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IMRN International Mathematics Research Notices 1998, No. 2 Cobordism of Nonspherical Links Vincent Blanlœil 0 Introduction and definitions 0.1. After the knot theory developed by Kervaire [K1]; [K2] and Levine [L1]; [L2]; in which they gave a classification of spherical knots up to cobordism; Le [Le] showed that the algebraic one-dimensional spherical knots have a particular behavior. More precisely; Le proved that cobordant algebraic knots of dimension one are isotopic. Some 20 years later; du Bois and Michel proved that in high dimensions (i.e.; 2n¡ 1 with n ‚ 3); things are completely different: du Bois and Michel found for; any n ‚ 3; examples of cobordant algebraic spherical knots; of dimension 2n¡ 1; which are not isotopic. Using the spherical knots given by du Bois and Michel; we construct the first examples of cobordant nonspherical links which are nonisotopic. The links we construct here are fibered and nonalgebraic. Let us be more precise. 0.2. A link is an (n¡ 2)-connected; oriented; smooth; closed; (2n¡ 1)-dimensional submanifold of S2nC1. A knot is a spherical link (i.e.; a link abstractly homeomorphic to S2n¡1).
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IMRN International Mathematics Research Notices 1998, No. 2 Cobordism of Nonspherical Links Vincent Blanlœil 0 Introduction and definitions 0.1. After the knot theory developed by Kervaire [K1]; [K2] and Levine [L1]; [L2]; in which they gave a classification of spherical knots up to cobordism; Le [Le] showed that the algebraic one-dimensional spherical knots have a particular behavior. More precisely; Le proved that cobordant algebraic knots of dimension one are isotopic. Some 20 years later; du Bois and Michel proved that in high dimensions (i.e.; 2n¡ 1 with n ‚ 3); things are completely different: du Bois and Michel found for; any n ‚ 3; examples of cobordant algebraic spherical knots; of dimension 2n¡ 1; which are not isotopic. Using the spherical knots given by du Bois and Michel; we construct the first examples of cobordant nonspherical links which are nonisotopic. The links we construct here are fibered and nonalgebraic. Let us be more precise. 0.2. A link is an (n¡ 2)-connected; oriented; smooth; closed; (2n¡ 1)-dimensional submanifold of S2nC1. A knot is a spherical link (i.e.; a link abstractly homeomorphic to S2n¡1).
- knots
- nonspherical links
- torsion free
- algebraically cobordant
- free z-module
- algebraic spherical
- modules hn
- seifert forms
- links
IMRNInternational Mathematics Research Notices 1998, No. 2
0 Introductionand definitions
Cobordism of Nonspherical Links
Vincent Blanlœil
0.1. After the knot theory developed by Kervaire [K1],[K2] and Levine [L1],[L2],in which they gave a classification of spherical knots up to cobordism,Leˆ [Leˆ] showed that the algebraic one-dimensional spherical knots have a particular behavior. More precisely, Leˆ provedthat cobordant algebraic knots of dimension one are isotopic. Some 20 years later,du Bois and Michel proved that in high dimensions (i.e.,2n¡1 withn¸3),things are completely different: du Bois and Michel found for,anyn¸3,examples of cobordant algebraic spherical knots,of dimension 2n¡1,which are not isotopic. Using the spherical knots given by du Bois and Michel,we construct the first examples of cobordant nonspherical links which are nonisotopic. The links we construct here are fibered and nonalgebraic. Let us be more precise. 0.2. Alinkis an (n¡2)-connected,oriented,smooth,closed,(2n¡1)-dimensional 2n+1 submanifold ofS. Aknotis a spherical link (i.e.,a link abstractly homeomorphic to 2n¡1 2n+1 S). For any linkK,there exists a smooth,compact,oriented 2n-submanifoldFofS , havingKas boundary;such a manifoldFis called aSeifert surfaceforK. 0.3. Following Kervaire [K1],we say that two linksK0andK1,abstractly dif-feomorphic to the same manifoldK,arecobordantif there exists an embedding©, 2n+1 :K£[0,1]→S£[0,1],such that
(K£ {0})=K0and©(K£ {1})= ¡K1,
where¡K1is the linkK1with the orientation reversed.
Received 2 October 1997.
118 VincentBlanlœil
2n+1 0.4. For any 2n-dimensional oriented,smooth,submanifoldFofS ,we denote 1 byGthe quotient of Hn(F) byitsZ-torsion. TheSeifert formassociated toFis the bilinear formA:G£G→Zdefined as follows (see also [K2,p. 88] or [L2,p. 185]): let (x, y) be inG£G;thenA(x, y) is the linking 2n+1 2n+1 number inSofxandi+(y),wherei+(y) is the cycley“pushed” in (S\F) by the 2n+1 positively oriented vector field normal toFinS. By definition,aSeifert form for a linkKis the Seifert form associated to a Seifert surface forK. 0.5. Asimple linkis a link which has an (n¡1)-connected Seifert surface. A link 2n+1 1 Kis asimple fibered linkif there exists a differentiable fibrationϕ:S\K→S , being trivial onU\K,whereUis a “small” open tubular neighbourhood ofK,and having (n¡1)-connected fibers,the adherence of which are Seifert surfaces forK. Analgebraic linkis a linkK(f) associated to a holomorphic germfwith an isolated singularity. Furthermore,Milnor’s theory of singular complex hypersurfaces implies that algebraic links are simple fibered links [M]. 0.6. LetAbe the set of bilinear forms defined on freeZ-modulesGof finite rank. T Letεbe+1 or¡1. IfAis inA,let us denote byAthe transpose ofA,byStheε-T¤ ¤¤ symmetric formA+εAassociated toA,byS:G→Gthe adjoint ofS(Gbeing the dual HomZ(G;Z) ofG),and byS:G£G→Ztheε-symmetric nondegenerated form induced bySonG=G¤. A submoduleMofGis pure ifGis torsion free. IfMis any /KerS /M ∧ submodule ofG,let us denote byMthe smallest pure submodule ofGwhich contains ∧ M. In fact, Mis equal to (MQ)∩G. For a submoduleMofG,we denote byMthe image ofMinG.
Definition. LetA:G£G→Zbe a bilinear form inA. The form A isWitt associated to zeroif the rankmofGis even and if there exists a pure submoduleMof rankm/2 inG such thatAvanishes onM;such a moduleMis called ametabolizerforA.
0.7. Definition.LetAi:Gi£Gi→Z, ,be two bilinear forms inA. LetGbeG0©G1 i=0,1 and letAbe (A0© ¡A1). The formA0isalgebraically cobordanttoA1if there exist a ¤ ¤ metabolizerMforAsuch thatMis pure inG,an isomorphismϕfrom KerSto KerS ,and 0 1 ¤ ¤ an isomorphismµfrom Tors(CokerS) to Tors(CokerS) which satisfy the two following 0 1 conditions: ¤ ¤ (c.1)M∩KerS= {(x, ϕ(x));x∈KerS}; 0 ¤ ∧¤ (c.2)d(S(M) )= {(x, µ(x));x∈Tors (CokerS)},wheredis the quotient map from 0 ¤ ¤ Gto CokerS.
1 We denote by Hn(F) the nth homology group ofFwith integer coefficients.
Cobordism of Nonspherical Links119
T¤ Remark. Inthe previous definition, Si=Ai+εAis the intersection form on Hn(Fi),KerS i i ¤ ˜ is the image of Hn(Ki) in Hn(Fi),and CokerSis isomorphic to Hn¡1(Ki). So for spherical i ¤ ¤ links,both KerSand CokerSare zero,and conditions (c.1) and (c.2) vanish. i i In order to prove that the links we will construct are cobordant,we will use the following theorem. 0.8. Theorem ([BM]).Ifn¸3,two simple fibered links,of dimension 2n¡1,are cobor-dant if and only if they have algebraically cobordant Seifert forms.
1 Statementof result and proof
1.1. We will prove the following proposition.
Proposition.For alln¸3,there exist cobordant nonspherical fibered links of dimen-sion 2n¡1 which are not isotopic.
Proof. Letus fixn¸3. We will use the spherical knotsK0andK1of dimension 2n¡1, constructed by du Bois and Michel in [DM]. These knots are the first example of cobordant and nonisotopic algebraic spherical knots. Now we will use them to construct some nonspherical fibered links. LetKi,withi=0,1,be the algebraic link of dimension 2n¡1 associated to the n+1 isolated singularity at zero of the germs of holomorphic functionshi: (C,0)→(C,0) defined by n X p q2 +x+x , hi(x0, . . . , xn)=gi(x0, x1)2+x3k k=4 (s+9)/2 (r+15)/2 2 32s+5 26 2r+10 )¡x¡4x x)((x x)¡x¡4x x),and withg0(x0, x1)=(x0¡x1)((x¡x0 10 0¡0 11 1 1 0 (r+17)/2 (s+7)/2 2 32r+14 25 2s+2 , x)=(x¡x)((x¡x)¡x xx)((x¡x)¡x¡4x x). g1(x01 00 10 1¡411 00 11 0 Heres¸11 ands6=r+8 are odd,andpandqare distinct prime numbers which do not divide the product²=330(30+r)(22+s) ([DM,p. 166]). We denote byAi, i=0,1 the Seifert form associated toKidefined on a freeZ-module of finite rankHi. LetLbe the algebraic link of dimension 2n¡1 associated to the isolated singu-larity at zero defined by the germ
n+1 f: (C,0)→(C,0) n X 2 (. .x , 0. , xn)7→xk. k=0
According to [D,Proposition 2.2],this algebraic link hasA=(( freeZ-module of rank oneG,as a Seifert matrix.
n(n+1)/2 1) ),defined on a
120 VincentBlanlœil
We constructLithe connected sum ofLandKifori=0,1. The Seifert form forLi is the integral bilinear formA©Aidefined on a freeZ-moduleGi=G©Hiof finite rank. The linksLiare simple fibered sinceA©Aiis unimodular (see [KW,Chapter V,§3]) and the linksLandKiare simple. According to [A,Theorem 4],the linksL0andL1,which are the connected sum of two algebraic links,cannot be algebraic. Using Theorem 0.8,we will prove thatL0is cobordant toL1. First let us get some n Tn T notation:B=A©A0© ¡(A©A1), S=B+(¡1)§B ,=A+(¡1)A. ¤ ¤ We haveA=(§1) so TorsCokerS6= {0}or KerS6= {0};henceLi, i=0,1 are not i i spherical links. LetMbe the metabolizer forA0© ¡A1given by du Bois and Michel. The module N=¢G©M,where¢G= {x©x, x∈G},is a metabolizer forB. In order to haveA©A0 algebraically cobordant toA©A1,we have to show thatNfulfills (c.1) and (c.2) in 0.7. ¤ (1) If KerS= {0},thenN=N;hence condition (c.1) of 0.7 is fulfilled. Furthermore, ¤ ¤ Tors CokerS=Coker§. This implies condition (c.2) of 0.7,and the two Seifert forms of L0andL1are algebraically cobordant. ¤ ¤ (2) If KerS=G,thenN∩KerS=¢G,and the metabolizerNfulfills (c.1) in 0.7; ¤ we also have thatN=M=Mis pure. Moreover,Tors KerS= {0},so the Seifert forms of L0andL1are algebraically cobordant. Now we are going to prove that the links considered are not isotopic. Let¿ibe the monodromy associated to the fibered linkLi;if there exists an integeresuch that e (¿¡1)Gi=0,theneis called anexponentforLi. i e2 Recall that thee-twistgroup forKiis defined as follows: assuming (t¡1)Hi=0, ifeis an exponent forKi,then thee-twist group associated toKiis the group denoted by e ee e GT(hi) (orGT(Ki)),which is theZ-torsion subgroup of the quotient Ker (t¡1) . i /(t¡1)Hi i According to the monodromy theorem (Breiskorn-Grothendieck),thee-twist group is well defined for one-dimensional algebraic links,and du Bois and Michel showed that (1)²is an even exponent for the algebraic knots associated tog0andg1;for all k k multipleskof²,the finite abelian groupsGT(g0) andGT(g1) have distinct orders; k k(p¡1)(q¡1) (2) thek-twist group forh0andh1are well defined,andGT(hi)=(GT(gi)) i=0,1. Letkbe a multiple of²=330(30+r)(22+s). For a fibered linkL,the matricesAof n¡1T Seifert form and¿of the monodromy are related together by :¿=(¡1)A A. Hence for k kk i=0,1 we have¿i=(§Id)©ti. ThusGT(Li) is well defined,and we haveGT(Li)=GT(hi). k k¡1 Finally, GT(L0) andGT(L1) have distinct order and theZ[t, t]-modules Hn(G0) and Hn(G1) are not isomorphic. Hence the linksL0andL1are not isotopic.
Cobordism of Nonspherical Links121
References [A] N.A’Campo,Le nombre de Lefschetz d’une monodromie,Indag. Math. (N.S.)35(1973),113– 118. [BM] V.Blanlœil and F. Michel,A theory of cobordism for nonspherical links,Comment. Math. Helv.72(1997),30–51. [DM] P.du Bois and F. Michel,Cobordism of algebraic knots via Seifert forms,Invent. Math.111 (1993),151–169. [D] A.Durfee,Fibered knots and algebraic singularities,Topology13(1974),47–59. [K1] M.Kervaire,Lseonuesdedidemionsupnsri´ereeu,Bull. Soc. Math. France93(1965),225–271. [K2],“Knot cobordism in codimension two” inManifolds–Amsterdam,1970,Lecture Notes in Math.197,Springer,Berlin,1971,83–105. [KW] M.Kervaire and C. Weber,“A survey of multidimensional knots” inKnot Theory(Proc. Sem. Plans-sur-Bex,Switzerland,1977),Lecture Notes in Math.685,Springer,Berlin,1978,61– 134. [Lˆe]D.T.Leˆ,Sur les noeuds alge´briques,Compositio Math.25(1972),281–321. [L1] J.Levine,Knot cobordism groups in codimension two,Comment Math. Helv.44(1969), 229–244. [L2],An algebraic classification of some knots of codimension two,Comment. Math. Helv. 45(1970),185–198. [M] J.Milnor,Singular Points of Complex Hypersurfaces,Annals of Math. Stud.61,Princeton Univ. Press,Princeton,1968.
Universit´eLouisPasteurStrasbourgI,De´ partement de Mathe´ matiques,IRMA,7,rue Rene´ Descartes, F-67084 Strasbourg ce´dex,blanloeil@math.u-strasbg.fr
