Annals of Mathematics

Link Groups Author(s): John Milnor Source: The Annals of Mathematics, Second Series, Vol. 59, No. 2 (Mar., 1954), pp. 177-195 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/1969685 . Accessed: 25/09/2011 16:06 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp 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].

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ANNALS

OF MATHEMATICS

Vol. 59, No. 2, March, 1954 Printed in U.S.A.

LINK GROUPS BY

JOHN MILNOR

(Received March 3, 1952) (Revised October16, 1953)

1. Summary By a linkhomotopyis meant a deformationof one link onto another,during which each componentof the link is allowed to cross itself,but such that no two componentsare allowed to intersect.The purposeof this paper is to study linksunder the relationof homotopy.The fundamentaltool in this study will be the linkgroup.The linkgroupof a linkis a factorgroupof the fundamental groupof its complement,whichis invariantunderhomotopy. To each conjugate class of elementsin the link group of a link with n componentsthere correspondsa link with n + 1 componentswhichis definedup to homotopy.A studyof this grouptherefore, not only gives a methodfordistinguishingbetweenlinkswhichare not homotopic,but also gives a procedure forobtaininga (possiblyredundant)list of all homotopyclasses of linkswitha givennumberof components.By means of the linkgroup,an effective procedure is given fordecidingwhetheror not a given link is trivial-that is homotopic to a collectionof "unlinked" circles. A complete homotopyclassificationis given for links with three componentsin euclidean space, and for links such that everypropersublinkis trivial. I am indebtedto R. H. Fox forassistancein the preparationof this paper. 2. The basic theorems Let M be an open 3-dimensionalmanifoldwhichpossessesa regulartriangulation. Let C be a circle. By an n-link L will be meant an ordered collection (11, - - - , In)of maps 1i: C -> M, wherethe images11(C), .- , l1(C) are to be disjoint.A link will be called properif the maps 11, , In are all homeomorphisms. Two links L and L' will be called homotopicif there exist homotopieshi, , hnt(C)are betweenthe maps 1i and the maps 1 so that the sets h1t(C), disjointforeach value of t. Clearly the relationof homotopyis reflexive,symmetricand transitive. Denote the image 11(C)u ... u ln(C)of a link L by I L . For each proper link L let G(L) denote the fundamentalgroup of the complementM - I L I . Let Lt = (11, ... , li-, kli+, ... , ln) denote the (n - 1)-sublink obtained by removingthe ithcomponent.To each such sublinktherecorrespondsa natural inclusionhomomorphism G(L) -> G(Lt). Denote thekernelofthishomomorphism by Ai(L), and denote the commutatorsubgroupof Ai by [Ai]. Since the [Ai] are normalsubgroupsof G(L), their product E = [A1][A2]... [An] is also a 177

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JOHN MILNOR

normal subgroup. By the link group p3(L) will be meant the factor group G(L)/E(L). For example fora link with one componentthe subgroupE of G is just the of G into commutatorsubgroup[A] of the kernelof the natural homomorphism the fundamentalgroup of M. Thus in this case 9 = G/E = G/[A]. (REMARK. The followingalternativedefinitionof the linkgroupis moreintuitive,althoughless practicalforcomputation.Considerthe set of all closed loops in M - I L I with base point xo. A multiplicationbetween loops is definedin the usual manner.Definetwo loopsf and g to be equivalent,if the (n + 1)-link (1l, .** In fg'-) is homotopicto a link (11, *, 1', x0),forwhichthe last componentconsistsof a singlepoint. The equivalence classes of loops now build a group.It willfollowfromCorollary1 to Theorem3 that thisgroupis isomorphic to 9). For theproofthat 9 is invariantunderhomotopy,threelemmaswill be needed. ofa one-dimensional A homeomorphism complexC intoM will be called polygonal if,forsomesubdivisionofC, each simplexis mapped linearlyintoa simplexofM. complexC intoM can be approxiLEMMA 1. Any map f of a one-dimensional of C into M, whichis matedarbitrarilycloselyby a polygonalhomeomorphism tof. homotopic Thus in particular,every link can be approximatedarbitrarilyclosely by a properlink. The proofis easily given. The next lemma is closely related to the Lefschetzduality theorem,and is proved in the same manner. LEMMA 2. Let U be an orientable n-dimensionalmanifoldwhichpossesses a oo denotetheone pointcompactification and let U of U. regulartriangulation; of U, and letV beany closedsubsetwhichis disjointfromX. Let X bea subcomplex Then thesingularhomology V, X) is naturallyisomorphicto the groupHr(U X _o, V Cech-Dowker co). cohomology groupHn- (ULet K denote an arbitrarilyfinesimplicialcomplexfor U and let K* denote the dual cell complex.Let V' be the smallestsubcomplexof K* whichcontains V, and let V" be the open star neighborhoodof V' withrespectto K. Let X' be the open star neighborhoodof X with respect to K*. By a standard combinatorial argument it follows that Hr(U - V", X) is isomorphic to o, V' , co). Furthermore the pair (U - X' o) To, V' Hn-r(U - X' is a deformation retract of (U - X coc). co, V' Now pass to the limit,as the meshof the complexK becomesarbitrarilyfine. Since the neighborhoodsV" convergeto the closed set V, it followsthat the singulargroupHr(U - V, X) is the directlimitof the groupsHr(U - V", X). Thus in order to complete the proof it is only necessary to show that H n-r(U_-X , co, V , co) is the direct limit of the corresponding groups for co are compact spaces with intersection the neighborhoodsV'. Since the V' V coc) is the direct limit of the groups H8(V' -, o). co, the Cech group H8(V I For the case n = 1, this factor group has been studied by R. H. Fox (see [2] in bibliography).

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179

Now usingthe exact cohomologysequencesof the pairs (U - X s, V co) o) is and (U - X o, V oc, V' co), it followsthat H-r(U -X the directlimitof the groupsHf-r(U - X , co, V' co); which completes the proof. LEMMA 3. Let Y and Z betopological spaces,and lethoand h1behomeomorphisms of Y into Z. A homotopybetweenho and h1 induces an isomorphismbetween Hr(Z, ho(Y)) and Hr(Z, h1(Y)). The proofwill be valid forany cohomologytheory.Let Qo and Q, denote the mapping cylindersof hoand h1 . A natural isomorphismbetweenHr(Z, h,(Y)) and Hr(Qi, Y) will be constructedfori = 0, 1. Since a homotopybetween ho and h1inducesa homotopyequivalence betweenthe pairs (Qo, Y) and (Qi, Y), this will completethe proof. The mappingcylinderQi is definedas theidentification space of Y X [0,1] Z in which(y, 1) is identifiedwithhi(y) foreach y E Y. The spaces Y (= Y X [0]) and Z can be consideredas subspacesofQi . Let Pi denotetheimageof Y X [0, 11 in Qi. The inclusionmap (Z, hi(Y)) -* (Qi, Pi) is clearlya homotopy equivalence.Sincehiis a homeomorphism, theinclusionmap Y -> Pi is also a homotopy equivalence. Using the exact sequences of the pairs (Qi, Y) and (Qj , Pi) it followsthat the inclusionmap (Qi, Y) -* (Qi, Pi) inducesisomorphismsof the cohomologygroups. The maps (Z, h%(Y)) -> (Qi, Pi) -- (Qi, Y) now induce the requiredisomorphism betweenHr(Z, hi(Y)) and Hr(Qi, Y); whichcompletes the proof. -

THEOREM 1. If twoproperlinks are homotopic, thentheirlink groupsare isomorphic. It will be assumed that thereis a fixedbase pointxoin M whichis not on the path of the homotopy.Clearly any given homotopycan be approximatedby one whichpermitssuch a point xo. CASE 1. Links with one component. For a linkL withone component,the subgroupA/[A] of the linkgroupG/[A] may be describedas follows.Let U denote the universalcoveringspace of M, and let V denote the inverseimage of I L I in U. Then U - V is a covering space of M - I L I, and its fundamentalgroup equals the subgroupA of the fundamentalgroup G of M - I L I . Thereforethe abelianized group A/[A] is isomorphicto the singularhomologygroupH1(U - V) withintegercoefficients. The fulllinkgroupG/[A]may be describedby a modification ofthisprocedure. Let X denote the inverseimage of xoin U, and let x' denote a base point in X. We will next definea homomorphism-qof G onto a certain group associated withthe singularhomologygroupH1(U - V, X). Each elementg of G is represented by a closed loop in M - I L I withbase pointxo. Such a loop is covered by a path in U - V which starts at x' and ends at some point x of X. This path representsan element-q(g)of H1(U - V, X). The image 71(G)clearlyconsistsof all elementsX of H1(U - V, X) such that the boundaryof X in Ho(X) has the formx - x . A groupoperationin 71(G)is definedas follows.To each X, with 8X1- xi- x there correspondsa unique

180

JOHN MILNOR

coveringtransformation (p of U - V over M - L which carriedx0 into xi. Definethe productof two such elementsby XlX2 = Xi + +1(X2).

Under this group operation the map 77becomes a homomorphism,and it is easily verifiedthat the kernelof -qis [A]. Note that the group of all covering transformations 0 is just the fundamentalgroup F of M. Thus -0definesan isomorphismof the link groupG/[A] onto a certaingroup 71(G)whichis defined in terms of H1(U - V, X), togetherwith the boundary homomorphism H1(U - V, X) -* Ho(X) and the operationsof F on these two groups. By Lemma 2, H1(U - V, X) is isomorphicto the Cech-Dowker group H2(U -X ?o, V coc).By Lemma 3, thisgroupis invariantunderhomotopies of V But a co. co. Certainly homotopyof L induces a homotopyof V the boundaryhomomorphism H1(U - V, X) -- Ho(X) and the operationsof F on these two groups are also invariantunder homotopiesof L. Therefore G/[A] is invariantunder homotopiesof L. CASE 2. L has more than one component,but only the ith componentis moved by the homotopy. ApplyingCase 1 to thelink 1iin themanifoldM|L it followsthatG/[Ai] is invariantunder homotopyof li. Furthermoreit is easily verifiedthat the kernelAj[Ai]/[Ai]ofthenaturalhomomorphism G(L)/[Ai(L)] -* G(L')[Ai(Lj)1 is invariant,foreach j # i. The commutatorsubgroupof thiskernelis [Aj[Ai]/[Ai]]= [Aj][Aij/[Ai]. The productover all j = i of the groups[Aj][Al]/[Ai]is just E/[Ai]. Therefore the factorgroup9 = G/E is invariantunderhomotopyof i. CASE 3. An arbitraryhomotopy. First suppose that,forsome subdivision0 = to < ti < ... < tk = 1 of [0, 1], the homotopyht satisfiesthe followingtwo conditions: (1) Only one componentof the link is moved duringeach interval[tj, tj+11. (2) The linkshto, ht1, .-- , htkare proper. Then the desired isomorphismis obtained by repeated application of Case 2. But, withthe use ofLemma 1, any homotopycan be approximatedby one which satisfiesthese conditions.This completesthe proofof Theorem 1. REMARK. The isomorphism whichis obtaineddoes not actuallydepend on the approximationwhichis chosen.Howeverthe proofof this assertionis somewhat complicated,and will not be givenhere. The meridiansto a properlink L are elementsof the link group of L, defined as follows.It will be assumed that M is an orientablemanifold,and that fixed orientationshave been chosenforM and C. Let p(t) be a path leading fromthe base point x0to a point p(l) of lQ(C),but not touchingL fort < 1. Choose a small neighborhoodN of p(l); and forma closed loop in M - i L I as follows.Traverse the path p fromxo to a point in

LINK

GROUPS

181

N; thentraversea closed loop in N whichhas linkingnumber+ 1 withli(C) n N consideredas a cycle of (N, N); and returnto xo along p. This definesan element ai of 9(L) which will be called the ith meridianof L with respectto the path p. Making use ofthe isomorphism Ai(L)/[A%] Hi(U - V) ofTheorem1, it is easy to see that ai is unique and well-defined forall sufficiently small neighborhoodsN. Let ai denote the kernelof the natural homomorphism 9(L) -* 9(Lt). An elementT3jof ?9(L') is definedas follows.Traverse the path p fromxo to p(l); then traverseli(C) in the positivedirection;and returnto xo along p. This element T3icorrespondsto a coset Ojai in ?3(L). This coset will be called the ith parallelof L withrespectto p. If the path p is replacedby some otherpath, thenthe pair (ai , Ojai) is clearly replacedby some conjugatepair (Xai)1, XAi3X1ai). THEOREM 2. The isomorphism of Theorem1 preserves thepairs (ai, 31a), , to na3n) up (an conjugations. More precisely:if the pair (i4 , ,O3(ai(L))in ?(L) correspondsto (a' , O3,ai(L')) in ?3(L') underthe isomorphism, and if(at' , 3'lai(L')) is a meridianand parallel pair in ?3(L'), then a'" = cXa>J-1 and 3'O'ai(L') = XOY>7ai(L') for some X in ?3(L'). The proofis easilygiven:it is onlynecessaryto checkthisassertionthrough each stage of the proofof Theorem 1. Nearly all of the above considerationscan stillbe carriedthroughifthe manifoldM is non-orientable. The only changeis that it is no longerpossibleto distinguishbetweenai and a-'j. If L is a polygonallink, then meridianscan be definednot only in the link group,but also in the fundamentalgroupG(L). (A closed loop in M - I L I is element definedjust as before.Howeverin this case it representsa well-defined of G(L).) LEMMA 4. Let L be a polygonallink,and letai e G(L) be a meridiantotheith component.Then thenormalsubgroupgeneratedby as is Ai. For an arbitrary properlinkL, thenormalsubgroupgenerated bya meridianas e ?3(L) is aj. Let N be a smoothtubular neighborhoodof lQ(C). The group G(L) can be persentedas the free product of the fundamentalgroups of N - li(C) and M - N - i L [, with relationscorrespondingto the boundaryof N. If the componentli is removed,the onlychangeis that the relationat = 1 is added in thegroupofN - l(C). Thereforethekernelofthehomomorphism G(L) -* G(Lt) assertionfor the is the normalsubgroupgeneratedby ai. The corresponding link group of an arbitrarylink followsimmediately,by passing to a polygonal approximation. L THEOREM 3. Let L bea propern-link,and letf and f' beclosedloopsin M -I withbase pointxo . If f and f' represent conjugateelements of thelink group?3(L), thenthe(n + 1)-links(L, f) and (L, f') are homotopic. It is sufficient to considerthe special case in whichf and f' representthe same elementof S. For if hfh-1is any loop conjugateto f,then the links (L, f) and (L, hfh1) are clearlyhomotopic. . ..

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JOHN MILNOR

g of [A], we will show that the For any closed loop f, and any representative (n + 1)-link (L, fg) is homotopicto (L, f). It will followby inductionthat (L, fga ... gn)is homotopicto (L, f), whereg9 ... gnrepresentsany elementof E(L). Thus loops whichrepresentthe same elementof 9 give rise to homotopic (n + 1)-links.This will completethe proof. lX. 1' This can Approximatethe maps 11, ,Inby polygonalmaps 1, be done so that the images 11(C), * I', (C) and f(C) g(C) remaindisjoint, and so that g representsan elementof [Ai(L')]. Let a, E G(L') be a meridianto the it',componentof L'. By Lemma 4 every elementof Ai(L') can be writtenas a productof conjugates of as and aT1. In otherwords everyelementof Ai(L') can be writtenas a productof meridians and theirinverses.Thereforethe representativeg of [Ai(L')] is homotopicto a , hr represent h71), where the loops h1, loop (h1h2hi h-1) ... (hr-1hrh-11 I, Pr or meridiansto the ith componentof L', correspondingto paths pi, the inversesof such meridians.By Lemma 1 it may be assumed that the paths pj are polygonal,and that the imagespj([O, 1]) and pk([O, 1]), j = k, have only the base point in common. Choose a 3-cellneighborhoodof the set pi([O,1]) - P2([O, 1]). In thisneighborshownin Figure 1 will occur. (The heavy lines represent hood the configuration and l'(C) as indicated in Figure 2, the portionsof l(C)). Deformingh1h2hi1hi1 loop hih2h'lhj1can be reduced to a point. It follows by induction that (L', fhlh2hjlhj1... hrlhrhlLihj1)is homotopicto (L', f). The proofis now completed by combiningthe homotopies ,

(L, fg)

-

(L', fg)

-

(L', fhlh2 ...

h-1)

-

(L', f)

-

(L, f).

A link will be called i-trivialif,forsome homotopiclink (1, *, I), the set l,(C) consistsof a singlepoint. CombiningTheorems 1, 2 and 3, the following resultis obtained. , In) bea properlink.A closedloopf in M LLetL =(l, COROLLARY1. f) is n theidentityelementof f(L) if and only if the link (11, * , In represents (n + 1)-trivial. This can also be stated as follows. is equal to Gti. 2. A linkis i-trivialif and onlyif its ith parallelf3cai COROLLARY The proofsare evident. 3. An example In orderto give a concretegeometricalillustrationof the precedingtheory, the followingtheoremwill be proved. Results in this section will not be used in the followingsections. A link is trivialif it is homotopicto some (1l, ** , l') wherethe components ' consistof singlepoints.By a knotis meant a proper1-link. 11(C)X **X(C) THEOREM 4. If a polygonalknotin euclideanspace is replacedby a collection of theoriginal,and having of parallel knotslyingwithina tubularneighborhood linkingnumberszero,thentheresultinglink is trivial. It is firstnecessaryto make some remarksabout the lowercentralseriesof a

LINK

183

GROUPS

group.For any groupG, the firstlower centralsubgroupG1is the groupitself. aba-1b-1with The nthlowercentralsubgroupGnis generatedby all commutators a E G and b E Gnl-.

r,9A

/4h2

Sy2a.

7 Pi' 2c.

\~~

r72 L.

1~/ Ft-2d

LEMMA 5. If L is a propern-linkin euclideanspace, thenthe(n + 1)-stlower centralsubgroupSgn+of q(L) containsonlytheidentityelement. This is clear forthe case n = 0. (The fundamentalgroup of euclidean space containsonly the identityelement.)If it has been proved forthe case n - 1, 2 Hence 9(L) can be considered as a factor group of G(L)/G,+, (L). This suggests some connection with the work of K. Chen [11 who has shown that, for a polygonal link L, the factor groups G(L)/G,(L) are invariants of the isotopy class of L for arbitrary values of the integer r.

JOHN MILNOR

184

then it followsthat the nth lowercentralsubgroupof 9(L') / 9(L)/ai(L) contains only the identityelement,foreach i. In otherwords the subgroup of 9(L) is containedin each ad. By the definitionof the link group,the subgroups , an generate9, it follows that ai are commutative.Since the groups a1,I elementsof gn commutewith all elementsof A, whichproves that 9"+i = 1. Let T denotethefundamentalgroupofthe complementM - N ofthe tubular neighborhood.By the Alexanderdualitytheorem,the abelianized group TIT2 g H1(M - N) is infinitecyclic.It followsthat T, = T2 forr _ 2. For if ao E T generatesT/T2, then every elementof T can be writtenin the formaj b with b E T2. Modulo the subgroupT3, all elementsof T commutewith elementsof T2 . Since powersof ao commutewith each other,it followsthat T/T3is commutative.But this impliesthat T3 D T2,hence T2 = T3 = T4= Si

34-

F;Y 3

Fij

Consider the natural homomorphismsT -3 G(L) >-* 9(L). The subgroup - 1 of S. This means that a = T.+, of T is mapped into the subgroup closed loop in M - N representsthe identityelementof9(L) wheneverit represents the identityelementof T/T2 H1(M - N): that is wheneverit has linkingnumberzero withN. The parallel jij to any componentof L can clearlybe representedby a loop in M - N having linkingnumberzero with N. This means that Aide equals ai , and hence that the link L is i-trivial.Since this is true forall values of i, it followseasily that L is trivial. To conclude this section,a counter-exampleto a possible generalizationof Theorem4 will be given. The 2-linkillustratedin Figure 3 is clearlytrivial.If each componentof this link is replaced by two parallel components,having linkingnumberszero, then the resulting4-link,illustratedin Figure 4, is not trivial.This can be proved by the techniquesof Section 4. (Compare also Section 5.) T2

pnal

4. Triviallinks Considerthe followingtwo problems. (I) To give a generalprocedurefordecidingwhetherany givenlinkis trivial. (II) To solve the word problemforthe link groupsof all triviallinks.

LINK

185

GROUPS

It followsfromCorollary1 that these problemsare equivalent: Suppose that problem(I) has been solved; and Let L be any trivialproperlink. Then a loop f in M - I L I representsthe identityelementof :(L) if and only if the link

(L, f) is trivial.

Now suppose that (II) has been solved, and suppose by inductionthat (I) has been solvedforlinkswithn - 1 components.Then a properlink (11, - - *, In) is trivialifand onlyifthe sublink(11, -, in-1) is trivial,and the loop Inrepresentsthe identityelementof ?(11, - , in-1). It will be shownin this sectionthat problem(II) can be solved wheneverthe wordproblemforthe fundamentalgroupF of M can be solved. For any properlink L let JG(L) denote the integralgroup ringof G(L). Let Ki(L) denote the kernelof the natural homomorphismJG(L) JG(Li). SimJ9(Lt). The ilarlylet XCi(L)denote the kernelof the homomorphismJ9(L) two-sidedideals KMand 3C2will be of particularinterest. JG/K2 + ... + K2 __ Jj3/3C2+ LEMMA 6. The natural homomorphism + S n is an isomorphism. *.. The ringsJG/K2 + ... + K2 + * + CW2 will be denoted by J(S/,3C2 @R(L). to prove that the subset E of JG is congruentto the It is evidentlysufficient + K2 . If a and a' are elements of Ai, identity modulo K2 + *.. + 1= (a- 1) (a' -1) is an elementofKi . Thereforeaa' thenaa'-a-a' a'a (mod K2). This means that [As] 1 (mod KM);hence a + a' - 1

E = [A1] ... [An]--1

(modKi

+ Kn)

which completesthe proof. Let as be a meridianto the ith componentof L. An elements of J9 is an expressionof the form Eej3yj, wherethe ej are integersand the -yjare elements of S. For each such s let a' denote the product

j jaaie

iny 1

Since all conjugatesof as commute,this productis well defined. in theforma' withs e J9(L). of ti(L) can be written LEMMA 7. Everyelement Two such elementsare equal whenevertheirexponentsare congruentmodulothe + 3W2). ideal 3Ci+ (3C2 + By Lemma 4, every elementof (ai(L) can be writtenas a productof conjugates of ai and a-'. Hence every elementof Es can be writtenin the forma' with s e Jo. Every elementof the ideal 3Qjcan be writtenas a sum of expressionsA-y withd- y (mod aj). For such an exponentwe have ...

a

e

a= iayfay',

fOaif

f

1

=

Hence as e Gj forall s e 3Cj. i we have For the special case j a.=

0a'i(0e)cfy

y' =

(07)aia

=

1,

(mod a3).

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JOHN MILNOR

since both -3'y and ai belong to the commutativegroup Gi. Thereforea = 1 for all s e XiX. y)s with Every elementof 3W2can be writtenas a sum of expressions(3A-y (mod aj) and s e3Cj. For such an exponentwe have a(-i)s

=

ga

(-1

=

)aS-l

0(0-1,y)aiaisf

=

since both a" and f-13belongto Qj . Thereforea" = 1 forall s that a, = as whenever

s _s' (mod 3C + (3C2+

1, e

32

It follows

+ 3C2)).

Note that the factorringJg/3Ci+ (3C2+ * + 3C2) is naturallyisomorphic to (R(Lt). Hence the expressionaX is definedforelementsa of (R(Lt). ofGi(L) can beexpressedin one element THEOREM 5. For an i-triviallinkL, every and only one way as a~ with a e (R(Lt). The group 3(L) and the ringR(LV)willnot be changedifL is replacedby some homotopiclink. Therefore,withoutloss of generality,we may assume that L is a polygonallinkand that its ith componentis a small unknottedcircle.Let N be a small euclidean neighborhoodof this circle,and let G' be the subgroupof G(L) generatedby all closed loops in M - I L I - N. Then G(L) is the free productof G' with the infinitecyclic group generatedby ai . (The group G' is naturallyisomorphicto G(Lt)). FollowingR. H. Fox [3] we introducethe concept of a derivationin the ring JG(L). Define the homomorphism-q:JG -- J by -q(Zejgj)

A derivationin JG is a map D: JG D(a + b) (1)

= Eej.

JG such that =

D(a) + D(b)

D(ab) = D(a)7(b) + aD(b). In particularthe derivationDi is definedby the furtherconditions (3) Di(g) = 0 (2)

(4)

forg EG'

Di(ai) = 1.

Since G(L) is a freeproduct,it is easily shown that thereis a unique function DTwhichsatisfiesthese conditions. It will next be proved that Di(KM) C KMforj 5 i, and that Di(K2) C Ki. For each j 5 i, a jth meridianaj to L can be chosenin the subgroupG' of G. It followsfromLemma 4 that everyelementof Kj is a sum of termsbkjc,where kj = aj - 1. For such a termwe have Di(bkjc) = Di(b)(1(kjc) + bDi(kj)'q(c) + bkj Di(c) = bkj Di(c) e Kj, since 7(k%)= 0 and Di(kj) = 0. ThereforeDi(Kj) C Kj forj 54 i . For any b, c e Kj we have -(c) = 0, hence Di(bc) = Di(b)-q(c) + bDi(c) = bDi(c).

LINK

For j

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GROUPS

i, sinceboth b and Di(c) are elementsofKj, thisprovesthat Di(K) c C Ki . Since Di is an additive homomorphism, it followsthat Di induces a homomorphismof 6R(L) = JG/K2 + ... + K2 into (R(Lt) ; JG/Ki + (K2 + *+ K2). . We Define the exponentialhomomorphism c: (R(L') - > ei(L) by E(o) = must provethat the functionE is one-one.Let X: ai(L) -> GI(L) be the function whichsends each -ye Gi(L) into -y- 1 reducedmodulo 3C2 + * + 3CW2.From thefactthat (a_ - 1)) (a e 3&,it follows 1 1+ that a' henceX -( + a') = XG(a) + X (o'). ai -1 (mod3C+ +3) It will now be provedthat the compositionc X E of the maps 54

K>2. For j = i we stillhave b e Ki, whichprovesthat Di(K)

(R(L)

i(L) X>((L) ---(R(Li)

-E+

is the identitymap of 6R(Lt).Since c X E is an additive homomorphism, it is sufficient to verifythat 6 Xc(-y)= -yforan element-yof (R(LV)whichcome from q(Li). Such an element-yis representedby an elementg of G'. The element X (cY) = -aiy - 1 of (R(L) is then representedby gaigq1- 1 in JG(L), and the elementc XE(-y)of (R(LV)by Di(gaig-F- 1) in JG(L). But Di(gaig-1 - 1) = Di(g)-q(aig-1)+ gDi(ai)-q(g'1) + gaiDi(g-') = g, sinceDi(g) = Di(g-1) = 0 and Di(ai) = -q(gq')= 1. This impliesthat 5X (-y)= -y,hence that c X E is the identitymap of 6R(L). It followsthat the functionE is one-one.Since E maps 6R(Lt)onto (Gi(L) by Lemma 7, thiscompletesthe proof of Theorem 5. Since 5X - is an isomorphismand since E is onto,it followsalso that X is oneone. This fact will be used to prove the followingtheorem. THEOREM 6. If thelink L is trivial,thenthecomposition of thenaturalhomo-> is an into 6R(L). morphisms 61(L) of 9(L) isomorphism q(L) Jq(L) For thecase n = 0 thisassertsthatthenaturalmap F -+ JF is an isomorphism whichis clear. Suppose that the case n - 1 has been proved.Considerthe multiplicativehomomorphisms R i(L)

q(L) 1j

q(Li)

{

P

Tt

(R(Li);

and let y be any elementof q(L) with p(y) = 1. By the inductionhypothesis, p' is an isomorphism. Since p'r(y) = r'p(y) = 1, thisimpliesthat r(-y)= 1, hence that -ye ei(L). Thereforethe functionX(-y)= p(-y) - 1 is defined.Since X is one-oneand since p(-y)- 1 = 0, this impliesthat y = 1, whichcompletesthe proof. Theorems5 and 6 suggestthe importanceof obtainingfurtherinformation about the ring(R(L). We will nextstudythis ringforthe special case of a trivial

188

JOHN MILNOR

link.It is evidentlysufficient to considera linkL forwhicheach componentis a smallunknottedcircle,so that G(L) can be presentedas a freeproduct. Suppose that G(L) is the free product of F with the infinitecyclic groups generatedby a,, *, an. Let ki be the elementai - 1 of JG. By a canonical p _ 0, wordof JG will be meant a product of the formspokj1splkcj,2e ...kljp1ppo, wherethe (piare arbitraryelementsof F and theji are distinctintegersbetween 1 and n. By a canonicalsentenceof JG will be meant a sum or difference of any numberof canonical words. each elementof 6= JG/K1 + *n*+ K2 THEOREM 7. Undertheseconditions, is represented by a unique canonicalsentencein JG. Let 8 be the ringwhoseelementsare canonicalsentencesof JG, with addition definedin the ordinaryway,but withmultiplicationdefinedby the rule .

(Spo kh, SPi

...

khP(P)

* kjqIq) = (do kj ...

0O if hi = ji,

f

(pokh,S(P ...

forsome i, i'

*1 * khPQp 1,t'o)kkj

kjqAlq

otherwise.

Using the distributivelaws, this multiplicationfor words extends to a unique multiplicationforarbitraryelementsof S. The associativelaw for multiplication is clear. A homomorphismI: 8 -* G is obtained by mapping each canonical word (pokej...*p into theactual productspokjl*.*pp reducedmoduloK2 + * + K 2 In orderto veryfythat v is a homomorphism, it is onlynecessaryto note that the identity *.*.p)( o0k j ***q)-q (5pokhl

0

*K + (modK1

+ K2)

holds in JG wheneverhi = ji, forsome i, i'. Theorem7 is clearlyequivalent to the propositionthat v is an isomorphismof 8 onto 61. A homomorphism 0: JG -- 8 is definedby mappingeach elementspof F into the canonicalwordso,and mappingeach meridianai into the canonicalsentence 1 + 1kil. Since G is a freeproduct,and since the element 1 + lkil of 8 has an is well defined.It clearlymaps JG onto S. inverse1 - l1kil, thishomomorphism The compositiont0 ofthesetwo maps is the naturalmap JG -i 61.(It is sufficient to verifythis forelementsof F and forthe meridiansat). Thereforev maps 8 onto 61;and the kernelof v is the image O(K2 + * + K2) of the kernelof I0. Every elementof the ideal Ki is a sum of elementsg(at - l)g'. It followsby direct computationthat every element of O(Ki) is a sum of canonical words whichcontainki . Since the productof two such wordsis zero by the definition of multiplicationin 8, this means that O(KM) = 0. It followsthat the kernel + K2) of v equals zero,whichcompletesthe proofof Theorem7. O(K2 + The precedingtheoremsgive two separate solutionsto the word problemfor to the trivialityproblemforlinks. the link groupof a triviallink,and therefore Using Theorems6 and 7, each elementof 9(L) correspondsto a unique element of (R(L) and therefore to a unique canonicalsentencein JG(L). Two elementsin if iftheircanonicalsentencesare equal. and are equal only 9

LINK

189

GROUPS

A moreefficient solutionin practiceis the following.Let Lnbe a trivialn-link, and let Lr be the sublinkformedby the firstr componentsof L. Then 9 (LrA) can be consideredas a subgroupof 9 (Lr). Furthermoreeach elementof 9 (Lr) can be expresseduniquelyas the productof an elementof 9(LrO) with an elementof 1r(Lr). By Theorem5, each elementof 1r(Lr) can be expresseduniquely in the forma with ar-1 e (R(Lri). It followsby inductionthat each element of ?(Ln) can be expresseduniquelyin the form ?r-l

~'O (pala2

0I

with so e F, ao- (R(Lo) = JF, olE 61(L1),

an

n- 1

X,on-,i

(R(L1_1). Since the word prob-

lems fortheseringsare solved by Theorem7, thisgivesa secondsolutionto the word problemforG(Ln). To concludethis section,the followingtheoremwill be proved. THEOREM 8. If 9(L) is isomorphicto thegroupq(L) ofa triviallink,in an isoto meridians,then morphismwhichpreservestheconjugateclasses corresponding L is trivial. be a meridianand parallel pair in 9(L), and let (ai, ?idi) be Let (ai, ,Oi3li) Then ai is a meridian the pair in 93(L) whichcorrespondunderthe isomorphism. in 9(L) (but fi3i is not necessarilya parallel). We may assume that the link L is polygonalso that lQ(C) has a smoothtubularneighborhood.If this neighbor1 holds. This implies hood is orientable,thenthe identitya.i1 _,iaiO 1ai that aq&-' = 1, hence by Theorem5, that T3irepresentsthe identityelementof Gi(LD). It followsby Theorem6 that $i representsthe identityelementof g(Lt), hence that Aii = di . This impliesthat 3i(i = (ai, hence that L is i-trivial. Since thisis trueforall values ofi, it followsthatL is trivial. 1 would If the neighborhoodwere non-orientable, then the identityadi+l = 1, hence that - 7i representedthe identity hold. This would implythat a elementof 6(Li). Since thisis impossible,the proofis complete. -

5. Almosttriviallinks Let L be an n-linkin euclideanspace such that everypropersublinkis trivial. in 9(L) correspondsto Such linkswillbe called almosttrivial.The nth parallel an(in an elementO3of9(L'). SincethelinkL'-' is trivial,it followsthat 3' Eaa,-(Ln); hence O3 can be writtenin the forma'-, with a- (R(Ln-'s). Since L' is trivial fori < n - 1, it followsthat everywordof the canonicalsentencecorresponding to a containsthe factorki . Thereforea can be writtenuniquelyin the form a- =

,(i1

n-2

n-

n)kil ...kin-2

i. .. wherethe summation extendsoverall permutations

in2

of the integers

1, 2,**, n-2. The (n - 2)! integers U(il . . . in-2 , n-1 n) are homotopyinvariantsof L: to prove that they are not alteredif each ai and By Theorem2 it is sufficient is replacedby a conjugate.For i = n thisis truesinceO3nis an element each ai3t6 ofthe centerofthe group9(Ln). For i < n it can be verifiedby a simplecompu-

190

JOHN MILNOR

tation. On the otherhand the homotopyclass of L is completelyspecifiedby theseintegersu. A homotopiclinkcan be constructedfromany trivial(n - 1)a0link by adjoiningthe loop a' ukil... ki2 . Thus we have obtaineda completeset of homotopyinvariantsfor almost triviallinks in euclideanspace. Every link withtwo componentsis almost trivial.The singleinvariant,u(12) is clearlythe linkingnumber:hence the linkingnumberis a completehomotopy invariantfor2-links.A linkwiththreecomponentsis almost trivialif each pair of componentshas linkingnumberzero. For such linkswe again obtain a single invariant,u(1,23). The case y = 1 turnsout to be the familiarlinkillustratedin Figure 5. Other values of /umay be obtained by traversingone componentof this link,utimes.The case y = 3 is illustratedin Figure 6. For linkswith four componentswe obtain two invariants.For example the link of Figure 4 has in3

23

Fil 5s

1

Fiy i,

2

3

n-2 fiq 7

variants/u(12,34) = ,u(21,34) = 1. As a finalexample,Figure 7 illustratesan n-linkwith invariants y(12... n-2, n-i n) = 1, , n - 2. of 1, in-2, n-1 n) = 0 forall otherpermutations (i... of the linkwill The behaviorof the invariantsu undersimpletransformations now be discussed.If the orientationof one componentofL is reversed,theneach invariantis multipliedby (- 1). (Hence if the orientationsof two components are reversed,thenthe resultinglinkis homotopicto the original).If the orientation ofeuclideanspace is reversed,theneach invariantis multipliedby (-_1)n1* In orderto studythe behaviorofthe invariantsunderpermutationofthe componentsof L, it is convenientto put the precedingdiscussionin the following more symmetricalform.For any two integersr # s the parallel 3' e,(L8) can ar8, wherethe element be expressedin the form s

LINK =

(rs

Z,'(il

191

GROUPS

r s)ki, ...

...*i2X

ki.-2

of cR(L78) is a completeinvariantforL. We mustnowfindthe relationsbetween the invariants7rs fordifferent values ofr and s. If i3: = el( jY7'1) thenit can be shownbya simplegeometricargumentthat 3. = fl(zy7'a~ei-yj). The additivehomomorphism he jj -> ieyj72' ofJq on itself induces an additive homomorphism w: 61-* 61. It followsthat so8,r=

(1)

(ars).

To findthe relationshipbetween0ro,and at,8 it is necessaryto solve the equation /3 = a'" = a"tS. Considerthe homomorphisms 61(Lt's)Et at(Ls) X a(

32> a L(Lt8s)

= at, and -t(at~s) = of Theorem 5. From the identities3t'Xzt(o7,,) obtain

(2)

Oa,8 =

Cr(ar,.)

we

8tXEr(0r,s).

In orderto make specificcomputations,it is necessaryto know the effectsof the additivehomomorphisms c, 8r , and XEron canonicalwordsof 61.These are given as follows w(kil * ki,) = (-1)mkim ***ki,

(3) (4)

br(ki,

if kiilje ki,) = gki. t0otherwise.

im=

r

The functionXsrcan be definedinductivelyby the rules xr(M) = kr and (5)

XEr(kil ki2 ...

kim)= kilr - Tkil

where T

= Xtr(ki2..

kim)

The composition t XEr can now be describedby the rule

(6)

bitXr(kil

...

kim) = -kil ...

ki-

Xor(kip+,...

kim)

where iv = t. Some simpleexampleswill illustratethese formulas.For the case n = 2 the invariant 712 = ,.(12) is an integer. Therefore ,4(21) = w(,4(12)) = ,u(12). Thus the linkingnumberis not changedif the componentsare interchanged. For n = 3 the invariant 023 is of the formjtk1. Therefore 032 =

and

0T13 =

81XE2(iukj) =

bl(/.tklk2-

jk2kj)

=

k2

(jk1) = -k

192

JOHN MILNOR

under permutationsof the Thus the invarianty = y(1, 23) is skew-symmetric components. has the form,.klk2+ 'k2ki, For n = 4 the invariant -34 hence = ,y'k 1k2+

043

=

(0a34)

024

=

62X3(034)

,yk2k1

and k3,-'k1

--kl

k3+ lt'k3ik.

Thus the permutations(34) and (12) replace the invariants (,., ,.') by (u', ju); whilethe permutation(23) replaces (,., ,.') by (-y - j', y'). This behaviorcan also be describedby the followingsymmetryrelations: i2i1) =

u(il i2

u(il i2 , i3i4) +

YU(i3il

u(i4 i3

,

,

i3i4) =

,

i2i4) + ,u(i2 i3

u(i4 il

,

i2i3)

,

il i4)

=

0.

The completeset of symmetryrelationsforarbitraryvalues of n is given by the rules

(7) (8)

,u(il i2 . . . in_22 in-1 in = ,u(in il i2 . . *.in_3 2 in-2in-1) y.(ii

...

ivr ji

. . .

jf.-v2s)

=

(-1)n-Z,.t(hi

hn_2 r

...

s),

wherethe summationis extendedover all sets h1... hn_2of integersformedby i .. i2 in that orderwithjn-v-2 . . . j2j1 in that order.(For examintermeshing ple .u(lr23s) = .u(132rs)+ ,u(312rs)+ ,u(321rs)). These relationsare obtained by manipulationof the formulas(1) through(6). Howeverthe details are too involvedto give here. The precedingmethodscan also be used to definecertain"self linkREMARK. ing numbers",whichare not invariantunderhomotopy.Let L be a trivialpolygonal link,and let L' be the linkobtainedby replacingeach componentof L by a collectionofparallelcomponentshavinglinkingnumberszero (compareSection 3). Suppose that L' is almost trivial.Then the invariants,uof L' may be consideredas describingthe selflinkingof L. For example the link of Figure 3 has the self linkinginvariant,u(I1, 22) = + 1. 6. Arbitrary links in euclidean space The linkgroupofan n-linkin euclideanspace has the presentation (al,

,

an/aiwiaY'wi'

=

1, ...

anwnajwTl

=

1 E

1)

, an whereai is a meridianto theith component,and wherewi is a wordin al, ith parallel i(ii. The symbol "E = 1" dewhichrepresentsthe corresponding notes the set of relationswhichspecifythat conjugatesof each ai commute. to prove this forthe special case of a polygonallink. It is evidentlysufficient

LINK

193

GROUPS

We will startwith the Wirtingerpresentation3 of the fundamentalgroupG(L). This presentationis in termsof generatorsai; i = 1, * , n; j = 1, ,ri to the componentsof the projectionof L on a plane, and relations corresponding (1)

a,

(2)

ax =

=wiJ ai wi a' Wij,r,

to the crossingsof the projection.The followingset of relations corresponding is clearlyequivalent: (1')

a2+l

(2')

1

ax

=

wij wiLY- ... -1

-1

wj ai wij

..*

wi,j

-1 1

.

... wj laiwij ... i wiWi Wi,riWi,ri_1

j > 1. It is natural to try taking the relations(1') as definitionsof the a which can occur is that the word wi,1... wj may contain The only difficulty aj+1 as a factor.But in the linkgroup9(L), such factorsmay be cancelledwithout alteringthe relation(1'). It followsby an obvious double inductionthat all of the ai can be definedin termsof the a'. Furthermorethe relationsof type (1') are completelyexhaustedduringthis process.The relations(2') can be put in the form ai Wia'Wi

=

1,

whereas = a' and Wi = WiJ ... Wi,ri . Since the wordwi = Wil ... Wi,ri clearly (ai, this completesthe proof. representsthe ithparallel f3s if L 2-link is withlinkingnumberju,thenthe parallelsare reprea For example = = and a'; hence9(L) has thepresentation 2 sentedby0, a2 (al

,

a2/ala2aia2 a a2a2

aiM

=

E

=

1).

For a trivial2-linkL', everyelementof G(L') can be put in the canonicalform For thelinkL, theadditionalrelaintegers. whereh,i, j are arbitrary a!2+, = 1 ai = introduces the single relation kd = 0 into tions ala2al a2 a2ala2 the commutator form. means that subgroupof q(L) is a canonical this (This order of i cyclicgroup 1.) A 3-linkmay be specifiedby choosinga conjugateclass of elementsin the link groupofan arbitrary2-link.An elementO'3ofthisconjugateclass has the canoni, wherej is only definedmodulothe linkingnumberpi(12). The cal forma a2 integersh and i are clearlyequal to pz(13)and pz(23).The effectof conjugating /33by a, or a2 is to replace the integerj by j + i or j - h. Hence an arbitrary 3-linkis specifiedby givingthe threelinkingnumberstogetherwiththe number j = ,(123) which need only be definedmodulo the greatestcommondivisor A = (h, i, jt(12)) = (/.z(13),jt(23),pz(12)).On the otherhand the residueclass of pz(123)modulo A is not changedif each meridianand parallel is replaced by a conjugate. Thereforethe three integersjz(12), /.i(13),,p(23) togetherwith the residueclass of pz(123)mod A give a completehomotopyinvariantfor 3-links. I

See for example [41page 44.

194

JOHN MILNOR

For a givenset of threelinkingnumbers,the numberof distinctlinkswhich occur is equal to A, unless A = 0 in whichcase an infinitenumberof distinctlinks occur.For exampleforthe linkingnumbers0, 2, 4 thereare two possible links. The case ,u(123) -0 (mod 2) is illustratedin Figure 8; the case ,u(123) = 1 (mod 2) in Figure 9. The threelinkingnumbersforma completeinvariantby themselvesonly if they are relativelyprime. The procedureswhichhave been used above to classifyspecial types of links can be generalizedto givea veryroughdescriptionforarbitrarylinksin euclidean space. The generaln-linkcan be built up as follows.Start withany properloop 11, and adjoina loop12 = a'P12). Thenadjoin13 = a" l3)aI(23)+a(l"23)k. Continuing by induction,foreach i < n it is necessaryto adjoin a loop 1 = a... a"-'1, whereTj = Ziy(hi ... hr, ji)khl ... khr; the summation being extended over all orderedcollectionshi ... h, of integersbetween1 and j - 1. (To makethisconstructionpreciseit would be necessaryto adopt some conventionas to how the

Fa7 8

Ej