DONALD M. DAVIS. 1. Main result

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1 v 1 -PERIODIC 2-EXPONENTS OF SU(2 e ) AND SU(2 e + 1) DONALD M. DAVIS Abstract. We determne precsely the largest v 1 -perodc homotopy groups of SU(2 e ) and SU(2 e +1). Ths gves new results about the largest actual homotopy groups of these spaces. Our proof reles on results about 2-dvsblty of restrcted sums of bnomal coeffcents tmes powers proved by the author n a companon paper. 1. Man result The 2-prmary v 1 -perodc homotopy groups, v 1 1 π (X), of a topologcal space X are a localzaton of a frst approxmaton to ts 2-prmary homotopy groups. They are roughly the porton of π (X) detected by 2-local K-theory.([2]) If X s a sphere or compact Le group, each v 1 -perodc homotopy group of X s a drect summand of some actual homotopy group of X.([6]) Let T j (k) = odd ( j ) k denote one famly of partal Strlng numbers. In [5], the author obtaned several results about ν(t j (k)), where ν(n) denotes the exponent of 2 n n. Some of those wll be used n ths paper, and wll be restated as needed. Let e(k, n) = mn(ν(t j (k)) : j n). It was proved n [1, 1.1] (see also [7, 1.4]) that v 1 1 π 2k (SU(n)) s somorphc to Z/2 e(k,n) ɛ drect sum wth possbly one or two Z/2 s. Here ɛ = 0 or 1, and ɛ = 0 f n s odd or f k n 1 mod 4, whch are the only cases requred here. Date: September 22, Key words and phrases. homotopy groups, specal untary groups, exponents, v 1 -perodcty Mathematcs Subject Classfcaton: 55Q52,11B73. 1

2 2 DONALD M. DAVIS Let s(n) = n 1 + ν([n/2]!). It was proved n [8] that e(n 1, n) s(n). Let e(n) = max(e(k, n) : k Z). Thus e(n) s what we mght call the v 1 -perodc 2-exponent of SU(n). Then clearly (1.1) s(n) e(n 1, n) e(n), and calculatons suggest that both of these nequaltes are usually qute close to beng equaltes. In [4, p.22], a table s gven comparng the numbers n (1.1) for n 38. Our man theorem verfes a conjecture of [4] regardng the values n (1.1) when n = 2 e or 2 e + 1. Theorem 1.2. a. If e 3, then e(k, 2 e ) 2 e + 2 e 1 1 wth equalty occurrng ff k 2 e 1 mod 2 2e 1 +e 1. b. If e 2, then e(k, 2 e + 1) 2 e + 2 e 1 wth equalty occurrng ff k 2 e + 2 2e 1 +e 1 mod 2 2e 1 +e. Thus the values n (1.1) for n = 2 e and 2 e + 1 are as n Table 1.3. Table 1.3. Comparson of values n s(n) e(n 1, n) e(n) 2 e 2 e + 2 e e + 2 e e + 2 e e e + 2 e e + 2 e e + 2 e 1 Note that e(n) exceeds s(n) by 1 n both cases, but for dfferent reasons. When n = 2 e, the largest value occurs for k = n 1, but s 1 larger than the general bound establshed n [8]. When n = 2 e + 1, the general bound for e(n 1, n) s sharp, but a larger group occurs when n 1 s altered n a specfc way. The numbers e(n) are nterestng, as they gve what are probably the largest 2-exponents n π (SU(n)), and ths s the frst tme that nfnte famles of these numbers have been computed precsely. The homotopy 2-exponent of a topologcal space X, denoted exp 2 (X), s the largest k such that π (X) contans an element of order 2 k. An mmedate corollary of Theorem 1.2 s

3 v 1-PERIODIC 2-EXPONENTS OF SU(2 e ) AND SU(2 e + 1) 3 Corollary 1.4. For ɛ {0, 1} and 2 e + ɛ 5, exp 2 (SU(2 e + ɛ)) 2 e + 2 e ɛ. Ths result s 1 stronger than the result noted n [8, Thm 1.1]. Theorem 1.2 s mpled by the followng two results. The frst wll be proved n Secton 2. The second s [5, Thm 1.1]. Theorem 1.5. Let e 3.. If ν(k) e 1, then ν(t 2 e(k)) = 2 e 1.. If j 2 e and ν(k (2 e 1)) 2 e 1 + e 1, then ν(t j (k)) 2 e + 2 e If j 2 e + 1 and ν(k 2 e ) = 2 e 1 + e 1, then ν(t j (k)) 2 e + 2 e 1. Theorem 1.6. ([5, 1.1]) Let e 2, n = 2 e + 1 or 2 e + 2, and 1 2 e 1. There s a 2-adc nteger x,n such that for all ntegers x ν(t n (2 e 1 x + 2 e 1 + )) = ν(x x,n ) + n 2. Moreover and ν(x,2 e +1) ν(x,2 e +2) { = f = 2 e 2 or 2 e 1 > otherwse. = 1 f 1 2 e 2 = f 2 e 2 < < 2 e 1 > f = 2 e 1. Regardng small values of e: [7, 8] and [5, Table 1.3] make t clear that the results stated n ths secton for T n ( ), e(, n) and SU(n) are vald for small values of n 5 but not for n < 5. Proof that Theorems 1.5 and 1.6 mply Theorem 1.2. For part (a): Let k 2 e 1 mod 2 2e 1 +e 1. Theorem 1.5() mples e(k, 2 e ) 2 e +2 e 1 1, and 1.6 wth n = 2 e +2, = 2 e 1 1, and ν(x) 2 e 1 mples that equalty s obtaned for such k. To see that e(k, 2 e ) < 2 e + 2 e 1 1 f k 2 e 1 mod 2 2e 1 +e 1, we wrte k = + 2 e 1 x + 2 e 1 wth 1 2 e 1. We must show that for each k there exsts some j 2 e for whch ν(t j (k)) < 2 e + 2 e 1 1.

4 4 DONALD M. DAVIS If = 2 e 1, we use 1.5(). If = 2 e 2, we use 1.6 wth n = 2 e + 1 f ν(x) < 2 e 2 and wth n = 2 e + 2 f ν(x) 2 e 2. For other values of, we use 1.6 wth n = 2 e + 1 f ν(x) and wth n = 2 e + 2 f ν(x) >, except n the excluded case = 2 e 1 1 and ν(x) >. For part (b): Let k 2 e + 2 2e 1 +e 1 mod 2 2e 1 +e. Theorem 1.5() mples e(k, 2 e + 1) 2 e + 2 e 1, and 1.6 wth n = 2 e + 2, = 2 e 1, and ν(x) = 2 e 1 mples that equalty s obtaned for such k. To see that e(k, 2 e ) < 2 e + 2 e 1 f k 2 e + 2 2e 1 +e 1 mod 2 2e 1 +e, we wrte k = + 2 e 1 x + 2 e 1 wth 1 2 e 1. If = 2 e 1, we use 1.6 wth n = 2 e +1 unless ν(x) = 2 e 1, whch case s excluded. If = 2 e 2, we use 1.6 wth n = 2 e + 2 f ν(x) = 2 e 2 and wth n = 2 e + 1 otherwse. If 1 < 2 e 2, we use 1.6 wth n = 2 e + 1 f ν(x) = 1 and wth n = 2 e + 2 otherwse. If 2 e 2 < < 2 e 1, we use 1.6 wth n = 2 e + 1 f ν(x) = and wth n = 2 e + 2 otherwse. The proof does not make t transparent why the largest value of e(k, n) occurs when k = n 1 f n = 2 e but not f n = 2 e + 1. The followng example may shed some lght. We consder the llustratve case e = 4. We wsh to see why e(k, 16) 23 wth equalty ff k 15 mod 2 11, whle e(k, 17) 24 wth equalty ff k mod Tables 1.7 and 1.8 gve relevant values of ν(t j (k)).

5 v 1-PERIODIC 2-EXPONENTS OF SU(2 e ) AND SU(2 e + 1) 5 Table 1.7. Values of ν(t j (k)) relevant to e(k, 16) j ν(k 15) Table 1.8. Values of ν(t j (k)) relevant to e(k, 17) j ν(k 16) The values e(k, 16) and e(k, 17) are the smallest entry n a row, and are lsted n boldface. The tables only nclude values of k for whch ν(k (n 1)) s rather large, as these gve the largest values of ν(t j (k)). Larger values of j than those tabulated wll gve larger values of ν(t j (n)). Note how each column has the same general form, levelng off after a jump. Ths reflects the ν(x x,n ) n Theorem 1.6. The prevalence of ths behavor s the central theme of [5]. The phenomenon whch we wsh to llumnate here s how the bold values ncrease steadly untl they level off n Table 1.7, whle n Table 1.8 they jump to a larger value before levelng off. Ths s a consequence of the synchroncty of where the jumps occur n columns 17 and 18 of the two tables. [5]. 2. Proof of Theorem 1.5 In ths secton, we prove Theorem 1.5. The proof uses the followng results from

6 6 DONALD M. DAVIS Proposton 2.1. ([8, 3.4] or [5, 2.1]) For any nonnegatve ntegers n and k, ν ( ( n 2+1) ) k ν([n/2]!). The next result s a refnement of Proposton 2.1. Here and throughout, S(n, k) denote Strlng numbers of the second knd. Proposton 2.2. ([5, 2.3]) Mod 4 S(k, n) + 2nS(k, n 1) ɛ = 0, b = 0 ( 1 2n+ɛ n! 2+b) k (2n + 1)S(k, n) + 2(n + 1)S(k, n 1) ɛ = 1, b = 0 2nS(k, n 1) ɛ = 0, b = 1 S(k, n) + 2(n + 1)S(k, n 1) ɛ = 1, b = 1. Proposton 2.3. ([5, 2.7]) For n 3, j > 0, and p Z, ν( ( n 2+1) (2 + 1) p j ) max(ν([ n ]!), n α(n) j) 2 wth equalty f n {2 e + 1, 2 e + 2} and j = 2 e 1. Other well-known facts that we wll use are (2.4) ( 1) j j!s(k, j) = ( j 2) (2) k T j (k) and (2.5) S(k +, k) ( k+2 1 k 1 ) mod 2. We also use that ν(n!) = n α(n), where α(n) denotes the bnary dgtal sum of n, and that ( m n) s odd ff, for all, m n, where these denote the th dgt n the bnary expansons of m and n. Proof of Theorem 1.5(). Usng (2.4), we have T 2 e(2 e 1 t) S(2 e 1 t, 2 e )(2 e )! mod 2 2e 1t, and we may assume t 2 usng the perodcty of ν(t n ( )) establshed n [3, 3.12]. But S(2 e 1 t, 2 e ) ( ) 2 e t 2 e+1 +2 e 1 2 e 1 1 mod 2. Snce ν(2 e!) = 2 e 1 < 2 e 1 t, we are done. Proof of parts () and () of Theorem 1.5. These parts follow from (a) and (b) below by lettng p = 2 e + ɛ 1 n (b), and addng.

7 v 1-PERIODIC 2-EXPONENTS OF SU(2 e ) AND SU(2 e + 1) 7 (a) Let ɛ {0, 1} and n 2 e + ɛ. { ν(t n (2 e = 2 e + 2 e 1 1 f ɛ = 1 and n = 2 e ɛ 1)) 2 e + 2 e 1 + ɛ 1 otherwse. (b) Let p Z, n 2 e, and ν(m) 2 e 1 + e 1. Then ( ( ν n ) ( 2+1 (2+1) p (2+1) m 1 )) = 2 e + 2 e 1 1 f n = 2 e + 1 and ν(m) = 2 e 1 + e 1 2 e + 2 e 1 otherwse. Frst we prove (a). Usng (2.4) and the fact that S(k, j) = 0 f k < j, t suffces to prove ν ( ( n ) ) { 2 2 e +ɛ 1 = 2 e 1 1 f ɛ = 1 and n = 2 e e 1 otherwse, and ths s mpled by Proposton 2.1 f n 2 e + 4. For ɛ = 0 and 2 e n 2 e + 3, by Proposton 2.2 ν( ( n 2) 2 e 1 ) 2 e mn(1, ν(s(2 e 1, 2 e 1 + δ))) wth δ {0, 1}. The Strlng number here s easly seen to be even by (2.5). Smlarly ν( ( ) 2 e e ) = 2 e 1 1 snce S(2 e, 2 e 1 ) s odd, and f n 2 e {2, 3}, then ν ( ( ) n 2) 2 e 2 e 1 snce S(2 e, 2 e 1 + 1) s even. Now we prove part (b). The sum equals j>0 T j, where T j = 2 j( ) ( m n j 2+1) (2 + 1) p j. We show that ν(t j ) = 2 e + 2 e 1 1 f n = 2 e + 1, j = 2 e 1, and ν(m) = 2 e 1 + e 1, whle n all other cases, ν(t j ) 2 e + 2 e 1. If j 2 e + 2 e 1, we use the 2 j -factor. Otherwse, ν( ( m j ) ) = ν(m) ν(j), and we use the frst part of the max n Proposton 2.3 f ν(j) e 1, and the second part of the max otherwse. References [1] M. Bendersky and D. M. Davs, 2-prmary v 1 -perodc homotopy groups of SU(n), Amer. J. Math. 114 (1991) [2] A. K. Bousfeld, On the 2-prmary v 1 -perodc homotopy groups of spaces, Topology 44 (2005) [3] M. C. Crabb and K. Knapp, The Hurewcz map on stunted complex projectve spaces, Amer Jour Math 110 (1988)

8 8 DONALD M. DAVIS [4] D. M. Davs, Dvsblty by 2 and 3 of certan Strlng numbers, Integers 8 (2008) A56, 25 pp. [5], Dvsblty by 2 of partal Strlng numbers, submtted, dmd1/partal5.pdf. [6] D. M. Davs and M. Mahowald, Some remarks on v 1 -perodc homotopy groups, London Math. Soc. Lect. Notes 176 (1992) [7] D. M. Davs and K. Potocka, 2-prmary v 1 -perodc homotopy groups of SU(n) revsted, Forum Math 19 (2007) [8] D. M. Davs and Z. W. Sun, A number-theoretc approach to homotopy exponents of SU(n), Jour Pure Appl Alg 209 (2007) Department of Mathematcs, Lehgh Unversty, Bethlehem, PA 18015, USA E-mal address: dmd1@lehgh.edu