, l if M is not spin. Q Next, let us calculate the linking forms. We restrict ourselves to the case a•‚0, because if a=0, then the homology groups have no torsion. H1(M; Z)~Z/a is generated by the meridian m. am is homologous to am+pll+r12 in the exterior X, which bounds the meridian disk DM of sT4. On the other hand, H2(M; Z)='Z/a is generated by the core T2 X{0} of sT4= T2•~D2. The linking number between the bases chosen above is 1/a. Dehn surgery along a torus T2-knot II 185 Since the linking forms in the other dimensions are trivial, we have the following. THEOREM 6.3. Let M be M(p, q, 0; a, 3, r) or M(p, q, q; a, p, r). Then the linking forms are trivial if a=0, trivial except for 1 and 2-dimensional if a.~0. The linking number of some bases of H1(M; Z) and H2(M; Z) is 1/a. U •˜ 7. Seifert invariants PROPOSITION 7.1. Let M be M(p, q, 0; a, /3, r) or M(p, q, q; a, /3, r). Then ir1M is representated by [u2, v]=1(i=0, 1, 2)>. PROOF. Use Proposition 4.1 and note that w=1. 0 ir1M is similar to the one of a Seifert fibered manifold but pgj3-a and 8 are not always coprime. In this chapter we assume that a=lpqj3aI~O and that the base orbi fold is Euclidean or hyperbolic, i.e. 1/p+1/q+1/o'<1. In such cases, the Seifert invariants are recovered from their funda mental groups. See for example, pp. 90-98 in [6]. We shall extend the proof to our case. LEMMA 7.2 (Lemma 1, p. 92 in [6]). The subgroup generated by v is the unique maximal cyclic normal subgroup of G and v has infinite order. PROOF. For the first half, see [6]. The second half is proved by con sidering the covering space MM in Proposition 5.1. [] In this case, 7r1M/center is infinite. On the other hand, if the base orbifold is elliptic, then Changing the notation slightly, we investigate the group (ad, /3j)'s and b above are called the normalized Seifert invariants. THEOREM 7.3. Let M, M' be the manifolds obtained by Dehn surgeries along torus T2-knots with a~0 and Euclidian or hyperbolic base orbifolds. If ,r1M~rc1M', then the normalized Seifert invariants of M and M' coincide. PROOF. By the argument similar to the one in [6], we have j~=p5/3'+ pA~az and A+pb-ob'=0, where p, 5E{-1,1}, A, AlEZ and A=A1+22+23. case (a) If p5=1, then a1>~~>0 implies Al=0, /32=/3?, b=b'. 186 IWASE, ZJUNICI case (b) If po=-1, then we have jZ+jai=pA1a~, which implies j~+9=0 or ai. case (b-1) If /3~+j2=0, then pA1=0, /3~=/2=0. Replace /z by ai then this case reduces to the case below. case (b-2) If /31+j2=a~, then pA~=1, j9=ai-j9z, we have oA1=-1, b'=-b+3. Replace (a1, j9) by (a1, a1 /91). 0 •˜ 8. The case a~0 with the Euclidean or hyperbolic base orbifold In this section, we try to classify the manifolds obtained by Dehn surgeries along torus fit-knots, we assume that a~0 and the base orbifold is Euclidean or hyperbolic throughout this section. Then, by Theorem 7.3, we can recover the normalized Seifert invariants (a1, (a2, /92), (a3, /93) and b. Choose two of the (a1, /91)'s, say (a1, j9), (a2, P2). Assume that a1=p and a2=q. If one of (a1, /31), (a2, /32), (a1, a2) is not coprime, then the assumption is false. Otherwise, go to the next step. Let po, qo is a solution of pp+qq=1. Then any solution is represented by p=i+kq, q=q0-kp where k i s an integer. On the other hand, /~ can be replaced only by/91 +1a1, where l is an integer. Therefore, /3°-i(mod p), /32=-po (mod q) must hold. If not, the assumption is false. Otherwise, go to the next step. Replace the Seifert invariants (a1, j1), (a2, /92) so that a1/32+a2/31=-1, b= 0. By Proposition 7.1, the (a3, 93) after this replacement becomes (Jpq/3-aJ, /3 sgn (pq/3a)). Therefore, (a, j9)=(~pgj93-a3J, /33sgn (pgj33-a3)). EXAMPLE 8.1. Assume that we obtained Seifert invariants (3, 1), (8, -3), (5, 1), b=0. Then, {p, q} must be {3, 8}, a=19, /3=1. EXAMPLE 8.2. Assume that we obtained Seifert invariants (2, -1), (3, 1), (8, -3), b=0. Then, {p, q}={2, 3}, a=26, j9=3, or {p, q}={3, 8}, a=26, j3=1 must hold. EXAMPLE 8.3. In Example 8.1 there is only one possibility, while in Example 8.2 there are two. Let us construct examples with three possibili ties. Assume that a1 holds. Therefore we have Dehn surgery along a torus T3-knot II 187 Since we have -l1-l2=1. Thus we have From these, we obtain Therefore, a3I(3a1a2a3-e3a3+e1a1-e2a2) holds . Since 0 EXAMPLE. Put k=5, e=1, then we have Seifert invariants (2, 1), (5, 2), (7 , 4), b=0. Assume that (p, q)=(2, 5). Then, p=3, q=-1 . Replace the invariants by (2, 1), (5, -3), (7,11). Then, we have a=J2X5X11-7J=103 , /3=11.A ssume that (p, q)=(5, 7). Then, p=3, q=-2 . Replace the invariants by (2, 3), (5, 2), (7, -3). Then, we have a=J5X7X32J=103 , /3=3.A ssume that (p, q)=(7, 2). Then, p=1 , q=-3. Replace the invariants by (2, -1), (5, -2), (7, -4). Replace them by (2, 1), (5, 7), 7, 3). Then we have a=17X2X(-7)-5J=103, /3=7. Therefore we have three possibilities; 188 TWASE, ZJUNICI Note that the argument used here is essentially for 3-dimensional Seifert manifolds. In the rest of this section, we restrict ourselves to special Dehn sur geries. Assume that {p, q} is known. Since a and are known, all we want to know is the knot;type and r. Note that gcd (a, 2j3)=2 if a is even, 1 if a is odd and apply Corollary 3.2, and it is reduced to one of the following. Here we quote the following theorem. THEOREM8.4. ([7], Theorem 3.1) Let Mo be a closed aspherical 3-mani fold. Then, the equivariant intersection forms on the universal covering N spaces o f Spin (M) and Spin (Mg) are not isomorphic. Therefore Spin (M) and Spin (M) are not diffeomophic. El PROPOSITION8.5. ([3], Proposition 3.9) M(p, q, 0; a, p,0) is diffeomor phic to Spin (Mg) and M(p, q, q ; a, j3, 0) is di feomorphic to Spin (M), where M4 is the manifold obtained by Dehn surgery of type (a, i) along k(p, q). L In our case, since Mo in Proposition 8.5 is aspherical, we can apply Theorem 8.4. Therefore (i) (resp. (iv)) in the table above is distinguished from (iii) (resp. (v)). (i) and (ii) are distinguished by spinness. In the argument above, we assumed that {p, q} is known. Now let us investigate the cases in which {p, q} has more than one possibilities. If M, the manifold given, is spin, then we may assume that r=0, and therefore, it is the untwisted or twisted spin of a Seifert 3-manifold. Hence any of the possible {p, q}'s yields the same manifold. Assume that M is not spin. {p, q} has two possibilities, say {a1, a2}, {al, a3}. If al is even, then a3=1ala2f-a is also even and it is a contradic tion. Therefore al is odd. First, assume that p=al, q=a2. Recall that p, q are integers with pi+qq=1. We may assume that q is even. Since ~3 is odd, M(p, q, 0; a, p, l) is diffeomorphic to M(p, q, 0; a, j3, j9) by Corollary 3.2. Then, by Proposition 4.1, M is obtained from Q•~T2, Tw and two sT4's Dehn surgery along a torus T2-knot II 189 by the gluing map r=-puo+qv s=w l1=-qu1+pv Replace (v, w> by (v', w'>, where v'=v+w, w'=w, r'=-puo+qv' s'=w' l1=-qu1+pv'-pw' Replacing u0, u2 by uo=uo+*v', uo=u2+*'v' where *, *' are some integers, we may assume that the gluing map is r'=-puo+qv' s'=w' l2=-qu2+pv, with pp'+q'q'=1. Assume that l1=a2ui+82v'+r'w'. Then, by Proposition 4.1, M is diffeomorphic to M(p, q', 0; a, j3', r') where a=pq'j32-a2!, P'= j2sgn (pq'j32-a2). Since M is not spin, r' is odd. Therefore M is diffeomor phic to M(p, q', 0; a, jS',1). We have proved the following. THEOREM8.6. Assume that a manifold obtained by a special Dehn surgery along a torus T2-knot is given and assume that o~0 and that the base orbifold is Euclidean or hyperbolic. Then, we can recover the normalized Seifert invariants from the fundamental group. Moreover, we can recover the knot type, the type of the surgery from spinness and the equivariant intersec tion form on the universal covering. 0 EXAMPLE8.7. Reconsider the example 8.1. The manifold is M(3, 8, 0; 43, 2, 0) or M(3, 8, 8; 43, 2, 0). They are not diffeomorphic. EXAMPLE8.8. Reconsider the example 8.2. The manifold is M(2, 3, 0; 26, 3, 0)~M(3, 8, 0; 26, 1, 0) or M(2, 3, 0; 26, 3,1)=' M(3, 8, 0; 26,1,1) or M(2, 3, 3; 26, 3, 0) M(3, 8, 8; 26, 1, 0). They are not diffeomorphic. EXAMPLE8.9. Reconsider the example 8.3. The manifold is M(2, 5, 0; 190 IWASE, ZJUNICI 103, 11, 0)M(5, 7, 0;103, 3, 0)~M(2, 7, 0;103, 7, 0) or M(2, 5, 5;103,11, 0) M(5, 7, 7; 103, 3, 0)M(2, 7, 7; 103, 7, 0). They are not diffeomorphic. •˜ 9. The case a=0 In this section, we treat the case a=0. Note that which has trivial center. Therefore 7r1M/center is Z/p*Z/q. The following Lemma can be proved straightforwardly. LEMMA 9.1. If cO, then 2r1M/center is not a free product of non-trivial finite cyclic groups. D If a=O, then we know p and q from ,r1M. Since a is known from H1(M; Z)=Z/a and pq/3aI=0,,8=a/pq is known. If we restrict ourselves to special Dehn surgeries, the type of the surgery is (pq,1, r). By Corollary 3.2, our manifold is K(p, q, 0; pq,1, 0) or K(p, q, 0; pq,1, l) or K(p, q, q; pq,1, 0). If pq is odd, then the second one is diffeomorphic to the first one. By Corollary 3.10 in [3], the first and the third one is Lp#Lq. By Prop osition 4.1, the second one is obtained from Q•~T2, Tw and two sT4's by the gluing map r=-puo+qv s=w l1=-qu1+pv 12=v+w. We assume that p is even and q is odd. Replace v by v'=v+qw, uo by uo= uo-w, u1 by u1=u1+w, then we have r=-puo+qv'-w s=w l1=-qui+pv' 12=v'-(q-1)w. This is the manifold M(p, q, 0; pq, 1, l q) with a twin removed and reglued. Note that M(p, q, 0; pq,1, 1 q) is diffeomorphic to M(p, q, 0; pq,1, 0) and the diffeomorphism maps this twin to itself. Now let us investigate how this twin is contained in M=M(p, q, 0; pq,1, 0). M is an untwisted spin of m, the manifold obtained by Dehn surgery of type (pq, l) along k(p, q). Recall that m is L(p, q) L(q, p) ([5]) and it is proved as follows. k(p, q) is Dehn surgery along a torus T2-knot II 191 contained in sTl (1 sT2 where we regard S3 as sT2UsT2. (sTinsT2)n aN(k(p, q)) consists of two circles, which are meridian circles in sT3 reglued. They bound disjoint meridian disks in sT3 and if we cut sT3 along the disks we obtain two 3-balls. Pasting them to each of sT i's, we obtain L(p, q)\Int B3, L(q, p)'\Int B3. Therefore the connecting sphere in L(p , q) L(q, p) it contained in sT3 U (s Tl fl sT2). The untwisted spin of this manifold is (Spin (L(p, q))\Int B3•~S1)U (L(q, p)\Int B3)•~S1. We may assume that B3 is in sT2. Since Spin (sT31) is a twin, Spin(L(p, q))\Int B3•~S1 is Tw U B3•~S1. Adding B3•~S1, we obtain Tw U sT4, which is Lp. Since this Tw is removed and reglued, this is L p or L p ([2]). Tnt B3•~S1 is contractible in this manifold, therefore it is contained in a 4-ball B4. Note that B4\Int B3•~S1 is diffeomorphic to 52•~D2\Int B4 . Since (S2 X D2\Int B4) U (L(q, p)\Int B3)•~S1 is Lq\Int B4, therefore M(p , q, 0; pq, 1,1) is diffeomorphic to Lp Lq or Lp#Lq. Since we know it is not spin, it must be L'p#Lq. We have proved the following. PROPOSITION 9.2. The manifolds obtained by special Dehn surgeries with a=4 along torus T2-knots are Lp#Lq~K(p, q, 0; pq ,1, 0)='K(p, q, q; pq, 1, 0) and Lp#Lq K(p, q, 0; pq, 1,1). They are diffeomorphic if p is odd, not diffeomorphic if p is even. EJ If c=0, the manifold does not have the structure of good torus fibration . But it seems natural to define the base orbifold as a disk with two cone points with cone angles 22r/p, 2ic/q. It is a hyperbolic orbifold. Then, we have proved the following. THEOREM 9.3. The difeomorphism types of the manifolds obtained by special Dehn surgeries along torus T2-knots with Euclidean or hyperbolic orbifolds are classified by the fundamental groups, spinness, and the equiva riant intersection forms on the universal covering spaces. Moreover, we can determine whether the base orbifold is Euclidean or hyperbolic or not and in the case of Euclidean or hyperbolic base orbifold whether the Dehn surgery is special or not by the fundamental group. Here we discuss the case gcd (a, j9)=0, i.e, the Dehn surgery of type (0, 0, •}1). Note that a=0. From the fundamental group we know that a=0 and {p, q} and that a=0. Therefore the surgery type is recovered . The knot type is recovered from the spinness. 192 IWASE, ZJUNICI •˜10. The case with elliptic base orbifolds In this section, we treat the case with elliptic base orbifolds. (a1, a2, a3) is (2, 2, n) or (2, 3, 3) or (2, 3, 4) or (2, 3, 5) up to order, where n is an odd integer with n>3. (a1i a2, a3) from ir1M/center. a is known from H1(M; Z). The case o f (2, 2, n). In this case, {p, q}={2, n} (n_??_3), a=2. Therefore p =(a•}2)/2n. Since n_??_3, only one of the double sign holds and a must be even. If we restrict ourselves to special Dehn surgeries, then gcd(a, (a•}2)/2n) =1 and the manifold is one of the following: The first and the third ones are spin, while the second one is not spin, we do not know whether the first and the third one are diffeomorphic or not. The case of (2, 3, 3). In this case, {p, q}={2, 3}, a=3. Therefore /3= (a•}3)/6. Both of the double sign hold and a must be odd. If we restrict ourselves to special Dehn surgeries, then gcd (a, (a•}3)/6) =1 and the manifold is one of the following: Let us prove that M(2, 3, q'; a, (a+3)/6, 0)~M(2, 3, q'; a, (a-3)/6, 0) for q'=0, q. For M(2, 3, q'; a, (a+3)/6, 0), we may assume p=-1, q=1 and the Seifert invariants are (2, -1), (3, 1), (3, (a+3)/6), b=0. Replacing /32's by -1~'s, we have (2, 1), (3, -1), (3, -(a+3)/6), which can be deformed to (2, -1), (3, -(a-3)/6), (3, 1), which are Seifert invariants for M(2, 3, q'; a, (a-3)/6, 0). We do not know whether K(2, 3, 0) and K(2, 3, 3) yields the same mani fold. The case of (2, 3, 4). In this case, {p, q}={2, 3}, a=4 or {p, q}={3, 4}, a=2. If {p, q}={2, 3}, a=4, then j3=(a•}4)/6. One of the double sign holds and a must be even. Dehn surgery akong a torus Ti-knot II 193 If we restrict ourselves to special Dehn surgeries , then gcd (a, (a•}4)/6) =1 and the manifold is one of the following: The first and the third ones are spin, while the second one is not spin . We do not know whether the first and the third one are diffeomorphic or not . If {p, q}={3, 4}, a=2, then /3=(a•}2)/12. One of the double sign holds and a must be even. If we restrict ourselves to special Dehn surgeries , then gcd(a, (a•}2)/12) =1 and the manifold is one of the following: The first and the third ones are spin, while the second one is not spin . We do not know whether the first and the third one are diffeomorphic or not . If we have two possibilities for {p, q}, then we have See the proof of Theorem 8.6. The case o f (2, 3, 5). In this case, {p, q}={2, 3}, c=5 or {p , q}={3, 5}, a=2 or {p, q}={2, 5}, a=3. If {p, q}={2, 3}, a=5, then /3=(a•}5)/6. One of the double sign holds and a must be odd. If we restrict ourselves to special Dehn surgeries, then gcd (a , (a•}5)/6) =1 and the manifold is one of the following: We do not know whether they are diffeomorphic or not. If {p, q}={3, 5}, a=2, then jS=(a•}2)/15. One of the double sign holds . If we restrict ourselves to special Dehn surgeries, then gcd (cc , (a•}2)/15) =1 and the manifold is one of the following: 194 IWASE, ZJUNICI If a is odd, the second one is diffeomorphic to the first one. If a is even, then the first and the third ones are spin, while the second one is not spin. In any case, we do not know whether the first and the third one are diffeo morphic or not. If {p, q}={2, 5}, a=3, then j9=(a•}3)/10. One of the double sign holds and a must be odd. If we restrict ourselves to special Dehn surgeries, then gcd (a, (a•}3)/10) =1 and the manifold is one of the following: We do not know whether they are diffeomorphic or not. If we have more than one possibilities for {p, q}, a must be odd and we have See the proof of Theorem 8.6 and Example 8.3. •˜ 11. Equivariant intersection forms on cyclic covering spaces of the manifolds obtained by Dehn surgeries along K(2, 3,0) or K(2, 3, 3) (Summary) Let M be a manifold obtained by Dehn surgery along K(2, 3, 0) or K(2, 3, 3). Since H1(M; Z) is cyclic, a cyclic covering space of M with a given degree is uniquely determined if it exists. M, the d-fold cyclic covering space of M, is constructed as follows. k(2, 3), the trefoil knot, is a fibered knot whose fiber is a punctured 2-torus T with monodromy A=[0 1. Define a relation on T20•~I by (x, 1) •`T201 1 (Adx, 0). Then, ((T20•~I/•`)\Int B3)•~S1US2•~D2 (d-1)S2•~S2 is the inverse N image of the exterior in M. m is lifted to a loop m. li (i=1, 2) are lifted to N d loops, one of which is denoted by l. The meridian of sT4, the inverse Dehn surgery along a torus T2-knot II 195 image of sT4, is glued to (a/d)m+1911+112. Since A has period 6, we have 4 cases corresponding to id, A or A, A2 or A-2, A3. Our result is the following. RESULT 11.1. For fixed a and /3, the equivariant intersection forms on the cyclic covering spaces of M(2, 3, 0; a, /3, r) or M(2, 3, 3; a, j3,1) with coeffi cients in an arbitrary cyclic group does not depend on r. •˜ A. The manifolds obtained by Dehn surgeries along T2-links DEFINITION A.1. A non-empty submanifold of a 4-manifold is called a T2-link if each of its connected components is diffeomorphic to the 2-dimen sional torus. A Dehn surgery means cutting off the tubular neighborhood of the T2-link and pasting it back via some diffeomorphism between the boundaries. THEOREM A.2. Any orientable closed 4-manifold is obtained from a connected sum of some S1•~S3's, CP2's and CP2's by a Dehn surgery along some T2-link in it. PROOF. Let M be an arbitrary orientable closed 4-manifold. We shall prove that a Dehn surgery along some T2-link in M yields a connected sum of some S1•~S3's, CP2's and -CP2's. Choose mutually disjoint embedded circles which represent generators of the fundamental group of M and consider the regular neighborhood of one of them, which is diffeomorphic to S1•~D3. Let U be an unknot in D3. S 1•~U is a T2 -knot in S1•~D3. If we attach the boundary diffeomorphic to S1•~aD2•~S1, then we obtain S3•~S1\S1•~Int B3. Since S1•~Int B3 is con tractible in S3•~S1, S3•~S1\S1•~Int B3=S3•~51#D2•~S2. The operation above is a Dehn surgery along S1•~U. After perfoming this operation to all the generators of the fundamental group of M, we have a connected sum of some 51•~53 's and simply-connected closed 4-manifold M'. A Dehn surgery along an unknotted T2-knot in M' yields M'#S1•~ S3#CP2#(-CP2) (see Remark 4.6 in [2]). Perform some Dehn surgeries along sufficiently many unknotted T2-knots which are disjoint from the sT4's taken above. Wall's theorem [9] completes the proof. References [1] G. Burde and H. Ziescha.ng, Knots, Walter de Gruyter,1985. [2] Z. Iwase, Good torus fibrations with twin singular fibers, Japan. J. Math., 10 (1984), 321-352. [3] Z. Iwase, Dehn-surgery along a torus T2-knot, Pacific J. Math., 133 (1988), 289 196 IWASE, ZJUNICI 299. [4] J. M. Montesinos, On twins in the four-sphere I, Quart J. Math. Oxford, 34-2 (1983), 171-199. [5] L. Maser, Elementary surgery along a torus knot, Pacific J. Math., 38 (1971), 737-745. [6] P. Orlik, Seifert manifolds, Lecture Notes in Math., ;291, Springer-Verlag, 1972. [7] S. P. Plotnick, Equivariant intersection forms, knots in 54, and rotations in 2 spheres, Trans. Amer. Math. Soc., 296 (1986), 543-575. [8] H. Seifert, Topology of 3-dimensional fibered spaces, Seifert-Threlfall: A text book of topology, Academic Press, (1980), 359-422. [9] C. T. C. Wall, On simply-connected 4-manifolds, J. London Math. Soc., 39 (1964), 141-149. This paper is the English version of the second half of the author's thesis in the University of Tokyo entitled "Torasu T2-musubime ni sotta Dehn syuzyutu" (written in Japanese). The first half, which was origi nally written in English, has already been published as [3]. The numbering of Theorems etc. in this paper coincides with the one in the original thesis except for Proposition 3.1, Corollary 3.2 and Theorem 7.3, which are 3.2, 3.3, 7.5 in the thesis. •˜12 in the thesis is completely omitted. The author's name was formerly spelt "Zyun'iti Iwase". During the preparation of this paper, the author was partially sup ported by Fellowships of the Japan Society for the Promotion of Science for Japanese Junior Scientists and Grant-in-Aid for Encouragement of Young Scientist, No. 61790113. DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE KANAZAWA UNIVERSITY MARUNOUCHI, KANAZAWA 920 JAPAN, <, '2>'