ARKOWITZ, M.

CURJEL, C. R.

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SOME PROPERTIES OF THE EXOTICMULTIPLICATIONS ON THE THREE-SPHEREBy M. ARKOWITZ and C. R. CURJELf[Received 14 June 1968]1. IntroductionIT is well known that up to homotopy the three-sphere S3 admits twelvedistinct multiplications or Z/-space structures [see (2), or Lemma 2].In this note we establish some properties of these different multiplica-tions on S3 by elementary group-theoretic arguments.We work with path-connected topologies! spaces with base point andhomotopy classes of base-point-preserving maps. As usual [A, E] standsfor the collection of such homotopy classes of A into B. For conveniencewe do not distinguish notationally between a map and its homotopyclass. If (0,m) and (H,n) are two iZ-spaces then a map/: (<•?, m)->- (H,n)is called an H-map if n(fxf) =fme[OxO, H]. If the #-space (0, m)has a two-sided homotopy inverse then a (two-fold) commutator map<f>:GxQ^>-G can be defined by <f>(x,y) = (xy^x'^y-1), where themultiplication is denoted by juxtaposition and the homotopy inverseby the exponent ' — 1 ' . A k-fold commutator map <f>k: G* -> 0 is theninductively denned by <j>x = the identity map 1, <f>t = <f>, andThe homotopical nilpotency of (0, m) is said to be equal to I if <£j+1 isnullhomotopic and <^  is not. We write nil((?,m) = I for i ^ / O and<£j+1 = 0, where 0 is the constant map. In the present paper we deter-mine the i7-maps and the homotopical nilpotency of S3 for all possiblemultiplications.If m0 is the standard multiplication on 8s then it is shown in Lemma 2that the twelve multiplications on 8s can be written asHere t = 0, 1,...,11, </> denotes the commutator map, and the exponentand juxtaposition are taken with respect to the group structure in[S3 X (S3, S3] induced by m0. Finally we denote by N the map of <S^  intoitself of degree N.•f The first author was supported by a Dartmouth College Faculty Fellowshipand the second author partially by the National Science Foundation (NSF-GP5874).Quart. J. Math. Oxford (2), 20 (1969), 171-6.172 M. ARKOWITZ AND C. R. CURJELTHEOBJEM A. The map N: (S3, mr) -*• (S3, m,) of degree N is an H-mapif and only if N*(2t+1) = N(2r+1) mod 24.We also proveTHEOREM B. If r = 1, 4, 7, or 10, then nil^.m,.) = 2. If r = 0, 2,3, 5, 6, 8, 9, or 11, *Aen nil^S3,»»,.) = 3.In the case of the standard multiplication (r = t = 0) Theorem Awas proved by us in (1). S. Y. Husseini informs us that he and J. Stasheffhave obtained Theorem A by different methods (unpublished). TheoremB was proved by G. J. Porter for m0 in (5). Some of the steps in theproof of Theorem B can be found in (3), 175-6 and (4), § 10, where theyare established in another way.Our method of proof consists of examining the group-theoreticproperties of [S3 x 8s, S3]. With this method it is also possible toretrieve rather easily some known theorems on the multiplications of 8*(see Remark 1). A similar discussion of the group [S7 X S7, S7] leads toresults on the multiplications of S7. I t would be interesting in ouropinion to extend our theorems to other i7-spaces.2. Proof of Theorem AWe write S for S3 and consider the set [A, S] for any space A as agroup, the group operation being induced from the standard multiplica-tion m0 of S. We write this group multiplicatively even though we denotethe identity of the group by 0. The commutator of two elements a and bis (a, b) = aba-1b-1.LEMMA 1. The group [SxS,S] is nilpotent of class < 2, and so forany a, b e [Sx S, S] and integer n > 0,(ab)n = (b, a)*"<n-%n&n.Proof. The first assertion is a result of G. W. Whitehead [see (6),Lemma 2.14 and § 3], and the second is an identity which is known tohold in a group of nilpotency class < 2.LEMMA 2. The twelve homotopy classes of multiplications on S can bewritten as m, = m ^ e [Sx S, S], t = 0,..., 11, where <f>e[SxS,8] isthe commutator map. Moreover, <j> has order 12.Proof. Consider the cofibre sequenceSvS -?-+ Sx S-?-+ S A S « S«.ON EXOTIC MULTIPLICATIONS ON THE THREE-SPHERE 173This gives a long exact sequence— [2(5 x S), S] -5£ [2(5 v 5), S] —+ [S*, 5]-^> [5x S, 5] - C [5 v 5, 5],where 2 denotes reduced suspension. It is well known and easily provedthat 2 j* and j * are epimorphisms. Thus we get a short exact sequenceof groups and homomorphisms0 > ne(S) JU[SxS,S]-£+ TT3(5) ® ns(S) • 0.A multiplication m on S is an element of [ 5 x 5 , 5 ] such thatj*(m) = 1 © 1,where 1 e 773(5) is the homotopy clasB of the identity map. Thus thecollection of multiplications is just the ooset of Kerj* which contains m0.Hence it suffices to show that Im q* is a cyclic group of order 12 generatedby (f>. Now it is known that TTS(5) is a cyclic group of order 12 generatedby <1,1), the Samelson product of 1 e TT3(5) with itself [see (3) 176].But by the definition of the Samelson product q*(l, 1) = (f>. This com-pletes the proof of the lemma.Proof proper of Theorem A. First note that m0 = p1pt&nd<f> = (p^Pi)in [5x5,5] where Pie[SxS,S] is the projection on to the ith factor.Assume that N > 0. Thenm,o(NxN) =since the commutator is biadditive in a group of nilpotency ^ 2. Onthe other hand,W, by Lemma 1,= 0= {P»Pi)iIiVt-'°P?P?<l>rN, by Lemma 1,Thus N is an £?-map if and only if ^ = ^ - ^ ( ^ - D . Since <f> hasorder 12 by Lemma 2, this is the case if and only if2V*(2t+l) = N(2r+l) mod24.IS N < 0 write N = (—N)(—1) and repeat the above argument.174 M. ARKOWITZ AND C. R. CURJEL3. Proof of Theorem BWe begin with a simple lemma on homotopy inverses.LEMMA 3. The map — le[8,S]of degree —\ is a two-sided identityfor every multiplication on 8.Proof. A left inverse vt e [S, S] for m, is characterized by the equationvt. 1 = 0 in [S, 8], where the dot ' . ' is the operation in [S, 8~\ inducedby TO,. But this operation coincides with the usual group operation inthe homotopy group [S, S] = ns(S). Thus vt= — 1 is the uniquesolution of the equation. Similarly the right inveroe for mt is — 1.Next we introduce the following notation: $.r) is the jfc-fold commutatormap with respect to the multiplication mr. It is well known that eachcj>kT) induces a unique homotopy class </4r): Ak8 -*• 8 with t/ikT) o qk = <j>kr\where AkS is the smashed product S A ••• A S (k times) « S3* and qk isthe projection of SX...XS on to AkS. For <xeirp(S) and ^ETTQ{S)denote the Samelson product relative to m, by (a, j3)r e TP+<1(S). Wewrite <f>k for <j>k0), <£(r) for «/4r), fa for $?>, ^  for ^{\ and («,/3) for <a,/3)0.LEMMA 4. (a) <jP> = <f>»+1 e [Sx 8,8];(c) <«,j8>r = (2r+l)<«,/3)/or a €TTP(8) and p e 7rg(S).Proof. Clearly (6) implies (c) since (a,/?)r = 0<r)(a£ A^3). Also (6) is animmediate consequence of (a). Thus it suffices to prove (a): Write ' . 'for the operation in [S X S, S] obtained from mr and use juxtapositionfor the standard multiplication in [SxS, S]. Then, by Lemma 3,where pt and p2 are the two projections SxS -*• S and the exponent' — 1 ' denotes the inverse in [8xS, S] with respect to the standardmultiplication in [SxS, 8]. Then= P1P2 <f>rpr1Pt1<f>r(PiP2<i>r<pr1pi1<f>r)rBut by repeatedly using the biadditivity of commutators in the groupf S X 8, S] of nilpotency class < 2 one easily sees that(PiP2<f>r,pr1pi~1<i'r) = 0.This completes the proof.ON EXOTIC MULTIPLICATIONS ON THE THREE-SPHERE 176Proof proper of Theorem B. We first note that $.r) = 0 if and only if$f> = 0. Now $,r> = 0« = <1, l ) r = (2r+l)( l , 1) for 1 e irs(S). But(1,1) £ TTt(S) is an element of order 12. Therefore,</#•> # 0 for all r.Next, ^ ) = « l l ) l )But ((1,1), 1) € iT9(/8f3) « Z8 is an element of order 3 [see for instance(2) § 3]. Therefore ip3r) (and consequently </4r)) is trivial if and only ifr = 1, 4, 7, or 10. Thus nil(»S>mr) = 2 for r = 1, 4, 7, or 10. Finallynote that «<1,1 >r, l ) r , l ) r is a multiple of <«1,1), 1), 1). To completethe proof of Theorem B it suffices to show that this latter Samelsonproduct is trivial.LEMMA 5. If 1 £ TT3(S) is the homotopy class of the identity map thenProof. Let 6 = «<1,1), 1), 1) and v = ((1,1), <1,1». First notethat 3((1,1), 1) = 0 by the Jacobi identity and so 3«a,/3),y) = 0 forany elements a, /?, y. Thus 3TJ = 0 and 30 = 0. But 77 = —77 by anti-commutativity, and so 2rj = 0. Hence it follows that 77 = 0. Now applythe Jacobi identity to the elements (1,1), 1, and 1 to obtain 20+ r\ = 0.Thus 26 = 0. But we observed earlier that 3d = 0. Therefore 6 = 0.Remark 1. I t is possible with our methods to retrieve some knowntheorems on the multiplications of 8. We can prove that all multiplica-tions on S satisfy the Moufang identity [see (4) § 9] and can determinewhich of them are homotopy-associative [see (2) Theorem 1.3]. Theseresults require an examination of the group [SxSx S, S], which isnilpotent of class < 3. For example, mr is homotopy-assooiatdve if andonly if <h. {q2. qa) = (q±. g2). q3 in [S X 8 X S, S], where qt is the projectionon to the tth factor. This equation, together with facts on commuta-tors in groups of nilpotency class < 3, reduces to a simple identity in[8 X S X S, S] in terms of m^. We note that the multiplications on 8which are homotopy-associative are precisely those whioh in Theorem Bwere shown to be of homotopical nilpotency 3.Remark 2. The definition of homotopical nilpotency for a non-homotopy-associative fl-space depended on the bracketing in thedefinition of a commutator map. It follows from our proof of Theorem Bthat the homotopical nilpotency with respect to one bracketing is thesame as that with respect to any other bracketing of the commutator.176 ON EXOTIC MULTIPLICATIONS ON THE T H R E E - S P H E R ER E F E R E N C E S1. M. Arkowitz and C. R. Curjel, 'On maps of H-spaoee', Topology, 6 (1967)137-18.2. I. M. James, 'Multiplications on spheres (II)', Trans. Amer. Math. Soc. 84(1957) 545-68.3. 'On ff-spaces and their homotopy groups', Quart. J. Math. (Oxford) (2)11 (1960) 161-79.4. C. W. Norman, 'Homotopy loops', Topology, 2 (1963) 23-43.5. G. J. Porter, 'Homotopical nilpotency of S*', Proc. Amer. Math. Soc. 16 (1964)681-2.6. Q. W. Whitehead, 'On mappings into group-like spaces', Oortvm. Math. Helv.28 (1964) 320-8.Dartmouth CollegeHanover, N.H., andMathematical InstituteOxfordUniversity of WashingtonSeattle, andForschungsinstitut fur Mathematik, E.T.H.Zurich

SOME PROPERTIES OF THE EXOTIC MULTIPLICATIONS ON THE THREE-SPHERE

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