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Intersection Theory of Manifolds with Operators with Applications to Knot Theory Author(S): Richard C

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Intersection Theory of Manifolds with Operators with Applications to Knot Theory Author(S): Richard C Annals of Mathematics Intersection Theory of Manifolds With Operators with Applications to Knot Theory Author(s): Richard C. Blanchfield Source: The Annals of Mathematics, Second Series, Vol. 65, No. 2 (Mar., 1957), pp. 340-356 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/1969966 . Accessed: 30/04/2011 08:58 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=annals. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to The Annals of Mathematics. http://www.jstor.org ANNALS OF MATHEMATICS Vol. 65, No. 2, March,1957 Printed in U.S.A. INTERSECTION THEORY OF MANIFOLDS WITH OPERATORS WITH APPLICATIONS TO KNOT THEORY BY RICHARD C. BLANCHFIELDt* (Received March 29, 1956) Introduction Let 9N be an orientedcombinatorial manifold with boundary,let 9)1,be any of its coveringcomplexes, and let G be any freeabelian groupof coveringtrans- formations.The homologygroups of 9N*are R-modules,where R denotes the integralgroup ring of the (multiplicative)group G. Each such homologymodule H has a well-definedtorsion sub-module T, and the correspondingBetti module is B = H/T. The automorphismy -> -1 of the groupG extendsto a unique automorphism a -a & of the ringR = R. FollowingReidemeister [4] thereis definedan inter- section S whichis a pairingof the homologymodules of dual dimensionto the ringR, and also a linkingV which is a pairingof dual torsionsub-modules to Ro/R,where Ro is the quotientfield of R. Two duality theoremsare proved: (1) S is a primitivepairing to R/rm of dual Betti modules with coefficients modulo tm and Tm respectively,where or is zero or a primeelement of R and m is any positive integer. (2) V is a primitivepairing of dual torsionmodules to Ro/R. These theoremsare analogous to the Burgerduality theorems [1]; in case R is the ringof integers,they specialize to the Poincar6-Lefschetzduality theorems for manifoldswith boundary. Althoughdual modules are not, in general, isomorphic,it is demonstrated that if one torsionmodule has elementarydivisors A0 c Al c ... then its dual has elementary divisors Ao C Al C *... (The numbering differsfrom that of [2] in that we beginwith the firstnon-zero ideal.) This result is applied to the maximal abelian coveringof a link in a closed 3-manifold.It is proved that the elementarydivisors Ai of the 1-dimensional torsionmodule are symmetricin the sensethat Ai = At . For the case of a knotor linkin Euclidean 3-space,the ideal A0is generatedby the Alexanderpolynomial, and the symmetryof A0has been proved previouslyby Seifert[6] forknots and Torres [7] forlinks. The "symmetry"of the Alexander polynomialwas proved by Seifertand Torres in a slightlymore precise form [8, Cors. 2 and 3]. The problemof similar extraprecision in the moregeneral case ofa linkin an arbitraryclosed 3-manifold remainsopen. * Died July 25, 1955. The revision of his Princeton 1954 Ph.D. dissertation presented here was made by J. W. Milnor and R. H. Fox. Aside from minor corrections the only changes were a strengthening of Lemma 4.3 and the insertion of Lemma 4.10 and Corollary 5.6. 340 INTERSECTION THEORY OF MANIFOLDS 341 1. Let 9M be an orientedn-dimensional manifold with boundaryS. Let G be a multiplicativefree abelian groupof orientation-preservinghomeomorphisms of WNonto itselfsuch that no point of W9is fixedunder any elementof G other than the identity.We assume that WNhas been triangulatedin such a way that G operateson 9 as a complex;that is to say, the elementsof G map cells onto cellspreserving the incidencerelations. The canonicalexample of such a manifold is a coveringcomplex of a triangulatedmanifold where the coveringtransforma- tions forma freeabelian group. Let 9) be the dual cell complexof 9W.Except for those cells of W9which lie entirelyin the boundaryQ3, each cell of 9N has a dual cell in W0.If the original triangulationof W9 is sufficientlyfine, the complex W9 will be a deformation retractof 9t. We use W9for the study of relativehomology and WNfor absolute homology. Let R be the group ringof G with integercoefficients. Define j = e-1 forall 7 e G. The mappingy -> jycan be extendedto an automorphismof R by linearity. Observe that for any a E R, a = a. Let Ci be the group of i-dimensionalchains of WNmodulo A3,and let Cn-i be the group of (n - i)-dimensionalchains of 9W,i = 0, *, n. The elementsof G operate naturallyas automorphismson Ci and C,2_j, and by linearitywe may regardthe elementsof R as operatorson Ci and C,,-i. Hence [3] we may regard Ci and Cni as R-modules.Note that Ci and C0-i are freemodules in the sense that each one is isomorphicwith the directsum of R with itselfa finitenumber of times. For any x E Cj let aix be the relative boundaryof x. Then a is an operator homomorphismof Ci into Cji_ and we have the usual relationaidai = 0 for i= 1, ***, n. Similarlywe have operatorhomomorphisms 4i: >- Where no confusioncan result,we writeaj and As in abbreviatedform as 0. In the usual way one can definerelative and absolute homologymodules. As abelian groups these homologymodules are the usual homologygroups. The operatorstructure of the modules is inherited from the correspondingstructure on the chain modules.The homologymodules are invariantsof the pair (9S, G), wherewe con- sider two pairs (9S, G) and (9M',G) equivalent if thereexists a homeomorphism opof 9onto 9' suchthat-ysp = eo forally EG. FollowingReidemeister [4] we introducean intersectionS as follows.Let e denote the ordinaryintersection of chains (disregardingoperators). For x E C and x e Czi define S (x,) -,,,GS(x, -x) y ER. All but a finitenumber of termson the rightare zero, so the sum is finite.The followingproperties of S are easily verified. (1) S(x + y, x) = S(x, x) + S(y, x). (2) S(x, x + y) = S(x, x) + S(x, g). (3) S(ax, ,3) = aS(x, )for all a, dER. (4) S(x, ax) = (-1)tS(ax, x) wherex E C0 and x e Cn-i+l . 342 RICHARD C. BLANCHFIELD (5) There exists a pair of dual bases xl, , Xr for Ci and xl, ., r forC.- such that S(xi, Tj) = bij fori, j = 1, ,r. It followsfrom (4) that the intersectionof a cycle with a boundingcycle is zero,and hencethe intersectionS givesrise to an intersectionof homology classes. We now abstract the algebraic propertiesof this setup whichare used in the followingsections. Let R be a Noetherianring with unit element 1. We assume that thereis given a fixedisomorphism of R onto a ringR. In the applicationwe willhave R = R, and the relationa = a willbe valid. DEFINITION. An R-moduleM is an additive abelian group with operatorsR such that: (1) (a +f)x = ax + #xfor all a, AER,xEM (2) 1 x = x forall x eM. Since R is a Noetherian ring it followsthat if Mllis finitelygenerated, then everysubmodule of M is finitelygenerated. DEFINITION. Let M1 and M3 be R-modulesand let M2 be an R-module.Then P is a pairing ofM1 and M2 to MN ifP assignsto everypair (X1 XX2)X X1 E NM1, X2 E M2, an elementP(x1 , X2) MEN3 such that (1) P(x1 + yl , x2) = P(x1 , x2) + P(yl , x2) forall xi , yl E M1, X2 E M2 (2) P(x1 , x2 + Y2) = P(x1 , x2) + P(x1 , Y2) forall xi E M1, X2, Y2 E M2 (3) P(ax, , j~x2) = afP(xi, x2) for all a, : ER, xi E M, XX2 E M2 . DEFINITION. Let P be a pairingof M1 and M2 to MN3.The annihilatorA1 of M2 is thesubmodule of all elementsxi EM1 forwhich P(xi, x2) = 0 forall X2 E M2 . Similarlythe annihilatorA2 of Ml is the submoduleof all elementsX2 E M2 for whichP(xl, x2) = 0 forall x e M1 . DEFINITION. A pairingP of M1 and M2 to M3 is primitiveif the annihilatorsof M1 and M2 are both zero. DEFINITION. Let M1 be a finitelygenerated free R-module and let M2 be a finitelygenerated free R-module. Let P be a pairingof M1 and M2 to R. Then bases xi, .*. , X?of M1 and Yi, Y,Yr of M2 are calleddual baseswhen P(xi, yj) = bij for i, j = 1, * * , r. Note that if thereexists a pair of dual bases, thenthe pairingis primitive. DEFINITION. An n-dimensionalchain complexwith coefficients in R is a system of finitelygenerated free R-modules and operatorhomomorphisms CM( < Co(R) ... C1((R) such that 9i1 i9 = 0 for i = 1, , in.The submodule of cyclesZj(R) is the kernelof at, i = 1, , n. Zo(R) = Co(R). The submoduleof boundingcycles Bi(R) is the image of diTl, i = 0. , n- 1. Bn(R) = 0. DEFINITION. Let I Ci(R) } and {C0.-(R) } be n-dimensionalchain complexes with coefficientsin R and I respectively.Then {CJ} and { n-i} are said to be dual chain complexeswith pairingS if: (1) S is a pairingof Ci and CO_ to R, fori = 0.
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