A NUMBER-THEORETIC APPROACH TO HOMOTOPY EXPONENTS OF SU (n) DONALD M. DAVIS AND ZHI-WEI SUN ( Version 1.9, 2005-08-11) Abstract.We use methods of combinatorial number theory to prove that, for each n 2 and any prime p, some homotopy group ssi(SU(n)) contains an element of order pn-1+ordp(bn=pc!), where ordp(m) denotes the largest integer ff such that pff| m. 1.Introduction Let p be a prime number. The homotopy p-exponent of a topological space X, denoted by expp(X), is defined to be the largest e 2 N = {0, 1, 2, . .}.such th* *at some homotopy group ssi(X) has an element of order pe. This concept has been studied* * by various topologists (cf. [3], [4], [12], [15], [16], and [5]). The most celebra* *ted result about homotopy exponents (proved by Cohen, Moore, and Neisendorfer in [3]) stat* *es that expp(S2n+1) = n if p 6= 2. The special unitary group SU (n) (of degree n) is the space of all n x n unit* *ary matrices (the conjugate transpose of such a complex matrix equals its inverse) * *with determinant one. (See, e.g., [10, p. 68].) It plays a central role in many area* *s of math- ematics and physics. The famous Bott Periodicity Theorem ([2]) describes ssi(SU* * (n)) with i < 2n. In this paper, we provide a strong and elegant lower bound for the homotopy p-exponent of SU(n). __________ Key words and phrases. homotopy group, unitary group, p-adic order, binomial coefficient. 2000 Mathematics Subject Classification: 55Q52, 57T20, 11A07, 11B65, 11S05. The second author is supported by the National Science Fund for Distinguished Young Scholars (No. 10425103) and the Key Program of NSF (No. 10331020) in P. R. China. 1 2 DONALD M. DAVIS AND ZHI-WEI SUN As in number theory, the integral part of a real number c is denoted by bcc. * *For a prime p and an integer m, the p-adic order of m is given by ordp(m) = sup{n 2 N* * : pn | m} (whence ordp(0) = +1). Here is our main result. Theorem 1.1. For any prime p and n = 2, 3, . .,.we have the inequality _$ % ! n expp(SU (n)) n - 1 + ordp __ ! . p We discuss in Section 2 the extent to which Theorem 1.1 might be sharp. Our reduction from homotopy theory to number theory involves Stirling numbers of the second kind. For n, k 2 N with n + k 2 Z+ = {1, 2, 3, . .}., the Stirlin* *g number S(n, k) of the second kind is the number of partitions of a set of cardinality * *n into k nonempty subsets; in addition, we define S(0, 0) = 1. In Sections 2 and 4 we pr* *ove the following standard result. Proposition 1.2. Let p be a prime, and let k, n 2 Z+ and k n. Set ep(n, k) = minm n ordp(m!S(k, m)). Then, we have expp(SU (n)) ep(n, k) unless p = 2 and n 0 (mod 2), in which case exp2(SU (n)) e2(n, k) - 1. Our innovation is to extend previous work ([13]) of the second author in com- binatorial number theory to prove the following result, which with Proposition * *1.2 immediately implies Theorem 1.1 when p or n is odd. In Section 4, we explain the extra ingredient required to deduce Theorem 1.1 from 1.2 and 1.3 when p = 2 and* * n is even. Theorem 1.3. Let p be any prime and n be a positive integer. (i) For any ff, h, l, m 2 N, we have _ l_ ! ! X l ordp m! (-1)kS (kh(p - 1)pff+ n - 1, m) ( k=0 k _$ % !) m min l(ff + 1), n - 1 + ordp __.! p (ii) If we define N = n - 1 + bn=(p(p - 1))c, then i j _ $n% ! ep n, (p - 1)pL + n - 1 n-1+ordp __! forL = N, N +1, . ... p A NUMBER-THEORETIC APPROACH TO HOMOTOPY EXPONENTS OF SU(n) 3 In Section 3, we prove the following broad generalization of Theorem 1.3, and* * in Section 2, we show that it implies Theorem 1.3. Theorem 1.4. Let p be a prime, ff, n 2 N and r 2 Z. Then for any polynomial f(x) 2 Z[x] we have _ _ ! _ !! _$ % ! X n k - r n ordp (-1)kf _____ff ordp __ff! . k r (mod pff)k p p i j Here we adopt the standard convention that nkis 0 if k is a negative integer. In Theorem 5.1, we give a strengthened version of Theorem 1.4, which we conje* *cture to be optimal in a certain sense. Our application to topology uses the case r =* * 0 of Theorem 1.4; the more technical Theorem 5.1 yields no improvement in this case. In [5], the first author used totally different, and much more complicated, m* *ethods to prove that $ % $ % n + 2p - 3 n + p2- p - 1 expp(SU (n)) n - 1 + _________ + ____________ , p2 p3 (1.5) where p is an odd prime and n is an integer greater than one. Since 1X$ m % ordp(m!) = __i for everym = 0, 1, 2, . . . i=1 p (a well-known fact in number theory), the inequality in Theorem 1.1 can be rest* *ated as 1 $ % X n expp(SU (n)) n - 1 + __i, i=2 p a nice improvement to (1.5). 2.Outline of proof In this section we present the deduction of Theorem 1.1 from Theorem 1.4, whi* *ch will then be proved in Section 3. We also present some comments regarding the e* *xtent to which Theorem 1.1 is sharp. Let p be any prime. In [8], the first author and Mahowald defined the (p-prim* *ary) v1-periodic homotopy groups v-11ss*(X; p) of a topological space X and proved t* *hat if X is a sphere or compact Lie group, such as SU(n), each group v-11ssi(X; p) is * *a direct summand of some actual homotopy group ssj(X). See also [7] for another exposito* *ry account of v1-periodic homotopy theory. 4 DONALD M. DAVIS AND ZHI-WEI SUN In [6, 1.4] and [1, 1.1a], it was proved that if p is odd, or if p = 2 and n * *is odd, then v-11ss2k(SU (n); p) Z=pep(n,k)Z (2.1) for all k n, where ep(n, k) is as defined in 1.2 and we use Z=mZ to denote the additive group of residue classes modulo m. Thus, unless p = 2 and n is even, f* *or any integer k n, we have expp(SU (n)) ep(n, k), establishing Proposition 1.2 in these cases. The situation when p = 2 and n is * *even is somewhat more technical, and will be discussed in Section 4. Next we show that Theorem 1.4 implies Theorem 1.3. Proof of Theorem 1.3.(i) By a well-known property of Stirling numbers of the se* *cond kind (cf. [11, pp. 125-126]), mX _m ! ff m!S(kh(p - 1)pff+ n - 1, m) = (-1)m-jjkh(p-1)p +n-1 j=0 j for any k 2 N. Thus Xl_ l! (-1)m m! (-1)kS(kh(p - 1)pff+ n - 1, m) = 1+ 2, k=0 k where Xl _ l! Xffm_m! _ j!n-1+kh(p-1)pff 1 = (-1)kpn-1+kh(p-1)p (-1)j _ k=0 k j=0 j p p|j and _ ! _ ! X m Xl l ff 2 = (-1)j (-1)kjn-1+kh(p-1)p j6 0(mod p)j k=0 k _ ! X m i jffl = (-1)jjn-1 1 - jh(p-1)p. j6 0(mod p)j Clearly ordp( 1) n - 1 + ordp(bm=pc!) by Theorem 1.4, and ordp( 2) l(ff + 1) by Euler's theorem in number theory. Therefore the first part of Theorem 1.3 ho* *lds. (ii) Observe that $ % _$ % ! n 1X n 1X n n N + 1 - (n - 1) > _______ = __> __ = ordp __ ! . p(p - 1) i=2pi i=2 pi p A NUMBER-THEORETIC APPROACH TO HOMOTOPY EXPONENTS OF SU(n) 5 By part (i) in the case l = h = 1 and ff = L N, if m n then i j ordp m!S(n - 1, m) - m!S((p - 1)pL + n - 1, m) _$ % ! n n - 1 + ordp __ ! . p Since S(n - 1, m) = 0 for m n, we finally have _$ % ! n ep(n, (p - 1)pL + n - 1) n - 1 + ordp __ ! p as required. || The following proposition, although not needed for our main results, sheds mo* *re light on the large exponents N and L which appear in Theorem 1.3(ii), and is us* *eful in our subsequent exposition. Proposition 2.2. Let p be a prime n > 1 be an integer. Then there exists a com- putable integer N0 0, only depending on p and n, such that ep(n, (p - 1)pL + * *n - 1) has the same value for all L N0. Proof.For integers m n and L 0, we write mX _m ! L (-1)m m!S((p-1)pL+n-1, m) = (-1)jj(p-1)p +n-1= Sm +S0m,L+S00m,L, j=0 j where _ ! X m Sm = (-1)jjn-1, j6 0(mod p)j _ ! X m L S0m,L= (-1)jjn-1(j(p-1)p- 1), j6 0(mod p)j _ ! X m L S00m,L= (-1)jj(p-1)p +n-1. j 0(mod p)j Note that both S0m,Land S00m,Lare divisible by pL+1. Assume that Sn, Sn+1, . .a.re not all zero. (This will be shown later.) Then L0 = minm n ordp(Sm ) is finite. Let m0 n satisfy ordp(Sm0) = L0. Whenever L L0, we have ordp(Sm + S0m,L+ S00m,L) L0 for every m n, and equality is 6 DONALD M. DAVIS AND ZHI-WEI SUN attained for m = m0. Thus, if L L0 then ep(n, (p - 1)pL + n -=1) minordp(m!S((p - 1)pL + n - 1, m)) m n = minordp(Sm + S0m,L+ S00m,L) = L0. m n Although L0 is finite, it may not be effectively computable. Instead of L0 we u* *se the p-adic order N0 of the first nonzero term in the sequence Sn, Sn+1, . ... This * *N0 is computable, also ep(n, (p - 1)pL + n - 1) = L0 for all L N0 since N0 L0. To complete the proof, we must show that Sm is nonzero for some m n. First * *note that this is clearly true for p = 2 since then Sm is a sum of negative terms. I* *f p is odd and Sm = 0 for all m n, then ep(n, (p-1)pL+n-1) = minm n ordp(S0m,L+S00m,L) L + 1 for any L 0. By (2.1), this would imply that v-11ss*(SU (n); p) has ele* *ments of arbitrarily large p-exponent. However, this is not true, for in [6, 5.8], it* * was shown that the v1-periodic p-exponent of SU(n) does not exceed e := b(n-1)(1+(p-1)-1+ (p - 1)-2)c; i.e., for this e, pev-11ss*(SU (n); p) = 0. || In the remainder of this section and in Section 4, once a prime p and an inte* *ger n > 1 is given, L will refer to any integer not smaller than max{N, N0} where N* * and N0 are described in Theorem 1.3(ii) and Proposition 2.2 respectively. We now comment on the extent to which Theorem 1.1 might be sharp. In Table 1, we present, for p = 3 and a representative set of values of n, three numbers. T* *he first, labeled exp3(v-11SU(n)), is the largest value of e3(n, k) over all values of k * * n; thus it is the largest exponent of the 3-primary v1-periodic homotopy groups of SU (* *n). The second number in the table is the exponent of the v1-periodic homotopy group on which we have been focusing, which is certainly the best universal choice, b* *ut not always quite the best. The third number is the nice estimate for this exponent * *given by Theorem 1.3. Note that, for more than half of the values of n in the table, the largest gr* *oup v-11ss2k(SU (n); 3) occurs when k = 2 . 3L + n - 1. In the worst case in the t* *able, n = 29, detailed Maple calculations suggest that if k 29 and k 10 (mod 18)* *, then e3(29, k) = min{ord3(k - 28 - 8 . 320) + 12, 34}. A NUMBER-THEORETIC APPROACH TO HOMOTOPY EXPONENTS OF SU(n) 7 Table 1. Comparison of exponents when p = 3 _n_||exp3(v-11SU(n))e3(n,_2_._3L_+_n_-n1)-_1_+_ord3(bn=3c!)_ 19 || 21 20 20 20 || 22 21 21 21 || 22 22 22 22 || 25 25 23 23 || 26 26 24 24 || 28 28 25 25 || 29 28 26 26 || 30 30 27 27 || 31 31 30 28 || 32 32 31 29 || 34 32 32 30 || 34 33 33 31 || 34 34 34 32 || 35 35 35 33 || 37 37 36 34 || 38 37 37 35 || 39 39 38 36 || 41 41 40 37 || 42 41 41 38 || 43 42 42 39 || 43 43 43 40 || 45 44 44 41 || 45 45 45 Shifts (as by 8 . 320) were already noted in [6, p. 543]. Note also that for mo* *re than half of the cases in the table, our estimate for e3(n, 2 . 3L + n - 1) is sharp* *, and it never misses by more than 3. The big question for topologists, though, is whether the v1-periodic p-expone* *nt agrees (or almost agrees) with the actual homotopy p-exponent. The fact that th* *ey agree for S2n+1when p is an odd prime ([3]) leads the first author to conjectur* *e that they will agree for SU(n) if p 6= 2, but we have no idea how to prove this. The* *riault ([15], [16]) has made good progress in proving that some of the first author's * *lower bounds for p-exponents of certain exceptional Lie groups are sharp. 8 DONALD M. DAVIS AND ZHI-WEI SUN 3. Proof of Theorem 1.4 In this section, we prove Theorem 1.4, which we have already shown to imply Theorem 1.1. Lemma 3.1. Let p be any prime, and let ff, n 2 N and r 2 Z. Then _ _ ! ! _$ % ! $ % _$ % ! X n n n n ordp (-1)k ordp ____ff-1!= __ff+ordp __ff!. k r (mod pff)k p p p Proof.The equality is easy, for, _$ % ! 1 $ % 1 $ % n X bn=pff-1c X n ordp ____ ! = ________ = __ pff-1 i=1 pi j=ffpj $ % 1 $ % $ % _$ % ! n X bn=pffc n n = __ + ______ = __ + ordp __ ! . pff i=1 pi pff pff When ff = 0 or n < pff-1, the desired inequality is obvious. Now let ff > 0 and m = bn=pff-1c 1. Observe that 1X$ m % 1X m m X1 1 m 1 m ordp(m!) = __i< __i= __ __j= __. _______-1= _____. i=1 p i=1p pj=0p p 1 - p p - 1 Thus (p - 1) ordp(m!) m - 1, and hence $ % $ % $ % m - 1 n=pff-1- 1 n - pff-1 ordp(m!) ______ = __________ = ________ , p - 1 p - 1 '(pff) where ' is Euler's totient function. By a result of Weisman [18], _ _ ! ! $ % X n n - pff-1 ordp (-1)k ________ff. k r (mod pff)k '(p ) (Weisman's proof is complicated, but an easy induction proof appeared in [13].)* * So we have the desired inequality. || Lemma 3.2. Let m, n 2 Z+ and r 2 Z, and let f(x) be a complex-valued function defined on Z. Then we have _ ! _ ! `~ ' _ ! X n k - r n - r X n (-1)kf _____ - f _____ (-1)k k r (mod m)k m m k r (mod m)k n-1X_n!X _j! X _n - j - 1! _k - r ! = - (-1)i (-1)k f _____j_, j=0 j m|i-ri m|k-rj k m where rj = r - j + m - 1 and f(x) = f(x + 1) - f(x). A NUMBER-THEORETIC APPROACH TO HOMOTOPY EXPONENTS OF SU(n) 9 Proof.Let i be a primitive mth root of unity. Clearly 8 m-1X

0 and assume the result for smaller values of l. We use induction* * on n to prove the inequality in Theorem 3.3. The case n = 0 is trivial. So we now let n > 0 and assume that the inequality holds with smaller values of n. Observe that _ ! _ ! l X n k - r (-1)k _____ff k r (mod pff)k p _ _ ! _ ! ! _ ! l X n - 1 n - 1 k - r = + (-1)k _____ff k r (mod pff) k k - 1 p _ ! _ ! l X n - 1 k - r = (-1)k _____ff k r (mod pff)k p _ ! _ ! l X n - 1 0 k0- (r - 1) - 0 (-1)k __________ff. k0 r-1(mod pff)k p In view of this, if pffdoes not divide n, then, by the induction hypothesis for* * n - 1, we have _ _ ! _ ! l! X n k - r ordp (-1)k _____ff k r (mod pff)k p _$ %! _$ %! n - 1 n ordp _____ = ordp __ . pff pff Below we let pff| n and set m = n=pff. A NUMBER-THEORETIC APPROACH TO HOMOTOPY EXPONENTS OF SU(n) 11 Case 1. r 0 (mod pff). In this case, _ ! _ ! _ ! l-1 _1_ X n k k r k - r (-1) __- __ _____ m! k r (mod pff)k pff pff pff _ ! _ ! l-1 n=pff X n - 1 k k - r = _____ (-1) _____ m! k r (mod pff)k - 1 pff _ ! _ ! l-1 r=pff X n k k - r -____ (-1) _____ m! k r (mod pff)k pff _ ! _ !l-1 1 X n - 1 k+1 k - (r - 1) = ____________ (-1) __________ b(n - 1)=pffc!k r-1(mod pff)k pff _ ! _ ! l-1 r=pff X n k k - r -_______ (-1) _____ . bn=pffc!k r (mod pff)k pff Thus, by the induction hypothesis for l - 1, _ ! _ ! l _1_ X n k k - r (-1) _____ m! k r (mod pff)k pff is a p-integer (i.e., its denominator is relatively prime to p) and hence the d* *esired inequality follows. P i0j i Case 2. r 6 0 (mod pff). Note that i r (mod pff)i(-1) = 0. Also, _ _ ! ! _ ! X n n ordp (-1)k ordp ____ff-1!= m + ordp(m!) k r (mod pff)k p by Lemma 3.1. Thus, in view of Lemma 3.2, it suffices to show that if 0 < j < n* * then the p-adic order of _ ! _ ! _ ! _ ! n X j i X n - j - 1 k k - rj (-1) (-1) f ______ j pff|i-ri pff|k-rj k pff is at least ordp(m!), where rj = r - j + pff- 1 and f(x) = xl. Let 0 < j n - 1 and write j = pffs + t, where s, t 2 N and t < pff. Note th* *at $ % $ % $ % j_ n - j - 1 t + 1 = s and ________ = m - s - _____= m - s - 1. pff pff pff 12 DONALD M. DAVIS AND ZHI-WEI SUN Since f(x) = (x+1)l-xl= P l-1ilji i=0i x , by Lemma 3.1 and the induction hypothesis we have __ ! _ ! _ ! _ !! n X j i X n - j - 1 k k - rj ordp (-1) (-1) f ______ j pff|i-ri pff|k-rj k pff _ ! _ _ ! ! n X j i = ordp + ordp (-1) j i r (mod pff)i _ _ ! _ !! X n - j - 1 k - rj + ordp (-1)k f ______ff k rj(mod pff) k p _ ! n ordp + (s + ordp(s!)) + ordp((m - s - 1)!) j _ ! n = ordp + s + ordp(s!) - ordp(m(m - 1) . .(.m - s)) + ordp(m!) j _ ! _ ! pffm m = ordp - ordp + s - ordp(m - s) + ordp(m!). pffs + t s Observe_that! _ ! pffm Xff pffm pffs pff(m - s) ordp = ____ - ___- _________ pffs i=1 pi pi pi 1X _ $pffm% $pffs% $pff(m - s)%! + ____i- ___i- _________i i=ff+1 p p p X1 _ $m % $s % $m - s%! _m ! = __i- __i- _____i = ordp . i=1 p p p s Define ordp(a=b) = ordp(a) - ordp(b) if a, b 2 Z and a is not divisible by b. T* *hen _ ! _ ! pffm m ordp - ordp pffs + t s ipffmj ffs+t (pffs)!(pff(m - s))! = ordp_p____ipffmj= ordp______________________ffff pffs (p s + t)!(p (m - s) - t)! pff(m - s) Y pff(m - s) - i = ordp_________+ ordp ____________. pffs + t 0 2, hence n=2 + 1 is even and* * not larger than n - 1. As first noted in [1, 1.1] and restated in [9, 6.5], for k =* * 2L + n - 1, d3 : E1,2k+13! E4,2k+33is nonzero if and only if e2(n, 2L + n - 1) = e2(n - 1, 2L + n - 1) + n - 1. We show at the end of the section that e2(n - 1, 2L + n - 1) = ord2((n - 1)!). (4.1) Thus, if the above d3 is nonzero, then e2(n, 2L + n - 1) = n - 1 + ord2((n - 1)* *!) and hence exp2(SU (n)) e2(n, 2L+n-1)-1 = n-1+ord2((n-1)!)-1 n-1+ord2(bn=2c!), as claimed in Theorem 1.1. Proof of (4.1).Putting p = 2, ff = L, l = h = 1 and m = n - 1 in the first part* * of Theorem 1.3, we get that i j ord2 (n - 1)!S(n - 1, n - 1) - (n - 1)!S(2L + n - 1, n - 1) `~n - 1 ' n - 1 + ord2 _____ ! n - 1 > ord2((n - 1)!). 2 Therefore ord2((n - 1)!S(2L + n - 1, n - 1)) = ord2((n - 1)!). On the other han* *d, by the second part of Theorem 1.3, ord2(m!S(2L + n - 1, m)) n - 1 + ord2(bn=2c!)* * for all m n. So we have (4.1). || 5.Strengthening and sharpness of Theorem 3.3 In this section, we give an example illustrating the extent to which Theorem * *3.3 is sharp when r = 0, which is the situation that is used in our application to top* *ology. Then we show in Theorem 5.1 that the lower bound in Theorem 3.3 can sometimes be increased slightly. We begin with a typical example of Theorem 3.3. Let p = ff = 2, r = 0 and n = 100. Then bn=pffc = 25 and ordp(bn=pffc!) = 22. For l 25, set _ _ ! _ !l! X n k ffi(l) = ord2 __ - 22. k 0(mod 4)k 4 A NUMBER-THEORETIC APPROACH TO HOMOTOPY EXPONENTS OF SU(n) 15 The range l bn=pffc = 25 is that in which we feel Theorem 3.3 to be very stro* *ng. (See Remark 5.3(2).) Clearly ffi(l) measures the amount by which the actual p-a* *dic order of the sum in Theorem 3.3 exceeds our bound for it. The values of ffi(l)* * for 25 l 45 are given in order as 0, 0, 0, 0, 2, 3, 2, 4, 1, 1, 1, 1, 2, 2, 4, 1, 0, 0, 0, 0, 3. When r = 0 and in many other situations, Theorem 3.3 appears to be sharp for infinitely many values of l. Before presenting our strengthening of Theorem 3.3 we need some notations. For a 2 Z and m 2 Z+, we let {a}m denote the least nonnegative residue of a modulo m. Given a prime p, for any a, b 2 N we let op(a, b) represent the number of ca* *rries occurring in the addition of a and b in base p; actually 1X_ $a + b% $a % $b %! _a + b! op(a, b) = _____i- __i- __i = ordp i=1 p p p a as observed by E. Kummer. Here is our strengthening of Theorem 3.3. The right hand side is the amount by which the bound in Theorem 3.3 can be improved. This amount does not exceed ff, by the definition of op. In Table 2, we illustrate this amount when p = 3 and f* *f = 2. Theorem 5.1. Let p be a prime, and let ff, l, n 2 N. Then, for all r 2 Z, we ha* *ve _ _ ! _ ! l! _$ % ! X n k - r n ordp (-1)k _____ff - ordp __ff! k r (mod pff)k p p _ ! {r}pff+ {n - r}pff op({r}pff, {n - r}pff) = ordp . {r}pff Proof.We use induction on n. In the case n = 0, whether r 0 (mod pff) or not, the desired result holds * *trivially. Now let n > 0 and assume the corresponding result for n - 1. Suppose that op({r}pff, {n - r}pff) > 0. Then neither r nor n - r is divisible by pff. Set _ ! _ ! l 1 X n k k - r R = _______ (-1) _____ bn=pffc!k r (mod pff)k pff 16 DONALD M. DAVIS AND ZHI-WEI SUN and _ ! _ ! l n=pff X n - 1 k k - (r - 1) R0= _______ (-1) __________ . bn=pffc!k r-1(mod pff)k pff Clearly _ ! _ ! l n=pff X n - 1 k k - r R0 = -_______ (-1) _____ bn=pffc!k r (mod pff)k - 1 pff _ ! _ !l 1 X n k k k - r = -_______ (-1) __ _____ , bn=pffc!k r (mod pff)k pff pff and thus _ ! _ ! l+1 r_ 0 1 X n k k - r R + R = -_______ (-1) _____ . pff bn=pffc!k r (mod pff)k pff This is a p-integer by Theorem 3.3; therefore ordp(rR + pffR0) ff. Let fi = ordp(n). We consider three cases. Case 1. fi ff. In this case, bn=pffc!=(n=pff) = b(n - 1)=pffc! and hence R0* *is a p-integer by Theorem 3.3. In view of the inequality ordp(rR + pffR0) ff, we h* *ave ordp(R) ff - ordp(r) = op({r}pff, {n - r}pff), where the last equality follows from the definition of op and the condition n * * 0 6 r (mod pff). Case 2. ordp(r) fi < ff. Since bn=pffc = b(n - 1)=pffc, the definition of R* *0implies that _ ! _ ! l pffR0_ 1 X n - 1 k k - (r - 1) = ____________ (-1) __________ . n b(n - 1)=pffc!k r-1(mod pff)k pff Applying the induction hypothesis, we find that ordp(pffR0)-fi op({r-1}pff, {n-1-(r-1)}pff) = op({r-1}pff, {n-r}pff). Since {r}pff+ {n - r}pff n 6 0 (mod pff) and _ ! _ ! {r}pff+ {n - r}pff {r}pff+ {n - r}pff{r}pff+ {n - r}pff- 1 = ________________ {r}pff {r}pff {r}pff- 1 _ ! {r}pff+ {n - r}pff{r - 1}pff+ {n - r}pff = ________________ , {r}pff {r - 1}pff A NUMBER-THEORETIC APPROACH TO HOMOTOPY EXPONENTS OF SU(n) 17 we have op({r}pff, {n - r}pff) = op({r - 1}pff, {n - r}pff) + fi - ordp(r). Thus ordp(pffR0) ordp(r) + op({r}pff, {n - r}pff). Clearly op({r}pff, {n - r}pff) ff - ordp(r) by the definition of op, so we al* *so have ordp(rR + pffR0) ordp(r) + op({r}pff, {n - r}pff). Therefore ordp(R) = ordp(rR) - ordp(r) op({r}pff, {n - r}pff). Case 3. fi < min{ff, ordp(r)}. In this case, ordp(~r) = fi < ff where ~r= n -* * r. Also, _ ! _ ! l _ ! _ !l X n k - ~r X n r - (n - k) (-1)k _____ff = (-1)k __________ff k ~r(mod pff)k p n-k r (mod pff)k p _ ! _ ! l X n k - r = (-1)l+n (-1)k _____ff. k r (mod pff)k p Thus, as in the second case, we have _ _ ! _ ! l! 1 X n k k - ~r ordp(R) = ordp _______ (-1) _____ bn=pffc!k ~r(mod pff)k pff op({~r}pff, {n - ~r}pff) = op({r}pff, {n - r}pff). The induction proof of Theorem 5.1 is now complete. || The following conjecture is based on extensive Maple calculations. Conjecture 5.2. Let p be any prime. And let ff, l 2 N, n 2 Z+ and r 2 Z. Then equality in Theorem 5.1 is attained if l bn=pffc and $ % $ % r n - r i blog(n=pff)cj l __ + _____ mod (p - 1)p p . pff pff Remark 5.3. (1) The conjecture, if proved, would show that Theorem 5.1 would be optimal in the sense that it is sharp for infinitely many values of l. (2) Note that the conjecture only deals with equality when l bn=pffc. For s* *maller values of l, our inequality is still true, but not so strong. In [14], we obtai* *n a stronger inequality when l < bn=pffc. 18 DONALD M. DAVIS AND ZHI-WEI SUN We close with a table showing the amount by which the bound in Theorem 5.1 improves on that of Theorem 3.3. That is, we tabulate op({r}pff, {n - r}pff) w* *hen p = 3 and ff = 2. Table 2. Values of o3({r}9, {n - r}9) {r}9 |0 1 2 3 4 5 6 7 8 | __|_______________________| 0 ||02 2 1 2 2 1 2 2 || 1 ||00 2 1 1 2 1 1 2 || 2 ||00 0 1 1 1 1 1 1 || {n}9 3 ||01 1 0 2 2 1 2 2 || 4 ||00 1 0 0 2 1 1 2 || 5 ||00 0 0 0 0 1 1 1 || 6 ||01 1 0 1 1 0 2 2 || 7 ||00 1 0 0 1 0 0 2 || _8_||00_0__0__0_0__0_0__0_|| References [1]M. Bendersky and D. M. Davis, 2-primary v1-periodic homotopy groups of SU(n), Amer. J. Math. 114 (1991) 529-544. [2]R. Bott, The stable homotopy of the classical groups, Annals of Math. 70 (1* *959) 313-337. [3]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. [4]________, Exponents in homotopy theory, Annals of Math. Studies 113 (1987) 3-34. [5]D. M. Davis, Elements of large order in ss*(SU(n)), Topology 37 (1998) 293- 327. [6]________, v1-periodic homotopy groups of SU(n) at odd primes, Proc. London Math. Soc. 43 (1991) 529-541. [7]________, Computing v1-periodic homotopy groups of spheres and certain Lie groups, Handbook of Algebraic Topology, Elsevier, 1995, pp. 993-1049. [8]D. M. Davis and M. Mahowald, Some remarks on v1-periodic homotopy groups, London Math. Soc. Lect. Notes 176 (1992) 55-72. [9]D. M. Davis and K. Potocka, 2-primary v1-periodic homotopy groups of SU(n) revisited, submitted. http://www.lehigh.edu/~dmd1/sun2long.pdf. [10]D. Husemoller, Fibre bundles, 2nd edition, Springer, 1975. [11]J.H. van Lint and R. M. Wilson, A Course in Combinatorics, 2nd ed., Cam- bridge Univ. Press, Cambridge, 2001. A NUMBER-THEORETIC APPROACH TO HOMOTOPY EXPONENTS OF SU(n) 19 [12]J. A. Neisendorfer, A survey of Anick-Gray-Theriault constructions and ap- plications to exponent theory of spheres and Moore spaces, Contemp. Math. Amer. Math. Soc. 265 (2000) 159-174. [13]Z. W. Sun, Polynomial extension of Fleck's congruence, preprint, 2005, http://arxiv.org/abs/math.NT/0507008. [14]Z. W. Sun and D. M. Davis, A combinatorial congruence on polynomials, preprint, 2005. [15]S. D. Theriault, 2-primary exponent bounds for Lie groups of low rank, Cana* *d. Math. Bull. 47 (2004) 119-132. [16]________, The 5-primary homotopy exponent of the exceptional Lie group E8, J. Math. Kyoto Univ 44 (2004) 569-593. [17]H. Toda, A topological proof of theorems of Bott and Borel-Hirzebruch for homotopy groups of unitary groups, Mem. Coll. Sci. Univ. Kyoto 32 (1959) 103-119. [18]C. S. Weisman, Some congruences for binomial coefficients, Michigan Math. J. 24 (1977) 141-151. Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA E-mail address: dmd1@lehigh.edu Department of Mathematics, Nanjing University, Nanjing 210093, People's Repub- lic of China E-mail address: zwsun@nju.edu.cn