v 1 -periodic 2-exponents of SU(2 e ) and SU(2 e + 1)

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1 Joural of Pure ad Appled Algebra 216 (2012) Cotets lsts avalable at ScVerse SceceDrect Joural of Pure ad Appled Algebra joural homepage: v 1 -perodc 2-expoets of SU(2 e ) ad SU(2 e + 1) Doald M. Davs Departmet of Mathematcs, Lehgh Uversty, Bethlehem, PA 18015, USA a r t c l e f o a b s t r a c t Artcle hstory: Receved 22 September 2010 Receved revsed form 2 December 2011 Avalable ole 23 Jauary 2012 Commucated by C.A. Webel We determe precsely the largest v 1 -perodc homotopy groups of SU(2 e ) ad SU(2 e +1). Ths gves ew results about the largest actual homotopy groups of these spaces. Our proof reles o results about 2-dvsblty of restrcted sums of bomal coeffcets tmes powers proved by the author a compao paper Elsever B.V. All rghts reserved. MSC: 55Q52; 11B73 1. Ma result The 2-prmary v 1 -perodc homotopy groups, v 1 π 1 (X), of a topologcal space X are a localzato of a frst approxmato to ts 2-prmary homotopy groups. They are roughly the porto 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 summad of some actual homotopy group of X [7]. T j (k) = j k odd deote oe famly of partal Strlg umbers. I [6], the author obtaed several results about ν(t j (k)), where ν() deotes the expoet of 2. Some of those wll be used ths paper, ad wll be restated as eeded. e(k, ) = m(ν(t j (k)) : j ). It was proved [1, 1.1] (see also [8, 1.4]) that v 1 1 π 2k(SU()) s somorphc to Z/2 e(k,) ϵ drect sum wth possbly oe or two Z/2 s. Here ϵ = 0 or 1, ad ϵ = 0 f s odd or f k 1 mod 4, whch are the oly cases requred here. s() = 1 + ν([/2]!). It was proved [9] that e( 1, ) s(). e() = max(e(k, ) : k Z). Thus e() s what we mght call the v 1 -perodc 2-expoet of SU(). The clearly s() e( 1, ) e(), ad calculatos suggest that both of these equaltes are usually qute close to beg equaltes. I [5, page 22], a table s gve comparg the umbers (1.1) for 38. (1.1) E-mal address: dmd1@lehgh.edu /$ see frot matter 2012 Elsever B.V. All rghts reserved. do: /j.jpaa

2 D.M. Davs / Joural of Pure ad Appled Algebra 216 (2012) Our ma theorem verfes a cojecture of [5] regardg the values (1.1) whe = 2 e or 2 e + 1. Theorem 1.2. a. If e 3, the e(k, 2 e ) 2 e + 2 e 1 1 wth equalty occurrg ff k 2 e 1 mod 2 2e 1 +e 1. b. If e 2, the e(k, 2 e + 1) 2 e + 2 e 1 wth equalty occurrg ff k 2 e + 2 2e 1 +e 1 mod 2 2e 1 +e. Thus the values (1.1) for = 2 e ad 2 e + 1 are as Table 1.3. Table 1.3 Comparso of values. s() e( 1, ) e() 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() exceeds s() by 1 both cases, but for dfferet reasos. Whe = 2 e, the largest value occurs for k = 1, but s 1 larger tha the geeral boud establshed [9]. Whe = 2 e + 1, the geeral boud for e( 1, ) s sharp, but a larger group occurs whe 1 s altered a specfc way. The umbers e() are terestg, as they gve what are qute possbly the largest 2-expoets π (SU()), ad ths s the frst tme that fte famles of these umbers have bee computed precsely. The homotopy 2-expoet of a topologcal space X, deoted exp 2 (X), s the largest k such that π (X) cotas a elemet of order 2 k. A mmedate corollary of Theorem 1.2 s as follows. Corollary 1.4. For ϵ {0, 1} ad 2 e + ϵ 5, exp 2 (SU(2 e + ϵ)) 2 e + 2 e ϵ. Ths result s 1 stroger tha the result oted [9, Theorem 1.1]. We remark that kow upper bouds for exp 2 (SU()) are much larger tha our lower bouds. All that oe ca really say s that exp 2 (SU()) 1 =1 exp 2(S 2+1 ), ad the use Selck s result [10] that exp 2 (S 2+1 ) [(3 + 1)/2]. Ths gves roughly 3 ( 1) as the upper boud, whereas our lower boud s roughly 3. It s expected that the actual 2-expoet for 4 2 S 2+1 s or + 1, depedg o the mod 4 value of, but a bgger ssue s that the largest expoets from the varous spheres probably do ot buld up addtvely SU(). The reaso that oe mght be optmstc that our boud s sharp s that f p s a odd prme, the p-expoet of S 2+1 equals the v 1 -perodc p-expoet, by Cohe et al. [3]. Theorem 1.2 s mpled by the followg two results. The frst wll be proved Secto 2. The secod s [6, Theorem 1.1]. Theorem 1.5. e 3.. If ν(k) e 1, the ν(t 2 e(k)) = 2 e 1.. If j 2 e ad ν(k (2 e 1)) 2 e 1 + e 1, the ν(t j (k)) 2 e + 2 e If j 2 e + 1 ad ν(k 2 e ) = 2 e 1 + e 1, the ν(t j (k)) 2 e + 2 e 1. Theorem 1.6 ([6, 1.1]). e 2, = 2 e + 1 or 2 e + 2, ad 1 2 e 1. There s a 2-adc teger x, such that for all tegers x ν(t (2 e 1 x + 2 e 1 + )) = ν(x x, ) + 2. Moreover = f = 2 e 2 or 2 e 1 ν(x,2 e +1) > otherwse. ad = 1 f 1 2 e 2 ν(x,2 e +2) = f 2 e 2 < < 2 e 1 > f = 2 e 1. Regardg small values of e, [8, 8] ad [6, Table 1.3] make t clear that the results stated ths secto for T ( ), e(, ) ad SU() are vald for small values of 5 but ot for < 5. Proof that Theorems 1.5 ad 1.6 mply Theorem 1.2. For part (a): k 2 e 1 mod 2 2e 1 +e 1. Theorems 1.5() mples e(k, 2 e ) 2 e + 2 e 1 1, ad 1.6 wth = 2 e + 2, = 2 e 1 1, ad ν(x) 2 e 1 mples that equalty s obtaed 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.

3 1270 D.M. Davs / Joural of Pure ad Appled Algebra 216 (2012) If = 2 e 1, we use 1.5(). If = 2 e 2, we use 1.6 wth = 2 e + 1 f ν(x) < 2 e 2 ad wth = 2 e + 2 f ν(x) 2 e 2. For other values of, we use 1.6 wth = 2 e + 1 f ν(x) ad wth = 2 e + 2 f ν(x) >, except the excluded case = 2 e 1 1 ad ν(x) >. For part (b): 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, ad 1.6 wth = 2 e +2, = 2 e 1, ad ν(x) = 2 e 1 mples that equalty s obtaed 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 = 2 e + 1 uless ν(x) = 2 e 1, whch case s excluded. If = 2 e 2, we use 1.6 wth = 2 e + 2 f ν(x) = 2 e 2 ad wth = 2 e + 1 otherwse. If 1 < 2 e 2, we use 1.6 wth = 2 e + 1 f ν(x) = 1 ad wth = 2 e + 2 otherwse. If 2 e 2 < < 2 e 1, we use 1.6 wth = 2 e + 1 f ν(x) = ad wth = 2 e + 2 otherwse. The proof does ot make t trasparet why the largest value of e(k, ) occurs whe k = 1 f = 2 e but ot f = 2 e + 1. The followg example may shed some lght. We cosder 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 ad 1.8 gve relevat values of ν(t j (k)). Table 1.7 Values of ν(t j (k)) relevat to e(k, 16). j ν(k 15) Table 1.8 Values of ν(t j (k)) relevat to e(k, 17). j ν(k 16) The values e(k, 16) ad e(k, 17) are the smallest etry a row, ad are lsted boldface. The tables oly clude values of k for whch ν(k ( 1)) s rather large, as these gve the largest values of ν(t j (k)). Larger values of j tha those tabulated wll gve larger values of ν(t j ()). Note how each colum has the same geeral form, levelg off after a jump. Ths reflects the ν(x x, ) Theorem 1.6. The prevalece of ths behavor s the cetral theme of [6]. The pheomeo whch we wsh to llumate here s how the bold values crease steadly utl they level off Table 1.7, whle Table 1.8 they jump to a larger value before levelg off. Ths s a cosequece of the sychrocty of where the jumps occur colums 17 ad 18 of the two tables. 2. Proof of Theorem 1.5 I ths secto, we prove Theorem 1.5. The proof uses the followg results from [6]. Proposto 2.1 ([9, 3.4] or [6, 2.1]). For ay oegatve tegers ad k, ν 2+1 k ν([/2]!).

4 D.M. Davs / Joural of Pure ad Appled Algebra 216 (2012) The ext result s a refemet of Proposto 2.1. Here ad throughout, S(, k) deote Strlg umbers of the secod kd, defed by k ( 1) k k!s(, k) = ( 1) k. =0 Proposto 2.2 ([6, 2.3]). Mod 4 S(k, ) + 2S(k, 1) ϵ = 0, b = ϵ! 2+b k (2 + 1)S(k, ) + 2( + 1)S(k, 1) ϵ = 1, b = 0 2S(k, 1) ϵ = 0, b = 1 S(k, ) + 2( + 1)S(k, 1) ϵ = 1, b = 1. Proposto 2.3 ([6, 2.7]). For 3, j > 0, ad p Z, ν 2+1 (2 + 1) p j max(ν([ ]!), α() j) 2 wth equalty f {2 e + 1, 2 e + 2} ad j = 2 e 1. ad Other well-kow facts that we wll use are ( 1) j j!s(k, j) = j 2 (2) k T j (k) S(k +, k) k+2 1 k 1 mod 2. We also use that ν(!) = α(), where α() deotes the bary dgtal sum of, ad that m s odd ff, for all, m, where these deote the th dgt the bary expasos of m ad. Proof of Theorem 1.5(). Usg (2.4), we have T 2 e(2 e 1 t) S(2 e 1 t, 2 e )(2 e )! mod 2 2e 1t, ad we may assume t 2 usg the perodcty of ν(t ( )) establshed [4, 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. Sce ν(2 e!) = 2 e 1 < 2 e 1 t, we are doe. Proof of parts () ad () of Theorem 1.5. These parts follow from (a) ad (b) below by lettg p = 2 e + ϵ 1 (b), ad addg. (a) ϵ {0, 1} ad 2 e + ϵ. ν(t (2 e =2e + 2 e 1 1 f ϵ = 1 ad = 2 e ϵ 1)) 2 e + 2 e 1 + ϵ 1 otherwse. (b) p Z, 2 e, ad ν(m) 2 e 1 + e 1. The =2 ν 2+1 e + 2 e 1 1 f = 2 e + 1 ad (2 + 1) p ((2 + 1) m 1) ν(m) = 2 e 1 + e 1 2 e + 2 e 1 otherwse. Frst we prove (a). Usg (2.4) ad the fact that S(k, j) = 0 f k < j, t suffces to prove ν 2 2 +ϵ 1 e =2 e 1 1 f ϵ = 1 ad = 2 e e 1 otherwse, ad ths s mpled by Proposto 2.1 f 2 e + 4. For ϵ = 0 ad 2 e 2 e + 3, by Proposto 2.2 ν e 2 e m(1, ν(s(2 e 1, 2 e 1 + δ))) wth δ {0, 1}. The Strlg umber here s easly see to be eve by (2.5). Smlarly ν( 2 e e ) = 2 e 1 1 sce S(2 e, 2 e 1 ) s odd, ad f 2 e S(2 e, 2 e 1 + 1) s eve. Now we prove part (b). The sum equals j>0 T j, where T j = 2 j m j 2+1 (2 + 1) p j. (2.4) (2.5) {2, 3}, the ν( 2 2 e ) 2 e 1 sce We show that ν(t j ) = 2 e +2 e 1 1 f = 2 e +1, j = 2 e 1, ad ν(m) = 2 e 1 +e 1, whle 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), ad we use the frst part of the max Proposto 2.3 f ν(j) e 1, ad the secod part of the max otherwse.

5 1272 D.M. Davs / Joural of Pure ad Appled Algebra 216 (2012) Refereces [1] M. Bedersky, D.M. Davs, 2-prmary v 1 -perodc homotopy groups of SU(), Amer. J. Math. 114 (1991) [2] A.K. Bousfeld, O the 2-prmary v 1 -perodc homotopy groups of spaces, Topology 44 (2005) [3] F.R. Cohe, J.C. Moore, J.A. Nesedorfer, The double suspeso ad expoets of the homotopy groups of spheres, Aals of Math. 110 (1979) [4] M.C. Crabb, K. Kapp, The Hurewcz map o stuted complex projectve spaces, Amer. J. Math. 110 (1988) [5] D.M. Davs, Dvsblty by 2 ad 3 of certa Strlg umbers, Itegers 8 (2008) A56, 25 pp. [6] D.M. Davs, Dvsblty by 2 of partal Strlg umbers, (submtted for publcato). [7] D.M. Davs, M. Mahowald, Some remarks o v 1 -perodc homotopy groups, Lodo Math. Soc. Lect. Notes 176 (1992) [8] D.M. Davs, K. Potocka, 2-prmary v 1 -perodc homotopy groups of SU() revsted, Forum Math. 19 (2007) [9] D.M. Davs, Z.W. Su, A umber-theoretc approach to homotopy expoets of SU(), J. Pure Appl. Alg. 209 (2007) [10] P. Selck, 2-prmary expoets for the homotopy groups of spheres, Topology 23 (1984)