ODD-PRIMARY HOMOTOPY EXPONENTS OF COMPACT SIMPLE LIE GROUPS DONALD M. DAVIS AND STEPHEN D. THERIAULT Abstract.We note that a recent result of the second author yields upper bounds for odd-primary homotopy exponents of com- pact simple Lie groups which are often quite close to the lower bounds obtained from v1-periodic homotopy theory. 1.Statement of results The homotopy p-exponent of a topological space X, denoted expp(X), is the lar* *gest e such that some homotopy group ssi(X) contains a Z=pe-summand.1 In work dat- ing back to 1989, the first author and collaborators have obtained lower bounds* * for expp(X) for all compact simple Lie groups X and all primes p by using v1-period* *ic homotopy theory. Recently, the second author ([11]) proved a general result, st* *ated here as Lemma 2.1, which can yield upper bounds for homotopy exponents of spaces which map to a sphere. In this paper, we show that these two bounds often lead * *to a quite narrow range of values for expp(X) when p is odd and X is a compact sim* *ple Lie group. Our first new result, which will be proved in Section 2, combines Lemma 2.1 w* *ith a classical result of Borel-Hirzebruch. Theorem 1.1. Let p be odd. a. If n < p2+ p, then expp(SU(n)) n - 1 + p((n - 1)!). ibn-2_c-p+2j b. If n p2+ 1, then expp(SU(n)) n + p - 3 + p-12 . Here and throughout, p(-) denotes the exponent of p in an integer, p is an o* *dd prime, and bxc denotes the integer part of x . All spaces are localized at p. * * It is __________ Date: January 17, 2006. Key words and phrases. Homotopy group, Lie group. 2000 Mathematics Subject Classification: 57T20, 55Q52. 1Some authors (e.g. [11]) say that peis the homotopy p-exponent. 1 2 DONALD M. DAVIS AND STEPHEN D. THERIAULT useful to note the elementary fact that p(m!) = bm_pc + b_m_p2c + . .,. and the well-known fact that p(m!) bm-1_p-1c. Theorem 1.1(a) compares nicely with the following known result. Theorem 1.2. a. ([7, 1.1]) For any prime p, expp(SU(n)) n - 1 + p(bn_pc!). b. ([8, 1.8]) If p is odd, 1 t < p, and tp-t+2 n tp+1, then expp(SU(n))* * n. Thus we have the following corollary, which gives the only values of n > p in* * which the precise value of expp(SU(n)) is known. Corollary 1.3. If p is an odd prime, and n = p+1 or n = 2p, then expp(SU(n)) = * *n. When n = p+1, this was known (although perhaps never published) since, locali* *zed at p, SU(p + 1) ' B(3, 2p + 1) x S5x . .x.S2p-1, the exponent of which follows * *from Proposition 1.4 together with the result of Cohen, Moore, and Neisendorfer ([5]* *) that if p is odd, then expp(S2n+1) = n. Here and throughout, B(2n+1, 2n+1+q) denotes an S2n+1-bundle over S2n+1+qwith attaching map ff1 a generator of ss2n+q(S2n+1)* *, and q = 2p - 2. Note also that the result of [5] implies that if n p, then expp(S* *U(n)) = expp(S3 x . .x.S2n-1) = n - 1. Proposition 1.4. If p is odd, then expp(B(3, 2p + 1)) = p + 1, while if n > 1, * *then n + p - 1 expp(B(2n + 1, 2n + 1 + q)) n + p. Proof.This just combines [3, 1.3] for the lower bound and [11, 2.1] for the upp* *er bound. || Upper and lower bounds for the p-exponents of Sp(n) and Spin(n) can be extrac* *ted from Theorems 1.1 and 1.2 using long-known relationships of their p-localizatio* *ns to that of appropriate SU(m). Indeed, Harris ([9]) showed that there are p-loc* *al equivalences SU(2n) ' Sp(n) x (SU(2n)=Sp(n)) (1.5) Spin(2n + 1) ' Sp(n) (1.6) Spin(2n + 2) ' Spin(2n + 1) x S2n+1. (1.7) Combining this with Theorems 1.1 and 1.2 leads to the following corollary. Corollary 1.8. Let p be odd. ODD-PRIMARY EXPONENTS OF LIE GROUPS 3 (1)expp(Spin(2n + 2)) = expp(Spin(2n + 1)) = expp(Sp(n)) expp(SU(2n)), which is bounded according to Theorem 1.1. (2)expp(Sp(n)) 2n - 1 + p(b2n_pc!). (3)If 1 t < p, and tp - t + 2 2n tp + 1, then expp(Sp(n)) 2n. Proof.The second and third parts of (1) are immediate from (1.6) and (1.5), whi* *le the first equality of (1) follows from (1.7) and the fact that expp(Spin(2n + 1* *)) expp(S2n+1), which is a consequence of part (2) and (1.6). For parts (2) and (3* *), we need to know that the homotopy classes yielding the lower bounds for expp(SU(2n* *)) given in Theorem 1.2 come from its Sp(n) factor in (1.5). To see this, we first* * note that in [2, 1.2] it was proved that, if p is odd and k is odd, then v-11ss2k(Sp(n); p) v-11ss2k(SU(2n); p).(1.9) These denote the p-primary v1-periodic homotopy groups, which appear as summands of actual homotopy groups. The proofs of [7, 1.1] and [8, 1.8], which yielded T* *heorem 1.2, were obtained by computing certain groups v-11ss2k(SU(n); p) with k n-1 * *mod 2. When applied to SU(2n), these groups are in v-11ss2k(SU(2n); p) with k odd, * *and so by (1.9) they appear in the Sp(n) factor. || For all (X, p) with X an exceptional Lie group and p an odd prime, except (E7* *, 3) and (E8, 3), we can make an excellent comparison of bounds for expp(X) using re* *sults in the literature. We use splittings of the torsion-free cases tabulated in [3,* * 1.1], but known much earlier.([10]) In Table 1, we list the range of possible values of e* *xpp(X) when the precise value is not known. We also list the factor in the product dec* *om- position which accounts for the exponent. Finally, in cases in which the expon* *ent bounds do not follow from results already discussed, we provide references. He* *re B(n1, . .,.nr) denotes a space built from fibrations involving p-local spheres * *of the indicated dimensions and equivalent to a factor in a p-localizaton of a special* * unitary group or quotient of same. Also, B2(3, 11) denotes a sphere-bundle with attach* *ing map ff2, and W denotes a space constructed by Wilkerson and shown in [12, 1.1] * *to fit into a fibration K5 ! B(27, 35) ! W . Finally, K3 and K5 denotes Harper's space as described in [1] and [11]. 4 DONALD M. DAVIS AND STEPHEN D. THERIAULT Theorem 1.10. The homotopy p-exponents of exceptional Lie groups are as in Table 1. Table 1. Homotopy exponents of exceptional Lie groups ___X______p____||expp(X)_________Factor___________Reference_____ G2 3 || 6 B2(3, 11) [3, 1.3],[11, 2.2] G2 5 || 6 B(3, 11) __G2_____>_5___||___5_____________S11___________________________ F4, E6 3 || 12 K3 [1, 1.6], [11, 1.2] F4, E6 5, 7 ||11, 12 B(23 - q, 23) F4, E6 11 || 12 B(3, 23) _F4,_E6__>_11__||___11____________S23___________________________ E7 5 ||18, 19, 20 B(3, 11, 19, 27, 35)factor of SU(18) E7 7 ||17, 18, 19 B(11, 23, 35) factor of SU(18) E7 11, 13 ||17, 18 B(35 - q, 35) E7 17 || 18 B(3, 35) ___E7____>_17__||___17____________S35___________________________ E8 5 || 30, 31 W [6, 1.1],[12, 1.2] E8 7 ||29, 30, 31, 32B(23, 35, 47,[59)3, 1.4],Proposition 2.3 E8 11 - 23 ||29, 30 B(59 - q, 59) E8 29 || 30 B(3, 59) E8 > 29 || 29 S59 2. Proof of Theorem 1.1 In [11, Lemma 2.2], the second author proved the following result. Lemma 2.1. ([11, 2.2,2.3]) Suppose there is a homotopy fibration q 2n+1 F ! E -!S q* 2n+1 where E is simply-connected or an H-space and | coker(ss2n+1(E) -! ss2n+1(S * *))| pr. Then expp(E) r + max(expp(F ), n). In [11, 2.2], it was required that E be an H-space, but [11, 2.3] noted that * *if E is not an H-space, the desired conclusion can be obtained by applying the loop-spa* *ce ODD-PRIMARY EXPONENTS OF LIE GROUPS 5 functor to the fibration. We require E to be simply-connected so that we do not* * loop away a large fundamental group. We now use this lemma to prove Theorem 1.1. Proof of Theorem 1.1.The proof is by induction on n. Let the odd prime p be im- plicit, and let SU0(n) denote the factor in the p-local product decomposition (* *[10]) of SU(n) which is built from spheres of dimension congruent to 2n - 1 mod q. By the induction hypothesis, the exponents of the other factors are the asserted amo* *unt. We will apply Lemma 2.1 to the fibration q 2n-1 SU0(n - p + 1) ! SU0(n) -!S . q* 2n-1 In order to determine | coker(ss2n-1(SU0(n)) -! ss2n-1(S ))|, we use the cla* *ssical result of Borel and Hirzebruch ([4, 26.7]) that ss2n-2(SU(n - 1)) Z=(n - 1)!. When localized at p, it is clear that its p-component Z=p p((n-1)!)must come fr* *om the SU0(n-p+1)-factor in the product decomposition of SU(n-1), since ss2n-2(SU(n-1)) is built from the classes ffi2 ss2n-2(S2n-1-iq)(p). Thus ss2n-2(SU0(n - p + 1)) Z=p p((n-1)!), and the exact sequence q* 2n-1 0 ss2n-1(SU0(n)) -! ss2n-1(S ) ! ss2n-2(SU (n - p + 1)) implies p(| coker(q*)|) p((n - 1)!). (2.2) (a.) By the induction hypothesis, expp(SU0(n - p + 1)) n - p + p((n - p)!)* *. By hypothesis, n-p < p2and hence p((n-p)!) p-1. Thus expp(SU0(n-p+1)) n-1, and so by 2.1 and (2.2) expp(SU0(n)) p(| coker(q*)|) + n - 1 p((n - 1)!) + n - 1, as claimed. (b.) By (a), part (b) is true if p2 + 1 n p2 + p - 1. Let n p2 + p, a* *nd assume the theorem is true for SU0(n-p+1). Then by Lemma 2.1 and the induction hypothesis _ n-p-1 ! b_____c - p + 2 expp(SU0(n)) ((n - 1)!) + n - p + 1 + p - 3 + p-1 . 2 6 DONALD M. DAVIS AND STEPHEN D. THERIAULT Note that even if expp(SU0(n - p + 1)) happened to be less than n - 1, our upper bound for it is n-1, and so this bound for expp(SU0(n)) is still a correct de* *duction from 2.1. Since p((n - 1)!) bn-2_p-1c, we obtain $ % _ ! n - 2 bn-2_c - p + 1 expp(SU0(n) _____ + n - 2 + p-1 p - 1 2 $ % _ ! _ ! n - 2 bn-2_c - p + 2 bn-2_c - p + 1 = _____ + n - 2 + p-1 - p-1 p - 1 2 1 _ ! bn-2_c - p + 2 = n + p - 3 + p-1 , 2 as desired. || The result in part (b) could be improved somewhat by a more delicate numerical argument. Part (b) of the following result was used in Table 1. Proposition 2.3. Let p = 7. a. exp7(B(23, 35, 47)) 25. b. exp7(B(23, 35, 47, 59)) 32. Proof.The thing that makes this require special attention is that these spaces * *are not a factor of an SU(n), because they do not contain an S11. There are fibrati* *ons B(23, 35) ! B(23, 35, 47) ! S47 and B(23, 35, 47) ! B(23, 35, 47, 59) ! S59. Since, localized at 7, ss46(S23) ss46(S35) Z=7, we have |ss46(B(23, 35))| * * 72, and similarly |ss58(B(23, 35, 47))| 73. (In fact, it is easily seen that these a* *re cyclic groups of the indicated order.) Using 2.1 and that exp7(B(23, 35)) 18 by 1.4,* * we obtain exp7(B(23, 35, 47)) 2 + max(18, 23) = 25, and then exp7(B(23, 35, 47, 59)) 3 + max(25, 29) = 32. ODD-PRIMARY EXPONENTS OF LIE GROUPS 7 || References [1]M. Bendersky and D. M. Davis, 3-primary v1-periodic homotopy groups of F4 and E6, Trans Amer Math Soc 344 (1994) 291-306. [2]________, The unstable Novikov spectral sequence for Sp(n), and the power series sinh-1(x), London Math Soc Lecture Notes 176 (1992) 73-86. [3]M. Bendersky, D. M. Davis, and M. Mimura, v1-periodic homotopy groups of exceptional Lie groups: torsion-free cases, Trans Amer Math Soc 333 (1992) 115-135. [4]A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces, II, American Jour Math 81 (1959) 313-382. [5]F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, The double suspension and exponents of the homotopy groups of spheres, Annals of Math 110 (1979) 549-565. [6]D. M. Davis, From representation theory to homotopy groups, Mem Amer Math Soc 759 (2002). [7]D. M. Davis and Z. W. Sun, A number-theoretic approach to homotopy expo- nents of SU(n), submitted, 2005. [8]D. M. Davis and H. Yang, Tractable formulas for v1-periodic homotopy groups of SU(n) when n p2- p + 1, Forum Math 8 (1996) 585-619. [9]B. Harris, On the homotopy groups of the classical groups, Annals of Math 74 (1961) 407-413. [10]M. Mimura, G. Nishida, and H. Toda, Mod p decomposition of compact Lie groups, Publ RIMS Kyoto Univ 13 (1977) 627-680. [11]S. D. Theriault, Homotopy exponents of Harper's spaces, Jour Math Kyoto Univ (2003). [12]S. D. Theriault, The 5-primary homotopy exponent of the exceptional Lie gro* *up E8, Jour Math Kyoto Univ 44 (2004) 569-593. Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA E-mail address: dmd1@lehigh.edu Department of Mathematical Sciences, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom E-mail address: s.theriault@maths.abdn.ac.uk