NON COMMUTATIVITY OF THE GROUP OF SELF HOMOTOPY CLASSES OF CLASSICAL SIMPLE LIE GROUPS MARTIN ARKOWITZ, HIDEAKI ~OSHIMA AND JEFFREY STROM Abstract.Let G be a topological group and let [G, G] be the group of homo* *topy classes of maps from G into G. For a large class of simple Lei groups, we prove t* *hat the group [G, G] is nonabelian. For certain Lie groups we show that nil[G, G] 3. 1.Introduction Let G be a compact, connected topological group with multiplication m : G x G* * ! G. There has been considerable work in homotopy theory to determine when m is homo* *topy- commutative [11, 12, 3, 7] that is, when m is homotopic to mop, where mopis def* *ined by mop(g, g0) = m(g0, g). For a compact, connected topological group G (and more g* *enerally, a group-like space) and any space X, the set of homotopy classes [X, G] inherit* *s a group structure from G, and this group is abelian when G is homotopy-commutative. Mor* *eover, it is easily seen that G is homotopy-commutative if and only if [X, G] is abelian * *for all spaces X. The most comprehensive result on the homotopy-commutativity of finite H-spac* *es is the theorem of Hubbuck [7] which asserts that the only compact, connected, non-cont* *ractible topological groups G which are homotopy-commutative are tori S1 x . .x.S1. Thus* * if G is not a torus, then [X, G] is non-abelian for some space X and nil[X, G] is of* * particular interest. (Here nil denotes the nilpotency class of the group , so that nil * * 2 if and only if is not commutative.) In fact, it is a classical result of G. Whitehea* *d [24] that nil[X, G] cat(X), where cat(X) is the reduced Lusternik-Schnirelmann category of X. An important * *special case occurs when X = G, for then [G, G] has a second binary operation obtained * *from composition of homotopy classes. It is a well known algebraic fact that if R is* * a set which satisfies all of the axioms for a ring except commutativity of addition, then a* *ddition is commutative. Now [G, G] satisfies all of the axioms of a ring except commutati* *vity of addition and one distributive law (this is sometimes called a near-ring), so it* * is reasonable to ask if addition in [G, G] is commutative when G is a compact, connected, non-co* *ntractible topological group other than a torus. One example that immediately comes to mi* *nd is [S3, S3] which is commutative, even though, by Hubbuck's theorem, the topologic* *al group S3 is not homotopy-commutative and consequently the group [S3 x S3, S3] is not * *abelian. ___________ 1991 Mathematics Subject Classification. 55Q05. Key words and phrases. Lie group, self map, self homotopy class, nilpotency c* *lass. 1 More generally, we would like to know nil[G, G]. The homotopical nilpotency nil* *(G) of a topological group G has been studied by Zabrodsky, Hopkins, Rao and others [25,* * 6, 22]. Since nil[G, G] is bounded above by nil(G), our results give a lower bound for * *nil(G). We have an upper bound, cat(G), for nil[G, G], but are there lower bounds? The fol* *lowing have been conjectured: Conjecture 1.1.If G is simple, then nil[G, G] rankG. Conjecture 1.2.If G is simple of rank 2, then nil[G, G] 2. If 1.1 is affirmative, then so is 1.2. Notice from Example 1.4 of [20] that the* *se conjectures are false in general without the assumption that G be simple. It is known that cat(* *SU(n)) = rankSU(n) [23], so in this case Conjecture 1.1 asserts that nil[SU(n), SU(n)] =* * rankSU(n). Recently two of us proved 1.2 for SU(n) (n 4) and Sp(n) (n 2) [1]. Recal* *l that any classical compact connected simple Lie group is a quotient of one of the gr* *oups SU(n), Sp(n) or Spin(n) by a central subgroup. The purpose of this note is twofold: (1* *) We extend the results of [A-S] on the noncommutativity of [G, G] for G = Sp(n) and G = SU* *(n) to the groups G = SU(n)=H, G = Sp(n)=H and G = Spin(4n)=H. This is achieved as follows* *: we give a simpler proof of the results of [1] and use Lemma 2.1 below to extend th* *ese results to G=H; we use a different method for Spin(4n)=H. (2) We obtain larger lower bo* *unds for nil[G, G] in some special cases. This requires a detailed study of certain low * *dimensional Lie groups. Theorem 1.3. If the universal covering group of G is SU(n) (n 3), Sp(n) (n * *2) or Spin(4n) (n 2), then nil[G, G] 2. There are results in [16, 19, 20, 21] supporting the above conjectures: (1) n* *il[G, G] equals 1 if G = S3, SO(3); nil[G, G] = 2 if G = SU(3), Sp(2), S3 x S3; nil[G, G] = 3 i* *f G = G2, SU(4), S3 x . .x.S3 (n-times) with n 3; (2) nil[E8, E8] 5, and nil[G, G* *] 3 for G = Spin(7), Spin(8), E6, F4. Conjecture 1.2 remains open in the following case* *s: (1) G = Spin(k)=H with k 6= 4n; (2) E6=Z3 and E7=H. We add to the evidence in suppo* *rt of Conjecture 1.1 by proving Proposition 1.4.nil[SU(4)=H, SU(4)=H] 3 for any central subgroup H of SU(4). Proposition 1.5.If G = S3 x8. .x.S3 (n-times) with n 2 and H is a central sub* *group <2 n = 2 of G, then nil[G=H, G=H] : . 3 n 3 We note that the conjectured inequality (1.1) and Whitehead's inequality, nam* *ely rank(G) nil[G, G] cat(G) can both be strict for a simple Lie group G. For example, for the exceptional L* *ie group G2 it is known that rank(G2) = 2, nil[G2, G2] = 3 and cat(G2) = 4. 2 We denote by eGthe universal covering group of G. In x2 we recall some genera* *l results and fix our terminology. In x3 we prove Theorem 1.3 for eG= SU(n) and in x4 we * *prove Proposition 1.4. We obtain Theorem 1.3 for eG= Sp(n) in x5 and in x6 we establ* *ish Proposition 1.5 and complete the proof of Theorem 1.3. We would like to thank the referee for his comments, which were helpful in im* *proving the exposition. (or this?) We would like to thank the referee for his comments, which led us * *to significantly improve the exposition of this paper. 2. General results In this paper, we do not distinguish notationally between a map and its homot* *opy class. We always let p denote an odd prime. The p-localization of a nilpotent group or* * a nilpotent space is denoted by (p)[5]. We write X 'p Y if X(p)' Y(p). Let ß : eG! eG=H * *be the canonical projection for any central subgroup H of eG. Then ß is a homomorphism* * of Lie groups and hence an H-map. Lemma 2.1. Let Y be a connected, homotopy associative CW H-space. (1) [A, Y ](p)~=[A(p), Y(p)] ~=[A, Y(p)] and nil[A, Y ] nil[A(p), Y(p)]. (2) If the Samelson product is not zero for some ff 2 ßm (Y ) and fi* * 2 ßn(Y ), then nil[Sm x Sn, Y ] 2. (3) If Y 'p X x Sm x Sn and the order of is a multiple of p for some* * ff 2 ßm (Y ) and fi 2 ßn(Y ), then nil[Y, Y ](p) 2. (4) If H is a central subgroup of eGsuch that H(p)= 0, then [Ge=H, eG=H](p)~=* *[Ge, eG](p). Proof.(1) is obvious and (2) is Lemma 6.1 of [1]. Since the localizing map e : Y ! Y(p)is an H-map, we have = e* 6= 0 under the assumption of (3). Hence nil[Sm x Sn, Y(p)] 2 by (2). Then (3) foll* *ows from (1) and the relations: nil[Y, Y ](p)= nil[X x Sm x Sn, Y(p)] nil[Sm x Sn, Y(p)]. Assume H(p)= 0. The map ß(p): eG(p)! (Ge=H)(p)is a weak homotopy equivalence * *so that it is a homotopy equivalence of H-spaces. Hence (4) follows. * * |___| For future reference we record the centers of the classical Lie groups (Theor* *ems 4.10, 4.14 of [18]): p__ Z(SU(n)) = Zn{e2ß -1=nIn}, Z(Sp(n)) = Z2{-In}, 8 >>Z2 Z2 n 0 (mod 4), n 4 >: Z2 n 1 (mod 2), n 3 where Zn{x} is the cyclic group of order n generated by x, and In is the identi* *ty matrix of rank n. Let Bn(p) be the standard S2n+1-bundle over S2n+2p-1[17] such that H*(Bn(p); Zp) = Zp(x2n+1, "1x2n+1), |x2n+1| = 2n + 1. 3 A Lie group G has type (2n1- 1, 2n2- 1, . .,.2nr- 1) if H*(G; Q) = Q(x2n1-1, x2n2-1, . .,.x2nr-1), |xk| = k, n1 n2 . . .nr. Q r The group G is p-regular if and only if G 'p i=1S2ni-1; G is quasi p-regular * *if and Q Q only if G 'p Bki(p) x S2li-1for some {ki} and {li}. If G is simply connect* *ed, then G is p-regular if and only if nr p [13]. If G has no p-torsion, then H** *(G; Zp) = Zp(x2n1-1, . .,.x2nr-1) and "1xs = xt holds (up to non-zero coefficient) if an* *d only if t - s = 2(p - 1) and (s - 1)=2 6 0 (mod p) [17]. 3. Nilpotency of the group [SU(n), SU(n)] Let bk(k) be a generator of ß2k-1(SU(k)) ~=Z for k 2 and write bk(m) = ik,m* **bk(k) for m k, where ik,m: SU(k) ! SU(m) is the inclusion. Then bk(m) is a generato* *r of ß2k-1(SU(m)) ~=Z. Recall from [2], [13], [17] the following. Theorem 3.1. (1) The order of 2 ß2k+2l-2(SU(k + * *l - 1)) is (k + l - 1)!=((k - 1)!(l - 1)!). (2) H*(SU(n); Z) = Z(x3, x5, . .,.x2n-1). (3) SU(n) is p-regular if and only if n p. (4) SU(n) is quasi p-regular if and only if n=2 < p. Corollary 3.2.For max{k, l} n k+l-1, the order of 2 ß2k+2l-2* *(SU(n)) is a multiple of (k + l - 1)!=((k - 1)!(l - 1)!). Proof.This follows from Theorem 3.1(1), the equality in,k+l-1* = * * and the fact that ß2k+2l-2(SU(n)) is finite. * * |___| Hence, if n=2 < p < n, then n-pY SU(n) 'p Bi(p) x (n, p) i=1 where 8 >><* (n + 1)=2 = p < n (n, p) = >S2p-1 (n + 2)=2 = p <.n >:Qp 2i-1 i=n-p+2S (n + 2)=2 < p < n Let n > 5 and choose p such that (n+2)=2 < p < n. Since the order of 2 ß2n+2(SU(n)) is a multiple of p by Corollary 3.2, it follows that nil[SU(n), SU* *(n)] nil[SU(n)(p), SU(n)(p)] 2 by Lemma 2.1. If H is a subgroup of Z(SU(n)), then * *H(p)= 0 since p does not divide n. Therefore nil[SU(n)=H, SU(n)=H] 2 by Lemma 2.1. We have SU(5) '7 S3 x S5 x S7 x S9 by 3.1(3). We proceed as in the preceding * *para- graph. The order of 2 ß14(SU(5)) is a multiple of 7 by Corollary* * 3.2. Hence nil[SU(5), SU(5)] nil[SU(5)(7), SU(5)(7)] 2 by Lemma 2.1. Since H(7)= 0 fo* *r any subgroup H of Z(SU(5)) = Z5, we have nil[SU(5)=H, SU(5)=H] 2 by Lemma 2.1. 4 Similar methods applied to the pairs (SU(3), 3) and (SU(4), 5) yield nil[SU(3* *), SU(3)] 2 and nil[SU(4)=H, SU(4)=H] 2 for any subgroup H of Z(SU(4)). Since Z(SU(3)) ~= Z3, to complete the proof of Theorem 1.3 for SU(n), n 3, * *it suffices to show that nil[PSU(3), PSU(3)] 2, where PSU(3) = SU(3)=Z(SU(3)). * *Let q : SU(3) ! S8 be the quotient map. Consider the following homomorphisms: * q* [PSU(3), PSU(3)]-ß*---[PSU(3), SU(3)]ß----![SU(3), SU(3)]----ß8(SU(3)). Since ß* is injective, it suffices to prove nilIm(ß*) 2. Let _3 : SU(n) ! SU* *(n) and pn : SU(n) ! S(Cn) be maps such that _3(x) = x3 and pn(A) is the first column o* *f the matrix A, where S(Cn) = S2n-1 is the unit sphere in Cn. Note that _3 = (id)3. * * Let f_3: PSU(3) ! SU(3) be a map such that f_3O ß = _3. We have a commutative diagr* *am: SU(3) --p3--!S5 --3'5--!S5b3(3)----!SU(3) ? ? fl ß?y ?yß flfl PSU(3) --fp3--!L5(3)-q---!S5 where L5(3) = S(C3)=Z3 is the mod 3 lens space of dimension 5 and the second ß * *is the projection. Then ß*[f_3, b3(3) O q O ep3] = [(id)3, (b3(3) O p3)3], where [ff, fi] = fffiff-1fi-1, the commutator of ff and fi. In any group, we ha* *ve [x, yz] = [x, y]y[x, z]y-1, [xy, z] = x[y, z]x-1[x, z]. Hence [x, ym ] = [x, y]y[x, ym-1]y-1 = . .=.([x, y]y)m-1[x, y]y-(m-1), [xm , z] = x[xm-1, z]x-1[x, z] = . .=.xm-1[x, z](x-1[x, z])m-1 Set x = id, y = b3(3) O p3 and m = 3. It follows from [21] and [5] (Lemma 6.4 * *p. 91) that [x, y] = q* is a central element; alternatively one could o* *bserve that, since nil[SU(3), SU(3)] cat(SU(3)) = 2 by (1.1) and [23], all commutators are* * central in [SU(3), SU(3)]. In any case, it follows that [x, y3] = [x3, y] = [x, y]3 and ß*[f_3, b3O q O ep3] = [x3, y3] = [x, y]9 = 9q*. Since the order of the last element is 4 by p. 85 of [16], it follows that nilI* *m(ß*) 2 as desired. 4.Proof that nil[SU(4)=H, SU(4)=H] 3 Let H = Zm be a subgroup of Z(SU(4)) = Z4 such that m = 2, 4. We use the foll* *owing notation in which M(n, C) denotes the set of n x n complex matrices and C{1, j}* * is the 5 division ring of quaternions. _ __! c0: Sp(n) ! SU(2n), c0(X + jY ) = X -Y_ , X, Y 2 M(n, C), Y X p0: SU(4) ! SU(4)=c0(Sp(2)) = S5, the projection, _ ! i : SU(3) ! SU(4), i : Sp(1) ! Sp(2), i(A) = A 0 , 0 1 q : SU(4) ! S15, the quotient map, ß5(SU(3)) = Z{b3(3)}, p3*b3(3) = 2'5, ß7(SU(4)) = Z{b4(4)}, p4*b4(4) = 6'* *7. Notice that c0is a monomorphism. In [21], we showed that the order of [id, [b3(4)Op0, b4(4)Op4]] = q*> 2 [SU(4), SU(4)] is a multiple of 3. We shall show that there exist a, b, c 2 [SU* *(4)=H, SU(4)] such that (id)m = a O ß, (b3(4) O p0)m=2 = b O ß, (b4(4) O p4)m = c O ß. If this is true, then ß*[a, [b, c]] = [(id)m , [(b3(4) O p0)m=2, (b4(4) O p4)m * *]]. The homotopy com- * * 3 mutative diagram (4.1) given below, where G = SU(4), implies the last element e* *quals m_2. q*> 6= 0. Thus nil[SU(4)=H, SU(4)] 3 and nil[SU(4)=H, S* *U(4)=H] 3. G --d--! G ^ G ^ G flfl ? fl ?yid^p0^p4 m^(b3(4))m=2^b4(4)m (4.1) G G ^ S5 ^ S7 -(id)---------------!G ^ G ^ G ? x ? q?y ??b2(4)^1^1 ?yC3 S15 ----!'S3 ^ S5 ^ S7-------------------! G > where d is the diagonal map and C3(x ^ y ^ z) = [x, [y, z]]. The existence of a is obvious. For c we use the following commutative diagram: SU(4) --p4--! S7 --m'7--!S7b4(4)----!SU(4) ? ? fl ß?y ?yß flfl SU(4)=H ----! L7(m) ----! S7 fp4 q where the second ß is the projection S(C4) ! S(C4)=Zm = L7(m). Then c := b4(4) * *O q O ep4 satisfies the desired property. In the rest of the proof we show the existence of b. 6 Lemma 4.1. There exists a homeomorphism h : SU(4)=c0(Sp(2)) S5 = S(C3) which makes the following square commutative: SU(4)=c0(Sp(2))-h---!S5 ? ? Lp__-1?y ?yL-1 SU(4)=c0(Sp(2))-h---!S5 Here Lffdenotes the multiplying by ff from the left. By identifying SU(4)=c0(Sp(2)) with S5 by h, we have the following commutativ* *e diagram: SU(4) ----! SU(4)=Z2----! SU(4)=Z4 ? ? ? p0?y ?y^p0 ?yep0 S5 _______ S5 ----! P5 flfl ? fl ?yq S5 ----! S5 2'5 where P5 = S(C3)=Z2 is the real projective space of dimension 5. When m = 2, le* *t b = b3(4) O ^p0. When m = 4, let b = b3(4) O q O ep0. Then these elements satisfy* * the desired properties. Proof of Lemma 4.1. Let i0: SU(2) ! SU(3) be a monomorphism defined by _ _! 0a 0 -_b1 i0 a -b_ = B@0 1 0 CA. b a _ b 0 a Then there exists a smooth map OE which makes the following diagram commutative: 0Oc0 Sp(1)-i---!SU(3) ----! SU(3)=i0c0(Sp(1)) ? ? ? i?y ?yi ?yOE Sp(2)----! SU(4) ----! SU(4)=c0(Sp(2)) c0 0 1 0 As is easily shown, OE is injective. Since the isotropy group at e2 = B@1CA2 S(* *C3) of the 0 standard action of SU(3) on C3 is i0c0(Sp(1)), the map h0: SU(3)=i0c0(Sp(1)) ! * *S(C3) = S5 defined by h0(Ai0c0(Sp(1))) = Ae2 = ~a2, A = (a~1, ~a2, ~a3) is a homeomorphism. Therefore OE is an embedding between smooth manifolds homeo* *morphic to S5. Hence OE is a homeomorphism. 7 Let _ : SU(3)=i0c0(Sp(1)) ! SU(3)=i0c0(Sp(1)) be defined by 0 1 -1 0 0 _(Ai0c0(Sp(1))) = A B@0 -1 0CAi0c0(Sp(1)). 0 0 1 The last element is (-a~1, -a~2, ~a3)i0c0(Sp(1)), where A = (a~1, ~a2, ~a3). Th* *e map _ is well- defined and makes the following diagram commutative: S(C3) --L-1--! S(C3) x x h0?? ??h0 SU(3)=i0c0(Sp(1))_----!SU(3)=i0c0(Sp(1)) ? ? OE?y ?yOE SU(4)=c0(Sp(2))----! SU(4)=c0(Sp(2)) Lp__-1 Then h := h0O OE-1 : SU(4)=c0(Sp(2)) ! S5 satisfies the desired property. * * |___| 5.Nilpotency of the group [Sp(n), Sp(n)] Let ck(k) be a generator of ß4k-1(Sp(k)) ~=Z for k 1 and write ck(m) = ik,m* **bk(k), where ik,m: Sp(k) ! Sp(m) is the inclusion. Then ck(m) is a generator of ß4k-1(* *Sp(m)) ~= Z. Recall from [2], [13], [17] the following. Theorem 5.1. (1) The order of 2 ß4k+4l-2(Sp(k + * *l - 1)) is (2k + 2l - 1)!a(k + l)={(2k - 1)!(2l - 1)!a(k)a(l)} where a(m) is 1 or 2 according as m is odd or even. (2) H*(Sp(n); Z) = Z(x3, x7, . .,.x4n-1). (3) Sp(n) is p-regular if and only if 2n p. (4) Sp(n) is quasi p-regular if and only if n < p. Corollary 5.2.For max{k, l} n k + l - 1 the order of 2 ß4k+4* *l-2(Sp(n)) is a multiple of (2k + 2l - 1)!a(k + l)={(2k - 1)!(2l - 1)!a(k)a(l)}. Proof.This follows from Theorem 5.1(1), the equality in,k+l-1* = * * and the fact that ß4k+4l-2(Sp(n)) is finite. * * |___| If n 4, then there exists p with n + 1 < p < 2n. For any such p n-(p-1)=2Y (p-1)=2Y (5.1) Sp(n) 'p B2i-1(p) x S4i-1 i=1 i=n+1-(p-1)=2 The order of 2 ß4n+2(Sp(n)) is a multiple of p b* *y Corol- lary 5.2. Hence nil[Sp(n), Sp(n)] nil[Sp(n)(p), Sp(n)(p)] 2 by Lemma 2.1. * * Since Z(Sp(n))(p)= 0, we have nil[PSp(n), PSp(n)] 2 by Lemma 2.1, where PSp(n) = Sp(n)=Z(Sp(n)). 8 Q n If n is 2 or 3 and p = 2n + 1, then Sp(n) 'p i=1S4i-1by Theorem 5.1(3) and the order of 2 ß4n+2(Sp(n)) is a multiple of p by Corollary 5.2.* * Hence nil[Sp(n)=H, Sp(n)=H] 2 for every subgroup H of Z(Sp(n)) ~=Z2 by Lemma 2.1. 6. Nilpotency of the group [Spin(n), Spin(n)] By [4], if n 0 (mod 2), then we have Spin(n) 'p Sp(n=2 - 1) x Sn-1. Consider the case: n = 4m with m 2. Choose p such that 2m - 1 < p < 4m - 2.* * Then, by (5.1), we have 2m-(p+1)=2Y (p-1)=2Y Spin(4m) 'p B2i-1(p) x S4i-1x S4m-1 i=1 i=2m-(p-1)=2 of which the last space has S4m-1 x S4m-1 as a direct factor, since 2m - (p - 1* *)=2 m (p - 1)=2. Let H be a central subgroup of Spin(4m). Let @'4m 2 ß4m-1(SO(4m)* *) = ß4m-1(Spin(4m)=H) be the characteristic element of the bundle SO(4m+1) ! S4m. I* *t fol- lows from James [9] (cf. [14], [15]) that the order of <@'4m, @'4m> 2 ß8m-2(SO(* *4m)) = ß8m-2(Spin(4m)=H) is a multiple of p. Hence nil[Spin(4m)=H, Spin(4m)=H] 2 by Lemma 2.1. This completes the proof of Theorem 1.3. Note that Spin(4) = S3 x S3. Let G = S3 x . .x.S3 (n-times) with n 2 and le* *t H be a central subgroup of G. Since ß* : [G=H, G] = [G=H, S3] . . .[G=H, S3] ! [G=* *H, G=H] is an injective homomorphism, we have nil[G=H, G=H] nil[G=H, S3]. Since H(3)=* * 0, it follows from Lemma 2.1 that nil[G=H, S3] nil[G, S3](3). If n = 2, then nil[G,* * S3](3)= 2 by (1.1), p. 176 of [10] and Lemma 2.1 (cf. Proposition 3.1 of [16]), and so nil[G* *=H, G=H] 2. Assume n 3. Let 0 pri G --ß--! S3 x S3 x S3----!S3 be defined by ß0(x1, . .,.xn) = (x1, x2, x3) and pri(x1, x2, x3) = xi for i = 1* *, 2, 3. Since ß0*: [S3xS3xS3, S3](3)! [G, S3](3)is an injective homomorphism, we have nil[G, * *S3](3) nil[S3 x S3 x S3, S3](3). Since the order of [pr1, [pr2, pr3]] 2 [S3 x S3 x S3,* * S3] is 3 by x4 of [20] (cf. x3 of [8]), it follows that nil[S3 x S3 x S3, S3](3) 3. Therefore ni* *l[G, S3](3) 3 and so nil[G=H, G=H] 3 as desired. This completes the proof of Proposition 1.* *5. References [1]M. Arkowitz and J. Strom, Homotopy classes that are trivial mod F, Alg. and* * Geom. Topology 1 (2001), 381-409. [2]R. Bott, A note on the Samelson product in the classical groups, Comment. M* *ath. Helv. 34 (1960), 249-256. [3]W. Browder, Homotopy-commutative H-spaces, Ann. Math. 75 (1962), 283-311. [4]B. Harris, On the homotopy groups of the classical groups, Ann. of Math. 74* * (1961), 407-413. [5]P. Hilton, G. Mislin and J. Roitberg, Localization of nilpotent groups and * *spaces, North-Holland, Am- sterdam (1975). [6]M. Hopkins, Nilpotence and finite H-spaces, Israel J. Math. 66 (1989), 238-* *246. 9 [7]J. Hubbuck, On homotopy commutative H-spaces, Topology 8 (1969), 119-126. [8]I. M. James, Multiplication on spheres II, Trans. Amer. Math. Soc. 84 (1957* *), 545-558. [9]I. M. James, Products on spheres, Mathematika 9 (1959), 1-13. [10]I. M. James, On H-spaces and their homotopy groups, Quart. J. Math. Oxford * *11 (1960), 161-179. [11]I. James, On homotopy commutativity, Topology 6 (1967), 405-410. [12]I. James and E. Thomas, On homotopy-commutativity, Ann. Math. 76 (1962), 9-* *17. [13]R. Kane, The homology of Hopf spaces, North-Holland, Amsterdam (1988). [14]A. T. Lundell, The embeddings O(n) U(n) and U(n) Sp(n), and a Samelson * *product, Mich. Math. J. 13 (1966), 133-145. [15]M. Mahowald, A Samelson product in SO(2n), Bol. Soc. Mat. Mexicana 10 (1965* *), 80-83. [16]M. Mimura and H. ~Oshima, Self homotopy groups of Hopf spaces with at most * *three cells, J. Math. Soc. Japan 51 (1999), 71-92. [17]M. Mimura and H. Toda, Cohomology operations and homotopy of compact Lie gr* *oups-I, Topology 9 (1970), 317-336. [18]M. Mimura and H. Toda, Topology of Lie groups, I, Trans. Math. Monographs v* *ol. 91, Amer. Math. Soc. (1991). [19]H. ~Oshima, Self homotopy set of a Hopf space, Quart. J. Math. Oxford 50 (1* *999), 483-495. [20]H. ~Oshima, Self homotopy group of the exceptional Lie group G2, J. Math. K* *yoto Univ. 40 (2000), 177-184. [21]H. ~Oshima and N. Yagita, Non commutativity of self homotopy groups, Kodai * *Math. J. , to appear. [22]V. Rao, Spin(n) is not homotopy nilpotent for n 7, Topology 32 (1993), 23* *9-249. [23]W. Singhof, On the Lusternik-Schnirelmann category of Lie groups, Math. Z. * *145 (1975), 111-116. [24]G. W. Whitehead, On mappings into group-like spaces. Comment. Math. Helv. 2* *8 (1954), 320-328. [25]A. Zabrodsky, Hopf Spaces, North-Holland Mathematics Studies 22, North-Holl* *and, Amsterdam (1976). Dartmouth College, Hanover, NH 03755, Ibaraki University, Mito, Ibaraki 310-8512, Japan, Dartmouth College, Hanover, NH 03755 E-mail address: martin.arkowitz@dartmouth.edu, ooshima@mito.ipc.ibaraki.ac.jp, jeffrey.strom@dartmouth.edu 10