TRACTABLE FORMULAS FOR v1-PERIODIC HOMOTOPY
GROUPS OF SU(n) WHEN n p2- p + 1
DONALD M. DAVIS AND HUAJIAN YANG
1.Statement of results
Let p be a fixed odd prime. The (p-local) v1-periodic homotopy groups, v-11ss*
**(X),
of a space X were defined in [8]. Roughly speaking, they tell the portion of th*
*e p-local
homotopy groups of X detected by K-theory. For a spherically resolved space X,
each group v-11ssi(X) is a direct summand of some actual homotopy group ssL(X).
We will focus on the case where X is one of the special unitary groups SU(n), w*
*hich
are spherically resolved by the fibrations
(1.1) SU(n - 1) ! SU(n) ! S2n-1:
Let (m) = p(m) denote the exponent of p in m. Let
(1.2) ep(k; n) = min{p(j!S(k; j)) : n j k};
where S(k; j) is the Stirling number of the second kind, which satisfies
Xj ij
(1.3) j!S(k; j) = (-1)j-i jiik:
i=1
The following result was proved in [7].
Theorem 1.4. If k n, then v-11ss2k(SU(n)) Z=pep(k;n), while v-11ss2k-1(SU(n))
has order pep(k;n), but is not necessarily cyclic.
Periodicity in v-11ss*(SU(n)) would allow one to determine v-11ss2k(SU(n)) for *
*smaller
or negative values of k from this, if one wished.
Theorem 1.4 at first glance appears to be all that one might want to know abo*
*ut
v-11ss*(SU(n)). However, it has two drawbacks. One is that it only gives the or*
*ders
__________
1991 Mathematics Subject Classification. 55T15.
Key words and phrases. v1-periodic homotopy groups, unitary groups, Stirling *
*numbers, unstable
Novikov spectral sequence.
1
2 D. DAVIS AND H. YANG
of the odd groups, and not their actual structure, and the other has to do with*
* the
intractability of the formula (1.2).
The numbers ep(k; n) which occur in Theorem 1.4 are in some sense given expli*
*citly
in (1.2), and some of them can be computed by a computer, but it seems to be ve*
*ry
difficult to obtain useful general formulas for them from (1.2). Indeed, despit*
*e efforts
in [11], [9], [6], and [7], the only general results obtained from (1.2) seem t*
*o be a
(sharp) lower bound for ep(k; n) when n p, and the inequality max {ep(k; n) : *
*k 2
Z} n - 1. These were proved in [6], using Fermat's Little Theorem. Since SU(n)
localized at p splits as a product of spheres when n p, the first result can a*
*lso be
obtained easily from results for spheres. The second result is more useful, si*
*nce it
implies (1.7).
The first main result of this paper is a tractable formula for v-11ss2k(SU(n)*
*), pro-
vided n p2- p + 1.
Theorem 1.5. Suppose p is odd, k = N + (p - 1)m with 1 N < p, and
N + 1 + (p - 1)i n < N + 1 + (p - 1)(i + 1)
with 0 i p - 1. Define m^ by 0 m^ < p and m m^ mod p. Then
v-11ss2k(SU(n)) Z=pe, where e is equal to
8
>>>i + 1 ifi < N and i < ^m
>>>
>>*>>min(Ni+ (p - 1)m^; i + (m - ^m) + 1) ifi < N and ^m i
>>>minN + (p - 1)m^+ 1;
>>: iNj j
i + (m - ^m+ (-1)m^^mm^pm^(p-1)) ifN i and 1 ^m i
The smallest n for which this theorem fails to give complete information is n =
N + 1 + (p - 1)i with N = 1 and i = p.
The proof of this result makes no use of formula (1.2); it is an independent *
*cal-
culation of v-11ss*(SU(n)). These separate calculations of the same homotopy gr*
*oup
give a topological proof of a result in number theory, a tractable evaluation of
min {p(j!S(k; j)) : n j k}
when n p2- p + 1, as given in Theorem 1.5. For example, the following corollary
is easily obtained from Theorem 1.5 and (1.2).
FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS 3
Corollary 1.6. Suppose p is an odd prime, 0 ^m i < N < p, and i + p(`) + 2
N + (p - 1)m^. Then
p(S(N + (p - 1)m^+ (p - 1)p`; N + (p - 1)i)) = p(p`):
We view this merely as an amusing offshoot of our work.
An application of Theorem 1.5 to homotopy theory is a slightly improved lower
bound for the p-exponent of SU(n). Recall that the p-exponent of a space X, den*
*oted
expp(X), is defined to be the largest e such that some homotopy group of X cont*
*ains
an element of order pe. As noted at the outset, if X is spherically resolved, *
*its p-
exponent will be at least as large as that of the order of any element of v-11s*
*s*(X). It
was shown in [7] that
(1.7) expp(SU(n)) n - 1:
This can be improved by 1 for certain values of n, as given in the following co*
*rollary.
Corollary 1.8. If p is odd, then expp(SU(n)) n if, for some i < p,
i(p - 1) + 2 n ip + 1:
Proof.We use Theorem 1.5 to determine max {ep(k; n) : k 2 Z}. In the notation of
Theorem 1.5, this maximum will occur when n = N + 1 + (p - 1)i and ^m= i. This
maximum will equal n - 1 if i < N, and n if i N. The expression in the corolla*
*ry
is obtained as N + 1 + (p - 1)i for 1 N i. ||
The other main result is an explicit determination of the groups v2k-1(SU(n))*
* when
n p2- p + 1. Recall that Theorem 1.4 only gave their order.
Theorem 1.9. Use the notation of Theorem 1.5, and let t = min{N; (m) + 1}.
Then
8
>>>Z=pe ifi N
>><
Z=p Z=pe-1 ifi > N and ^m6= 0
v-11ss2k-1(SU(n)) > t i-t
>>>Z=p Z=p ifi > N and ^m= 0 and t i - N
>: N i-N
Z=p Z=p ifi > N and ^m= 0 and t i - N
A description of the conditions under which these groups are cyclic can be gi*
*ven
without resorting to all the special notation of Theorem 1.5. The following res*
*ult is
easily obtained from Theorem 1.9.
4 D. DAVIS AND H. YANG
__ __ __
Corollary 1.10. Let k be defined by 1 k p - 1 and k k mod p - 1. Then
__
v-11ss2k-1(SU(n)) is cyclic if n < (k + 1)p, and is the direct sum of two cycli*
*c sum-
mands if
__ 2
(k + 1)p n p - p + 1:
The proofs of both of our main theorems will involve a delicate analysis of t*
*he
unstable Novikov spectral sequence (UNSS) based on BP . We let Es;t2(X) denote
the E2-term of this spectral sequence. The first part of Theorem 1.4 was an imm*
*ediate
consequence of the following result, the first part of which was proved in [3],*
* and the
second part in [7].
Theorem 1.11. (1)If k n, then E1;2k+12(SU(n)) Z=pep(k;n).
(2)If k n, then v-11ss2k(SU(n)) E1;2k+12(SU(n)).
Theorem 1.5 is proved by computing E1;2k+12(SU(n)) by increasing induction on*
* n.
By contrast, Theorem 1.11(1) was proved by computing E1;2k+12(SU(n)) by downward
induction on n, starting with E1;2k+12(SU(k)) Z=k!. The methods of calculation
of the UNSS used in proving Theorems 1.5 and 1.9 extend those of [5]; we think
that the methods introduced here of calculating this spectral sequence for mult*
*icell
complexes should be useful for other computations.
In Section 2, we present the requisite background on the UNSS. In Section 3, *
*we
outline the proof of Theorem 1.5, with details relegated to Section 4. In Secti*
*on 5,
we prove Theorem 1.9. This paper overlaps substantially with the second author's
thesis, [12].
2.Background in the UNSS
Let BP be the Brown-Peterson spectrum corresponding to the prime p. Then
BP* = ss*(BP ) Z(p)[v1; v2; : :]:;
where viare the Hazewinkel generators of BP*. Let = BP*(BP ) BP*[t1; t2; : :]*
*:,
where tiare Quillen's generators. We have |vi| = |ti| = 2(pi-1). Let c : BP*(BP*
* ) !
BP*(BP ) be the conjugation, and define hi= c(ti). Then = BP*[h1; h2; : :]:: L*
*et
j = jR : BP* ! BP*(BP ) be the right unit. We write hivj interchangeably with
j(vj)hi; this is the right action of BP* on .
FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS 5
Let M be a -comodule with coaction map M : M! BP*M. The stable cobar
complex C(T (M)) is defined as follows:
Cs(T (M)) = BP* . .B.P* BP*M;
with s copies of . We use the cobar notation, and write fl0[fl1| : :|:fls]m fo*
*r fl0
fl1 : : :fls m. The differential d is given by
X 0h 00 i
d(fl0[fl1| . .|.fls]m) = fl0 fl0| . .|.flsm
Xs h 0 00 i X h *
* 0i00
(2.1) + (-1)jfl0 fl1| . .|.flj|flj|m.+.|.fls(-1)s+1 fl0 fl1| . .|.fls|f*
*lm
j=1
P 0 00 P 0 00
where (fli) = flj fljfor 0 j s and M (m) = fl m :
The unstable cobar complex {C*(U(M)); d} is a subcomplex of {C*(T (M)); d},
consisting of terms satisfying an unstable condition, introduced in the followi*
*ng def-
inition.
Definition 2.2.[4, p. 243] If M is a nonnegatively graded free left A-module, t*
*hen
U(M) is defined to be the BP*-span of
{hI m : 2(i1+ i2+ i3+ . .).< |m|} BP*M;
where I = (i1; i2; : :):is a sequence containing only finitely many nonzero ij'*
*s, and
hI = hi11hi22... .
Define Us(M) = U(Us-1(M)), and Cs(U(M)) = Us(M). If M is a -comodule,
then U(M) T (M) = M as a sub--comodule, and so the differential d of the
stable cobar complex {Cs(T (M)); d} induces a differential on {Cs(U(M))} which
defines a complex {Cs(U(M)); d}. We will usually replace it by the chain-equiva*
*lent
reduced complex obtained by replacing U(M) by ker(U(M)-!fflM). This has the
effect of only looking at terms which have positive grading in each position. *
*The
homology groups of this unstable cobar complex are denoted by Exts;tU(M).
It was proved in [4] that, if X is a simply-connected CW -space, there is a s*
*pectral
sequence {Es;tr(X); dr} which converges to the homotopy groups of X localized a*
*t p,
and if the integral cohomology H*(X) is a free algebra, then
Es;t2(X) = Exts;tU(P (BP*X));
where P (BP*X) denotes the sub--comodule of BP*X consisting of the primitives
under the coproduct. This is the UNSS for the space X.
6 D. DAVIS AND H. YANG
The following basic formulas were used or proved in [5].
Lemma 2.3. (1)v1 = ph1+ j(v1) and (h1) = h1 1 + 1 h1;
P p p+1-ii i
(2)v2 = ph2 + (1 - pp-1)hp1v1 + j(v2) - (p + 1)vp1h1 + i=2aiv1 p h1, whe*
*re
ai2 Z;
(3)d(v1) = j(vn1) - vn1and
d(vahbvc) = (j(va) - va) hbvc- va (hb)vc- vahb (j(vc) - vc)
where (hb) = (hb) - hb 1 - 1 hb.
The first part of this lemma will be used very frequently in the context of rep*
*lacing
ph1 by v1- j(v1).
P F* P F P F
Let hi = c( c(hi)) = c( ti), where x +F y is the formal group sum
P i+1 P i+1
defined by x +F y = exp(logx + logy) with logx = i0mix , expx = i0bix ,
and exp(logx) = log(exp x) = x. Here {mi} and {bi} are two different polynomial
generators sets for H*BP with |mi| = |bi| = 2(pi- 1). Then the following lemma *
*of
Bendersky ([3]) is useful.
Lemma 2.4. The primitives P (BP*(SU(n)) form a free BP*-module generated by
elements {x3; x5; x7; :::; x2n-1} with coaction given by
X X F* j
(x2k+1) = ( hi)k-j x2j+1:
j i
The subscript k - j refers to the component in grading q(k - j). Here we have
introduced the notation q = 2(p - 1), which will be used frequently throughout *
*the
paper.
We will need the following explicit computation.
Proposition 2.5. Mod terms of degree greater than 3q,
X F* p-1
hi= 1 + h1- h1v1+ h1v21- ___2h21v1:
i
P F*
The only terms hj1which appear in hiare 1 + h1.
FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS 7
Proof.From
x = exp(logx)
= b0logx + b1(logx)2+ b2(logx)3+ . . .
X i X i X i
= b0 mixp + b1( mixp )2+ b2( mixp )3+ . .;.
i0 i0 i0
we deduceithatjthe first nonzero bi's are b0 = 1, bp-1 = -m1, b2p-2= pm21, and
b3p-3= p2- p(2p - 1). Then, mod terms of degree greater than 3q,
X F* X F
hi = c( ti)
i i i
X X pj j
= c bi( mjtk )i+1
i0 j;k0
i
= c 1 + m1+ t1- m1(1 + m1+ t1)p+ pm21(1 + m1+ t1)2p-1
i j j
+( p2- p(2p - 1))m31
i ij
= c 1 + m1+ t1- m1(1 + pm1+ pt1+ p2(m21+ t21) + p(p - 1)m1t1)
i j j
+pm21(1 + (2p - 1)(m1+ t1)) + ( p2- p(2p - 1))m31)
i ij j
= c 1 + t1- pm1t1+ p2m21t1- p2m1t21:
The desired result is now obtained using v1 = pm1 and c(v1) = j(v1).
The second statement follows since the only powers of t1appearing in the expa*
*nsion
P P pj i+1
of bi( mjtk ) are 1 + t1. ||
This coaction formula in SU(n) will be extremely important, as it determines *
*the
boundary homomorphism in the exact sequence associated to the fibration (1.1).
Indeed, there is an exact sequence
! Es;t2(SU(n - 1)) ! Es;t2(SU(n)) ! Es;t2(S2n-1)-@!Es+1;t2(SU(n - 1)) !;
with
(2.6) @(y) = y (x2n-1):
This boundary formula is true since gives the coboundary for the unstable cob*
*ar
complex of SU(n). The term 1 x2n-1will not appear in our reduced complex.
8 D. DAVIS AND H. YANG
We prefer to work with the v1-periodic UNSS of [1]. This has the advantage th*
*at
for X = SU(n) or S2n+1,
8
1, H0(hn+p-11 h12n+1) = -hp1by Lemma 2.10. By Lemma 2.11, this
equals -vp-11h1 6= 0, so that hn+p-11 h12n+1does not double desuspend. However,
H0(hn+p-11 h12n+1+ vp-11h1 hn12n+1) = -vp-11h1+ vp-11h1 = 0;
and so the sum double desuspends. ||
The following lemma will also be used many times. It is the one place where t*
*he
number of v1's on the left is important. We begin a policy of writing h for h1,*
* and v
for v1.
10 D. DAVIS AND H. YANG
Lemma 2.13. For j 0,
d(v`hn+1vj) -(` + n + 1)v`h hnvj+ jv`+1hn vj-1h mod S2n-1-qj:
Proof.We evaluate the desired boundary using Lemma 2.3, expanding
d(va) = (v - ph)a - va;
to obtain
X` ij Xn i j
` `-i i n+1 j n+1 ` i n+1-i j
iv (-ph) h v - i v h h v
i=1 i=1
Xj ij
- jiv`hn+1 vj-i(-ph)i:
i=1
For each term in each sum, we study whether it satisfies the unstable condition*
* on
S2n-1-qj(number of h's 1_2degree of stuff to the right of it) for both the par*
*t on
the left side of the and the part on the right side of the . In the first sum,*
* hion
the left of the satisfies the unstable condition on any sphere since pi with i*
*t can
be used (via ph = v - jv) to make the exponent of h small. Terms with i > 1 can
use p2 to bring the hn+1 on the right side of down to hn-1; then hn-1vj is def*
*ined
on S2n-1-qj. The (i = 1)-term is, mod S2n-1-qj,
-`v`-1h (v - jv)hnvj -`v`-1(v - ph)h hnvj -`v`h hnvj:
All terms in the second sum with i > 1 are defined on S2n-1-qj. In the third
sum, terms with i > 1 can use p2 to bring hn+1 down to hn-1, so that it satisfi*
*es
the unstable condition. Remaining p's can be used to bring hi down to h2, so th*
*at
it satisfies the unstable condition. The term with i = 1 gives the second term *
*in the
lemma since phn+1 = vhn - hnv. ||
The following corollary will be useful.
Corollary 2.14. If k + n 6 0 mod p, and z = (vkhn+p-1 h + pw0)2n+1is a cycle
with w02n+1satisfying the unstable condition 2.2, then
z = d(__1__k+n+pvk+p-1hn+12n+1+ w2n+1)
with w2n+1unstable.
FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS 11
Proof.Lemmas 2.12 and 2.13 imply that the leading term is correct. The group
E2n+1+q(k+n+p)2(S2n+1) has order p, and by Lemma 2.12 and Theorem 2.8(4), both *
*of
these leading terms yield nonzero classes. Since the other terms pw0 are p time*
*s an
unstable class, they cannot cancel the leading term, and the tail terms on the *
*second
representation must be an unstable boundary. ||
Our final preliminary is to introduce the spaces which are the factors in the*
* de-
composition of [10] of the p-localization of SU(n) and its quotients as a produ*
*ct of
p - 1 spaces.
Definition 2.15.If 1 N p - 1, let Xij(N) denote the direct factor space of the
p-localization of the space SU(N + i(p - 1) + 1)=SU(N + j(p - 1)) which is buil*
*t up
from fibrations involving p-local spheres S2N+1+kqfor j k i.
The UNSS of Xij(N) has E2-term Exts;tU(P (BP*(Xij(N)))), where P (BP*(Xij(N)))
has BP*-basis {x2N+jq+1; x2N+(j+1)q+1; : :;:x2N+iq+1}, with coaction induced fr*
*om
Lemma 2.4. Because of the sparseness results for spheres given in (2.7) and Th*
*e-
orem 2.8, the following is immediate.
Proposition 2.16. (1) For s = 1 or 2, v-11ss2k+1-s(Xij(N)) Es;2k+12(Xij(N)),
and is 0 unless k N mod p - 1.
(2)In the notation of Theorems 1.5 and 1.9,
v-11ss2k+1-s(SU(n)) v-11ss2k+1-s(Xi0(N)):
Thus our efforts in the remainder of the paper will be to show that for s = 1*
* or 2,
Es;2N+1+qm2(Xi0(N)) is as stated in Theorems 1.5 and 1.9. The fibrations whose *
*exact
sequence in E2(-) will be studied are of the form
Xjk(N) ! Xik(N) ! Xij+1(N)
for k j < i.
3. The cellular spectral sequence
In this section, we sketch the proof of Theorem 1.5 by showing (modulo details
postponed until Section 4) that E1;2N+1+qm2(Xi0(N)) Z=pe, where e is as in that
theorem. Here Xi0(N) is the space introduced at the end of the previous section*
*, and
all notation of Theorem 1.5 is in effect; in particular, N < p and i < p.
12 D. DAVIS AND H. YANG
Throughout the remainder of the paper, we consider N and m to be fixed, so th*
*at
they may be omitted from some notation. The expression Es2(-) will always mean
Es;2N+1+qm2(-), and we let Xij= Xij(N). Note that Xjj= S2N+1+qj. Our goal is to
determine Es2(Xi0) for s = 1 and 2.
We can organize this computation of Es2(Xi0) as a (cellular) spectral sequenc*
*e with
Es;j1= Es2(Xjj) for 1 s 2 and 0 j i, and dr : E1;jr! E2;j-rrinduced by pull*
*ing
the class in E12(Xjj) back to E12(Xjj-r+1) and then applying -@!E22(Xj-rj-r). O*
*ur desired
Es2(Xi0) is filtered with subquotients Es;j1, 0 j i. For s = 1, we know E12(X*
*i0) is
cyclic by Theorem 1.11, and so we just need to compute the sum
Xi
(3.1) (|E12(Xi0)|) = (|E1;j1|):
j=0
Note that for 1 s 2
8
0. We begin by calculating the
cellular spectral sequence when ^m= 0, this being the easiest of the three case*
*s. It is
easiest to describe the spectral sequence when i = p - 1, the maximal value tha*
*t we
are considering, and then to obtain the spectral sequence for smaller values of*
* i by
restriction.
Theorem 3.2. If ^m= 0, then in the cellular spectral sequence for Xp-10,
8
>>>Z=pmin(N;(m)+1)s = 1; j = 0
>><
Z=p s = 1; 1 j < N - (m)
Es;j1= Es;j1= >
>>>Z=p s = 1; N < j p - 1
>:
Z=p s = 2; 1 j p - 1:
dj 2;0 2;0
If max (1; N - (m)) j N, then Z=p E1;jj-!Ej is nonzero. Hence E1 = 0,
and E1;j1= 0 if max (1; N - (m)) j N.
If i < p - 1, then E1;j1(Xi0) will equal the group described in Theorem 3.2 i*
*f j i,
and will equal 0 if j > i. The same holds for E2;j1(Xi0) if j > 0 or i N; if i*
* < N,
FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS 13
Table 1. Spectral sequence when p = 11, N = 6, ^m= 0
| j
|| |
__________|0__1_2__3__4_5__6_7__8__9_10_|_
(m) 1 ||21 1 1 1 /1/1 1 1 1 1 ||
2 ||31 1 1 /1 /1/1 1 1 1 1 ||
3 ||41 1 /1 /1 /1/1 1 1 1 1 ||
4 ||51 /1/1 /1 /1/1 1 1 1 1 ||
________5_||6_/1_/1/1_/1_/1/1_1__1_1__1__||
then E2;01(Xi0) = Z=pmin(N-i;(m)+1). The reader can easily verify that the sum *
*(3.1)
when applied to 3.2 yields the desired result
8
**>>Z=p if0 j < N
>>>
>>>0 ifj = N
><
Z=p ifN < j < ^m
E1;j1= > min(m^(p-2)+N+2;+1)
>>>Z=p ifj = ^m
>>>
>>>Z=p if^m< j ^m(p - 1) + N + 1 -
:0 ifj > ^mand j > ^m(p - 1) + N + 1 - ;
provided j i.
14 D. DAVIS AND H. YANG
It is then an easy exercise to verify that the sum (3.1) has the desired value
8
>>**>:i ifN i < ^m
min(m^(p - 1) + N + 1; i + (m - ^m))if^m i:
Thus Theorem 1.5 in the case N < ^mwill be proved once we have proved Propositi*
*on
3.3.
In Table 2, we illustrate the spectral sequence in a particular case, namely *
*p = 11,
N = 2, and ^m= 7. Let = (m - ^m) = (m - 7). For each value of and each j
from 0 to 10, we list
(|E1;j1|) = (|E1;5+20m2(S5+20j)|):
Directly beneath this number, we list a number k if there is a nonzero differen*
*tial
E1;j! E2;k. Thus for example if = 68, there are nonzero differentials
E1;22! E2;02; E1;74! E2;34; E1;76! E2;16; E1;84! E2;44; E1;94! E2;54; *
* E1;104! E2;64:
In this case, we have 8
>>Z=p68 if5 r 6
>: 67
Z=p if7 r 1:
Proposition 3.3 is an immediate consequence of the following result, which de*
*scribes
the nonzero differentials in the spectral sequence when N < ^m. In Section 4 we*
* will
prove this result and its analogues in the other cases. We will frequently omit*
* writing
the subscript of the differential and the E-groups. It just equals the differen*
*ce of the
second superscripts.
Theorem 3.4. If N < ^m, the nonzero differentials in the spectral sequence con*
*verg-
ing to E2(Xi0) are those described below on groups E1;jwith j i.
(1)dN 6= 0 : E1;NN! E2;0N.
(2)For t > 1, there is a nonzero differential from xm^(p-2)+N+1+tto zt, where
8
< element of order pk in E1;m^ ifk (|E1;m^|)
xk = : 1;m^+ 1 1 1;m^
generator of E1 if = k - (|E1 |) > 0
8
>> generator of E2;t1 ifN < t < ^m
>: t-m^+1 2;m^
element of order p in E1if^m t
FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS 15
Table 2. Spectral sequence when p = 11, N = 2, ^m= 7
| j
|| |
|0__1_2__3__4_5__6____7____8_9__10_|
63 ||1 1 1 1 1 1 1 + 1 1 1 1 ||
| 0 |
|__________________________________|
= 64 ||1 1 1 1 1 1 1 65 1 1 1 ||
| 0 1 |
|__________________________________|
65 ||11 1 1 1 1 1 66 1 1 1 ||
| 0 1 3 |
|__________________________________|
66 ||11 1 1 1 1 1 67 1 1 1 ||
| 0 1 3 4 |
|__________________________________|
67 ||11 1 1 1 1 1 68 1 1 1 ||
| 0 1 3 4 5 |
|__________________________________|
68 ||11 1 1 1 1 1 69 1 1 1 ||
| 0 1,3 4 5 6 |
|__________________________________|
69 ||11 1 1 1 1 1 70 1 1 1 ||
| 0 1,3,4 5 6 7 |
|__________________________________|
70 ||11 1 1 1 1 1 71 1 1 1 ||
| 0 1,3,4,5 6 7 7 |
|__________________________________|
71 ||1 1 1 1 1 1 1 72 1 1 1 ||
| 0 1,3-6 7 7 7 |
_______|__________________________________|
16 D. DAVIS AND H. YANG
We take a little liberty with notation here; although the elements xk and zt are
defined as elements of E1, the differentials generally involve the projections *
*of these
elements in Er with r > 1.
Next we present the analogues of Proposition 3.3 and Theorem 3.4 when N ^m>
0. The condition
i j
(3.5) m ^m- (-1)m^^mN^mpm^(p-1)mod pm^(p-1)+1
will be important here.
Proposition 3.6. If N ^m > 0, then the nonzero groups E1;j1in the spectral se-
quence converging to E12(Xi0) are the following groups which also satisfy j i.*
* Here
= (m - ^m).
8
>>>Z=p j < ^m
>>>
>>>Z=pmin(+1;N+m^(p-2))j = ^m
>>>
>>>Z=p ^m(p - 1) and ^m< j < N
><
Z=p ^m(p - 1) < ^m(p - 2) + N - 2
>>> andm^ < j < ^m(p - 1) + N -
>>>
>>>Z=p j = N + 1 and not (3:5)
>>>
>>>Z=p j = N and (3:5)
>:
Z=p < ^m(p - 1) and N < j ^m(p - 1) + N + 1 -
As in the previous two cases, one can easily show that the sum (3.1) when app*
*lied
to the groups of 3.6 yields the desired result, Theorem 1.5, when N ^m> 0. Als*
*o as
in the previous case, Proposition 3.6 is easily derived from a listing of the d*
*ifferentials
in the cellular spectral sequence, which we now describe. The proof of the foll*
*owing
result will be given in Section 4.
Theorem 3.7. If N m^ > 0, the nonzero differentials in the spectral sequence
converging to E2(Xi0) are those described below on groups E1;jwith j i.
(1)If (m - ^m) < ^m(p - 1), then dN 6= 0 : E1;NN! E2;0N.
(2)If (3:5) is satisfied, then dN+1 6= 0 : E1;N+1N+1! E2;0N+1.
(3)If t 1 or if t = 0 and (m - ^m) ^m(p - 1) and (3:5) is not satisfied, t*
*hen
there is a nonzero differential from xm^(p-2)+N+1+tto zt, where
FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS 17
Table 3. Spectral sequence when p = 11, N = 7, ^m= 3
| j
|| |
_____________________|0_1__2___3___4__5__6_7__8_9__10_|
28 ||1 1 1 + 1 1 1 1 1 1 1 1 ||
| 0 |
|________________________________|_
= 29 ||1 1 1 30 1 1 1 1 1 1 1 ||
| 0 1 |
|_________________________________|
m 3 + 6p30 mod p31 ||1 1 1 31 1 1 1 1 1 1 1 ||
| 0 1 2 |
|_________________________________|
= 30; m 6 3 + 6p30 ||11 1 31 1 1 1 1 1 1 1 ||
| 0 1 2 |
|_________________________________|
= 31 ||1 1 1 32 1 1 1 1 1 1 1 ||
| 0 1 2 3 |
|_________________________________|
= 32 ||1 1 1 33 1 1 1 1 1 1 1 ||
| 0 1 2 3 3 |
|_________________________________|
= 33 ||1 1 1 34 1 1 1 1 1 1 1 ||
| 0 1 2 3 3 3 |
|_________________________________|
= 34 ||1 1 1 35 1 1 1 1 1 1 1 ||
| 0 1 2 3 3 3 3 |
|_________________________________|
= 35 ||1 1 1 36 1 1 1 1 1 1 1 ||
| 0,1 2 3 3 3 3 3 |
|_________________________________|
36 ||1 1 1 37 1 1 1 1 1 1 1 ||
| 0,1,2 3 3 3 3 3 3 |
_____________________|________________________________ |
8
>>gen of E1;m^+1 if = k - (|E1;m^1|) > 0 and ^m+ N
>: 1;m^++1 1;m^
gen of E1 if = k - (|E1 |) > 0 and ^m+ > N
8
j, and (3) if
a = 0, then j < N. Thinking of j and N as being fixed, let
8
<1 if` N < j
(4.3) ffl`= :
0 otherwise:
Then there is an element z 2 E12(Xja) satisfying
Xj
(4.4) z hj+1-`-ffl`y` mod L
`=a
which projects to a generator of E12(Xjj) Z=p. Hence dr = 0 : E1;jr! E2;j-rrf*
*or
r j - a.
FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS 19
Proof.The proof is by downward induction on a, and is true for a = j by parts 1
and 2 of Theorem 2.8. Assume that it is true for a certain a satisfying a > N +*
* 1 if
N < j, and also satisfying a - 1 > ^m. Let za denote the sum (4.4), which will *
*have
all ffl`= 0. We will show below that
@ : E12(Xja) ! E22(Xa-1a-1)
satisfies
Xj i j
@(za) N+(p-1)(a-1)`-a+1hj+1-` h`-a+1ya-1
`=a
(4.5) h hj-a+1ya-1 d(hj-a+2ya-1) mod L:
Thus za- hj-a+2ya-1is a cycle in E12(Xja-1), which extends the induction in thi*
*s case
(a > N + 1). (Remember that we are not usually worrying about unit coefficients*
*.)
Now we prove the first in (4.5). By (2.6), Lemma 2.4, and Proposition 2.5,
@(hj+1-`y`) = hj+1-` Cya-1, where C is the component of
i j
(1 + h - hv + hv2 - p-12h2v)N+(a-1)(p-1)
in degree q(` - a + 1). We will see that terms of degree greater than 3q in the*
* sum of
2.5 would not play an important role here. The term that appears in the sum of *
*(4.5)
is obtained from choosing (`-a+1) h's. If any other terms of 2.5 are chosen, th*
*e term
will desuspend farther because of the v's on the right. For terms in the sum of*
* 2.5
of degree greater than 3q, only pure powers of h1 could yield terms which desus*
*pend
as far as the one already considered, and such terms were shown in Proposition *
*2.5
not to exist. Note that a term such as h2 occurring in 2.5 desuspends no farthe*
*r than
h1, but it uses up many more degrees. If Ly` is a term that desuspends lower th*
*an
hj+1-`y`, then L is effectively htfor some t < j + 1 - `, and so @(Ly`) will de*
*suspend
farther than hj+1-` h`-a+1.
The second of (4.5) is true because the unstable condition easily implies th*
*at
other terms desuspend farther than the (` = j)-term. Also we need that the bino*
*mial
coefficient is a unit. This follows from Lemma 4.1 and the hypothesis that we d*
*o not
have a - 1 N < j. The third is true by Lemma 2.13, in which the coefficient of
the first term is
1_
(4.6) q((2N + 1 + qm) - (2N + 1 + q(a - 1))) = m - (a - 1);
which is not a multiple of p, since a - 1 > ^m.
20 D. DAVIS AND H. YANG
i j
The proof is similar when a - 1 N < j, except that N+(p-1)(a-1)j-a+1is now p
times a unit, by Lemma 4.1. Thus the last terms of (4.5) become ph hj-a+1ya-1
h hj-aya-1 d(hj-a+1ya-1), where we have used ph = v - j(v) to "cancel" the ph,
and then argued as before.
The condition that j + 1 N if a = 0 is required in order that all terms sati*
*sfy the
unstable condition, Definition 2.2. The term hj+1-aya is closest to failing. It*
* requires
j + 1 - a N + a(p - 1), which is satisfied if a > 0 or j + 1 N.
The statement about dr's being 0 follows from the definition of the spectral *
*se-
quence. If an element in E12(Xjj) pulls back to E12(Xja), then it survives to *
*E1;jj-a.
||
Lemma 4.2 must be modified as follows when a = ^m.
Lemma 4.7. Define = (m - ^m) and ffl` as in (4.3). Suppose ^m< j and
(4.8) j + 1 + - fflm^ N + ^mp:
Then, under the projection map,
Xj
hj+1-m^-ffl^m+ym^+ hj+1-`-ffl`y`2 E12(Xj^m)
`=m^+1
maps to a generator of E2(Xjj) Z=p. Hence dr = 0 : E1;jr! E1;j-rrfor r j - ^m.
The statement of this lemma initiates another abuse of notation which we will
allow ourselves. The class here is really only a cycle mod L. We mean here only*
* the
sort of thing that was stated explicitly in Lemma 4.2.
Proof.The generator of E12(Xjj) pulls back to z 2 E12(Xj^m+1) as in Lemma 4.2, *
*with
i j
@(z) N+(p-1)m^j-m^h hj-m^ym^ h hj-m^-ffl^mym^:
When we try to apply Lemma 2.13 to write this as a boundary, the first coeffici*
*ent
is m - ^m= sp , where s is a unit is Z(p). We don't worry about units, but the p
is missing from our term. To accommodate this, we note that, with e = j - ^m- f*
*flm^
and t + 1 + e = m - ^m,
vth he vt- h v he vt- h p he+
FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS 21
mod L. Here the first follows from the remarks after Theorem 2.8, and the sec-
ond follows from expanding p h = (v - j(v)) , noting that terms with j(v)'s wi*
*ll
desuspend farther than other terms. We obtain
@(z) p h hj-m^-ffl^m+ym^ d(hj+1-m^-ffl^m+ym^);
which satisfies the unstable condition by (4.8). The argument is completed as *
*in
Lemma 4.2. ||
A similar lemma tells how classes in E1;m^can be pulled back.
Lemma 4.9. Let ffl` = 1 if ` N < ^m, and ffl` = 0 otherwise. Suppose 0 a ^m
and that t - ffla N + a. Then hm^(p-2)+tym^2 E12(Xm^^m) pulls back, mod L, to *
*a cycle
P ^m `(p-2)+t-ffl`1 m^ 1;m^ 1;m^-r
`=ah y`2 E2(Xa ). Also, dr = 0 : Er ! Er for r ^m- a.
Proof.We assume that the formula is true for a (with class za), and will deduce*
* it
for a - 1. We assume at first that it is not true that a - 1 N < ^m, so that a*
*ll ffl's
are 0. We claim
X^mi j
@(za) N+(a-1)(p-1)`-a+1h`(p-2)+t h`-a+1ya-1
`=a
(4.10) ha(p-2)+t hya-1 h h(a-1)(p-2)+t-1ya-1 d(h(a-1)(p-2)+tya-1):
Then za-1 za - h(a-1)(p-2)+tya-1 is a cycle, extending the induction. The cla*
*im
about dr = 0 is immediate, as in the proof of Lemma 4.2.
The first in (4.10) follows by the argument used for the first of (4.5). The
second says that the term with ` = a desuspends least far of all terms, and has
coefficient a unit mod p. The coefficient is a unit since a - 1 6= N. The measu*
*re of
the size of the smallest sphere on which hi hj is defined is
exc(hi hj) = min(j; i - (p - 1)j):
For the terms in the sum of (4.10), this is largest when ` = a. The third foll*
*ows
from Lemma 2.12, and the fourth from Lemma 2.13, where the first coefficient is,
similarly to (4.6), m - (a - 1) 6 0 mod p.
This completes the proof of the lemma as longias it isjnot true that a-1 N <*
* ^m.
If a - 1 = N, then the binomial coefficient N+(a-1)(p-1)1is p times a unit, an*
*d this p
can be used to cancel one h, as in the proof of Lemma 4.2; this is accommodated*
* by
22 D. DAVIS AND H. YANG
ffl = 1. If a - 1 < N, then the t in the exponent has already been decreased to*
* t - 1,
and subsequent steps will leave it there.
The condition t-ffla N +a is necessary in order that all terms satisfy the u*
*nstable
condition. ||
Now we are ready to establish the differentials claimed in Theorems 3.2, 3.4,*
* and
3.7. The reader is encouraged to refer to Table 1 when considering this proof *
*of
Theorem 3.2.
Proof of Theorem 3.2. We emphasize that this is the case ^m= 0. Let 1 j p-1,
= (m), and ffl` be as in (4.3). By Lemma 4.2, a generator of E12(Xjj) pulls ba*
*ck
P j j+1-`-ffl 1;j 2;r-j
to z1 `=1h y``, and dr = 0 : Er ! Er for r < j. By Lemma 4.7,
dj = 0 : E1;jj! E2;0jif j < N - . If max(1; N - ) j N, then, as in the proof *
*of
Lemma 4.7, @(z1) hhjy0, and by Theorem 2.8, this is an element of order pN+1-j
in E2;01= E22(S2N+1). Thus E2;0N+1= 0, completing the proof. ||
The reader is encouraged to refer to Table 2 in the proof of Theorem 3.4 which
follows.
Proof of Theorem 3.4. If j < N, then dr = 0 on E1;jrfor all r by Lemma 4.2. A*
*lso
by Lemma 4.2, we have E1;jj= E1;j1if N j < ^m. We will show that
(4.11) dN 6= 0 : E1;NN! E2;0N:
Then E2;0N+1= 0 and hence dj is 0 on E1;jjwhen j > N, since there is nothing fo*
*r it to
hit. To prove (4.11), we use Lemma 4.2 to pull the generator of E1;N1back to
NX
z1 hN+1-`y`2 E12(XN1);
`=1
which satisfies @(z1) h hN y0 by part of (4.5). This is nonzero by Lemmas 2.10
and 2.11.
Next we consider the differentials from E1;m^. We first show that elements x*
* =
xm^(p-2)+N+1+tof order pm^(p-2)+N+1+twith N < t < ^msupport a nonzero differen-
tial to E2;t. By Theorem 2.8(2), x is represented, mod L, by hm^(p-2)+N+1+tym^*
* 2
E12(S2N+1+qm^). By Lemma 4.9, x can be extended, mod L, to
^mX
xt+1= h`(p-2)+N+1+ty`2 E12(Xm^t+1);
`=t+1
FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS 23
which implies that differentials from x into E2;jwith j t + 1 are 0. Then
(4.12) @(xt+1) (N + (p - 1)t)h(t+1)(p-2)+N+1+t hyt6 0 2 E22(S2N+1+tq):
The first here is obtained similarly to the first two steps of (4.10), while t*
*he last
step is from Lemma 2.12.
We have now completed the proof of the differential from xm^(p-2)+N+1+tto zt
asserted in Theorem 3.4 when t > N and ^m(p - 2) + N + 1 + t (|E1;m^1|). If in*
*stead
we have t N, but still ^m(p - 2) + N + 1 + t (|E1;m^1|), then in Lemma 4.9 ff*
*l`= 1
for ` N, and so x extends to
X^m XN
x0 h`(p-2)+N+1+ty`+ h`(p-2)+N+ty`:
`=N+1 `=max(t;1)
If t 1, this shows that dr(x) = 0 for r < ^m, and hence x is a permanent cycle,
since d2;0^m= 0. If t > 1, then, by part of (4.10), @(x0) ht(p-2)+N+t hyt-1, w*
*hich is
nonzero by Lemma 2.12.
This completes the analysis of the differentials of Theorem 3.4 on E1;jrwith *
*j ^m.
Now we will establish the differential on x = xm^(p-2)+N+1+twhen
(4.13) ^m(p - 2) + N + 1 + t > (|E1;m^1|):
Let e = (|E1;m^1|). Then x hym^(p-1)+N+1+t-e, and can be extended by Lemma 4.2
to
(4.14) x0 hym^(p-1)+N+1+t-e+ . .+.hm^(p-2)+N+1+t-eym^+1;
with
(4.15) @(x0) h hm^(p-2)+N+1+t-eym^:
By Corollary 2.9, this equals the element of order pt-m^+1on S2N+1+qm^, establi*
*shing
the claimed differential to ztwhen t ^m(so that the order is greater than 1).
Now we assume t < ^m. Let = (m - ^m). Recall that
e = (|E1;m^1|) = min( + 1; N + ^m(p - 1)):
Combining t < ^mwith (4.13) yields e < N + ^m(p - 1), and hence e = + 1. By
Lemma 4.7, the class x0of (4.14) extends to
x00 hym^(p-1)+N+t-+ . .+.hm^(p-2)+N+t-ym^+1+ hm^(p-2)+N+t+1ym^:
24 D. DAVIS AND H. YANG
By Lemma 4.9, x00can be extended to x000satisfying
8
N
x000 : P`=t+1^m-1 PN
x00+ `=N+1 h`(p-2)+N+t+1y`+ `=th`(p-2)+N+ty`ift N:
Then 8
N
@(x000) :
ht(p-1)+N hyt-1 ift N:
These are nonzero by Lemma 2.12, establishing the final differentials of Theore*
*m 3.4,
namely those from the second type of xk to the first two types of zt. ||
The proof of Theorem 3.7 proceeds similarly, except that in a certain case we
must keep track of unit coefficients because two terms are trying to have cance*
*lling
differentials. It is recommended that the reader consult with Table 3 while stu*
*dying
this proof.
Proof of Theorem 3.7. We divide into cases determined by the value of j, where *
*the
differential emanates from E1;j.
Case 1: j ^m. If j < ^m, then all differentials on E1;jare 0 by Lemma 4.2. F*
*or
j = ^m, we wish to show the differential asserted in 3.7 from xm^(p-2)+N+1+tto *
*the
generator of E2;t1. Here we must have t < ^min order that E1;m^1has an element *
*of the
asserted order. By Lemma 4.9 and (4.10), this xm^(p-2)+N+1+tpulls back to
x0 hm^(p-2)+N+1+tym^+ . .+.h(t+1)(p-2)+N+1+tyt+1;
which has @(x0) hN+(t+1)(p-1) hyt6 0, by Lemma 2.12.
Case 2: < ^m(p - 1) and ^m< j N. Here and throughout the remainder of
this proof, = (m - ^m). The claim here is that E1;j1consists of permanent cycl*
*es if
j < N, while dN 6= 0 : E1;NN! E2;0N. To establish this, we begin by using Lemma*
* 4.7
to pull the generator of E1;j1back to
(4.16) x0 hyj+ . .+.hj-m^ym^+1+ hj-m^++1ym^:
Let k be the integer satisfying k(p - 1) < (k + 1)(p - 1). Note that k satisf*
*ies
0 k < ^m. Next we show that x0 pulls back to a cycle x0in E12(Xj1) of the form
^m-1X ^m-k-1X
(4.17) x0 x0+ hj-m^++1-(m^-`)(p-2)y`+ hj+1-`y`:
`=m^-k `=1
(Actually, as will be discussed in the third bullet below, it is possible that *
*one of the
terms in this sum is incorrect, but this turns out to be inconsequential.)
FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS 25
To obtain (4.17), we note that if x0 has been pulled back to x002 E12(Xj`+1) *
*with
` + 1 ^m- k, with terms like those in (4.16) and the first sum of (4.17), then*
* @(x00)
can be evaluated by applying (4.5) to the terms of (4.16), and (4.10) to the te*
*rms of
(4.17). The leading terms will be hhj-`y`and hhj-m^+-(m^-`)(p-2)y`, respectivel*
*y.
Now there are three possibilities.
o If ` ^m- k and it is not the case that ` = ^m- k and = k(p - 1), then
j - ^m+ - (m^- `)(p - 2) > j - `, and so the leading term of @(x00) will*
* be
the second of those listed above. Thus, mod L, the extension of x00to E12*
*(Xj`)
is obtained by extending the first sum of (4.17).
o If ` = ^m- k - 1, then the leading term of @(x00) will be the first of th*
*ose listed
above. Thus, mod L, the extension of x00to E12(Xj`) contains the top term
in the last sum of (4.17). From this point on, subsequent extensions will*
* be
determined as they were in (4.5), by the very top term, hyj.
o If = k(p-1) and ` = ^m-k, these two leading terms will be equal. Actuall*
*y,
they may have different units as coefficients. If, when the units are tak*
*en into
account, these terms do not sum to 0, then the extension of x00to E12(Xj`*
*) is
obtained by extending the first sum of (4.17), just as it was in the firs*
*t case.
If, when the units are taken into account, these terms sum to 0 mod p, th*
*en
the powers of p in this coefficient can be used to cancel some of the h's*
* in the
ym^-k-term, rendering the expression (4.17) incorrect in this term. Howev*
*er, at
the next step, i.e., in going to ym^-k-1, and in subsequent steps, the ve*
*ry top
term will provide the leading term, just as it did in the second bullet a*
*bove,
and so this ambiguity in the ym^-k-term will be inconsequential.
If x0is as in (4.17), then the leading term in @(x0) will be h hjy0 2 E12(S2*
*N+1).
This is a consequence of the assumption that < ^m(p - 1), using the same sort *
*of
excess comparisons that occurred in the three possibilities above. This @(x0) i*
*s 0 if
j < N, since h hj2N+1 desuspends. Here and elsewhere, we use freely the fact t*
*hat
when E22(S2N+1) Z=p, then elements in it which desuspend are 0. If j = N, then
h hN 2N+1 6= 0 by Lemmas 2.10 and 2.11. This completes the proof of this case.
Case 3: = ^m(p - 1) and j < N. Here E1;j1consists of permanent cycles by the
argument just completed. Indeed, in this case, the first sum of (4.17) extends *
*down
to ` = 0 with all terms satisfying the unstable condition.
Case 4: = ^m(p - 1) and j = N. This is the delicate case, requiring that we
26 D. DAVIS AND H. YANG
keep track of units. We will show that, under these hypotheses, dN : E1;NN! E2;*
*0Nis
zero if and only if (3.5) is satisfied.
As in (4.17), a generator of E1;N1pulls back to
(4.18) hyN + . .+.hN-m^ym^+1+ hN+m^(p-2)+1ym^+ . .+.hN+p-1y1;
but now both the first and last terms hit a nonzero element of E22(X00) when @ *
*is
applied. We will show that if the coefficient of hyN is chosen to be 1, and if *
*m - ^m=
sp , with s 6 0 mod p, then, for 0 k < ^m, the coefficient of hN+(m^-k)(p-2)+*
*1ym^-k
in (4.18) is, mod p,
!
1 N - ^m+ k
(4.19) uk = (-1)k_ :
s k
Then E12(XN1)-@!E22(X00) satisfies, mod L,
i j
(4.20) @(hyN + um^-1hN+p-1y1) (h hN + um^-1N1 hN+p-1 h)y0
(1 - um^-1N)h hN 2N+1:
We have used Lemma 2.12(2) at the last step.
When @ is applied to (4.18), modified so as to properly incorporate unit coef*
*ficients
on all terms, the intermediate terms desuspend farther than the end terms, and *
*the
desired dN is determined by (4.20). Incorporating (4.19), we find that the imag*
*e of
this dN is
i j
1 + (-1)m^1_sN N-1^m-12 Z=p;
and this is 0 if and only if (3.5), as claimed.
It remains to prove (4.19), which we do by induction on k. For k = 0, we note
that, similarly to the proof of Lemma 4.7, for any numbers c`,
N-m^X i j
@(hyN + c`h`yN+1-`) N+m^(p-1)N-m^h hN-m^ym^
`=2
1_s(m - ^m)h hN-m^+ ym^ -1_sd(hN-m^++1 ym^);
and so adding 1_shN-m^++1 ym^extends the cycle, with u0 = 1_s.
FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS 27
Assume now that uk satisfies (4.19). Let z = hyN + . .+.ukhN+(m^-k)(p-2)+1ym^*
*-k.
Then, mod L, we have
i j
@(z) uk N+(m^-k-1)(p-1)1hN+(m^-k)(p-2)+1 hym^-k-1
-uk(N - ^m+ k + 1)h hN+(m^-k-1)(p-2)ym^-k-1
___1____m-m^+k+1uk(N - ^m+ k + 1)d(hN+(m^-k-1)(p-2)+1ym^-k-1)
i j
(-1)k1_sN-m^+k+1k+1d(hN+(m^-k-1)(p-2)+1ym^-k-1):
i j
Thus adding (-1)k+11_sN-m^+k+1k+1hN+(m^-k-1)(p-2)+1ym^-k-1to z yields a cycle m*
*od L
in E12(XN^m-k-1) which projects to z. The coefficient here is just what we have*
* claimed
is uk+1, and so this extends the induction.
Case 5: > ^m(p - 1) and ^m< j N. These are the second type of elements
xk in Theorem 3.7. As before, let e = (|E1;m^1|) = min( + 1; N + ^m(p - 1)). Th*
*en,
in the notation of Theorem 3.7, our j becomes ^m(p - 1) + N + 1 + t - e. We show
in the next paragraph that if t ^m, then the differential from E1;jhits the el*
*ement
of order pt-m^+1in E2;m^1. Of course, this hit element should really be viewed *
*as the
element of order p in E2;m^j-m^, elements of order less than pt-m^+1already hav*
*ing been
hit by shorter differentials.
By the first sentence of the proof of Lemma 4.7, a generator of E12(Xjj) pull*
*s back
to an element x 2 E12(Xj^m+1) satisfying @(x) h hj-m^ym^. This latter element
has order pt-m^+1in E22(S2N+1+qm^) by Corollary 2.9. Indeed, the equation of t*
*hat
corollary becomes
(m^(p - 1) + N + 1 + t - e - ^m) + e = (t - ^m+ 1) + (N + ^m(p - 1)):
If, on the other hand, t < ^m, then by Lemma 4.7 the generator of E1;jpulls b*
*ack
to a cycle
Xj
x0 hj+1-`y`+ hm^(p-2)+N+t-e++2ym^2 E12(Xj^m):
`=m^+1
We can simplify this by noting that the hypotheses imply e = + 1. (If not, then
e = ^m(p - 1) + N, so j = t + 1, contradicting j > ^m> t.) Now by Lemma 4.9 we
can pull x0back to
^m-1X
x00 x0+ h`(p-2)+N+t+1y`2 E12(Xjt+1):
`=t+1
28 D. DAVIS AND H. YANG
As in (4.10), @(x00) h(t+1)(p-1)+N hyt, which is nonzero by Lemma 2.12. We
observe that @(hyj) cannot interfere here, since this yields a term which desus*
*pends
unless j = N and t = 0, and this is impossible since, if t = 0,
j = ^m(p - 1) + N + 1 - e = ^m(p - 1) + N - < N:
Case 6: j > N. In the notation of the theorem, we have
j = m^+ + 1 = ^m+ k - e + 1
(4.21) = m^+ (m^(p - 2) + N + 1 + t) - e + 1 = ^m(p - 1) + N + 2 + t - e:
The generator of E12(Xjj) can be pulled back as in Lemma 4.2 to a class x 2 E12*
*(Xj^m+1)
satisfying @(x) h hj-m^-1ym^, similarly to part of (4.5) except that ffl = 1.*
* Using
Corollary 2.9 and (4.21), we obtain that the order of hhj-m^-12N+1+qm^is pt-m^+*
*1, as
claimed, provided t ^m. Note that if j = N +1, then the class which this diffe*
*rential
would like to hit has already been killed by a differential from E1;N.
If t < ^m, then by Lemma 4.7 a generator of E12(Xjj) pulls back to
x hyj+ . .+.hj-m^-1ym^+1+ hj-m^+ym^2 E12(Xj^m):
By Lemma 4.9, x can be extended farther, to
^m-1X
x0 x + h`(p-2)+N+2+t-e+y`2 E12(Xjt+1);
`=t+1
with @(x0) h(t+1)(p-1)+N hyt 6= 0 by Lemma 2.12. We must worry here about
possible cancellation from @(hyj) ch hj-tyt. This will desuspend, and hence be
0, if t > 0. Thus it remains to consider the possible cancellation when t = 0.
If > ^m(p - 1), then t must be greater than 0. This can be deduced using j >*
* N,
(4.21), and e = min( + 1; N + ^m(p - 1)).
If ^m(p - 1) and (3.5) is not satisfied, then it was proved in Cases 2 and 4
that E2;0N+1= 0, and so we need not worry about the case t = 0 here. Finally, *
*we
must establish that dN+1 6= 0 : E1;N+1N+1! E2;0N+1if (3.5) is satisfied. To pro*
*ve this by
the methods employed so far in this section is more delicate than we care to pr*
*esent.
Instead, we deduce it using differentials already determined, together with res*
*ults for
cyclicity of certain E2-groups which will be established in the next section.
So, we are assuming (3.5). In particular, = ^m(p - 1) and e = + 1. Earlier *
*in
Case 6, we verified that under this hypothesis dN+1 6= 0 : E1;N+2N+1! E2;1N+1. *
*This says
FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS 29
that the composite ae O @ 6= 0 in (4.22), and that | coker(i2)| = p.
ae 2 1
(4.22) E12(XN2)-i1!E12(XN+12)-i2!E12(XN+22)-@!E22(X10)-! E2(X1)
By Proposition 5.2, ae is a surjection from Z=p2 to Z=p. Thus @ is surjective,*
* and
hence @ O i2 6= 0, hitting E22(X00) E22(X10). On the other hand, the 0-differe*
*ntial
from E1;NNto E2;0Nestablished in Case 4 implies that the composite @ O i2O i1 i*
*n (4.22)
is 0. Thus @ O i2 is nonzero on an element which projects nontrivially to E12(X*
*N+1N+1),
and this establishes dN+1 6= 0 : E1;N+1N+1! E2;0N+1, as desired. ||
5.Group structure
In this section, we prove Theorem 1.9; we determine the group structure on the
2-line of the UNSS of SU(n) for n p2 - p + 1. We also prove a few subsidiary
results about group structure of 1-line groups E12(Xba).
The conventions of the previous sections are still in force. Thus p, N, and m*
* are
fixed, with 1 N < p. The second superscript of all E2-groups is 2N + 1 + qm
(unless specifically stated to the contrary), and ^m is the least nonnegative r*
*esidue
of m mod p. Also, Xbais a direct factor of a complex Stiefel manifold (quotient*
* of
SU(n)'s) built from spheres 2N +1+qj for a j b. Also, we continue the practice
of omitting v1's on the left and units in Z(p)whenever they are unimportant.
Although Theorem 1.11 implies that E12(Xb0) is cyclic, it will not always be *
*the
case that E12(Xba) is cyclic. The following result presents one case in which E*
*12(Xba)
will be cyclic.
Proposition 5.1. The group E12(Xba) is cyclic if there are no numbers j congrue*
*nt
to N mod p satisfying a j < b.
Proof.The proof is by induction on b. The result is true when b = a by Theorem
2.8(1). We assume it is true for Xba, and will deduce it for Xb+1a, provided b*
* 6 N
mod p. We consider the commutative diagram of exact sequences below.
j* 1 b+1 ae* 1 b+1 @ 2 b
0 ! E12(Xba)-!? E2(Xa? ) -! E2(Xb+1)? -! E2(Xa)
?? ? ?
y f* ?yg* ?y=
j1* 1 b+1 1 b+1
0 ! E12(Xbb)-! E2(Xb ) -! E2(Xb+1)
If j* is surjective, then E12(Xb+1a) is cyclic by the induction hypothesis, and*
* so we are
done. Thus we may assume that there is an element fi such that ae*(fi) = ff, wh*
*ere
30 D. DAVIS AND H. YANG
ff has order p. Since b 6 N mod p, the attaching map in Xb+1bis nontrivial, an*
*d so
[5, 2.23] implies that there is a generator fl of E12(Xbb) such that j1*(fl) = *
*p . g*(fi).
Exactness implies that there is y such that j*(y) = pfi. Then j1*(f*(y) - fl) *
*= 0,
so that f*(y) = fl, and hence y is a generator of the cyclic group E12(Xba). Th*
*us the
extension is cyclic. ||
Next we prove a result that implies the first case of Theorem 1.9.
Proposition 5.2. The group E22(Xba) is cyclic if either b N or N < a b < p.
Proof.The proof is by induction on b and is true for b = a by 2.8(1). We assume*
* the
result for Xba, and consider the exact sequence
ae* 2 b+1 @
E22(Xba) ! E22(Xb+1a) -! E2(Xb+1)-! 0;
with b 6= N. If
(5.3) E22(Xb+1b+1) Z=pN+(b+1)(p-1);
then by Theorems 3.4 and 3.7, E22(Xba) = 0, and so we are done in this case, si*
*nce ae*
will be iso. The easiest way to believe this claim that E22(Xba) = 0 is to look*
* at the
last row of Tables 2 and 3. In the first one, it is the assertion that E1;7kill*
*s E2;jfor
3 j 6, and in the second one, that E1;3kills E2;jfor 0 j 2.
Thus we may assume that (5.3) is not true, and so d(hN+(b+1)(p-1)+1)yb+1has o*
*rder
p in E22(Xb+1b+1), by Theorem 2.8(4). Since @ annihilates this class, there is *
*w 2 E21(Xba)
such that z = d(hN+(b+1)(p-1)+1)yb+1- w is a cycle in E22(Xb+1a). We wish to sh*
*ow
that pz is the image of a generator of E22(Xba).
We have
(5.4) d(hty) = d(ht)y + htd(y);
where d(ht) is as in Lemma 2.3(3). Note here that the minus sign which is attac*
*hed
to (fl1) in (2.1) has been incorporated into d(ht) (which is really d(vsht)) i*
*n Lemmas
2.3(3) and 2.13. Now, mod L,
pz d(hN+(b+1)(p-1)yb+1) - (N + b(p - 1))hN+(b+1)(p-1) hyb- pw:
Here we have used ph = v - j(v) and (5.4). The first term is a boundary, and so*
* is
ignored. Note that the exponent of h had to be brought down to N + (b + 1)(p - *
*1)
in order that it be placed in front of yb+1. Since b 6= N, the coefficient of t*
*he second
FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS 31
term is a unit, and so will be ignored. Since w was defined on Xba, pw desuspen*
*ds
below SN+b(p-1), and so may be incorporated into L. Thus, by Lemma 2.12(2), pz
generates E22(Xbb) mod L, and so pz is the image of a generator of E22(Xba), as*
* desired.
||
Next we prove the second part of Theorem 1.9, which we restate as follows.
Theorem 5.5. Suppose i > N and ^m> 0. If e is as in Theorem 1.5, then E22(Xi0)
Z=pe-1 Z=p.
Proof.In the exact sequence
OE 2 i ae 2 i
(5.6) E12(XiN+1)-@!E22(XN0)-! E2(X0)-! E2(XN+1) ! 0;
the groups E22(XN0) and E22(XiN+1) Z=pa are cyclic by Proposition 5.2. We will
show that if x 2 E22(Xi0) is such that ae(x) is a generator, then pax = pOE(g),*
* with g
a generator of E22(XN0). The result is immediate from this, with x and OE(g) - *
*pa-1x
generating the summands.
Case 1: N = ^m. The thing that distinguishes this case is that pg lives on t*
*he
top cell of XN0, i.e., E22(XN0) ! E22(XNN) sends pg nontrivially. Thus the exte*
*nsion
question can be studied in the exact sequence
0 ! E22(XNN) ! E22(XiN) ! E22(XiN+1) ! 0:
We let x also equal the image of a generator of E22(Xi0) in E22(XiN). As in the*
* proof
of 5.2, we can write
pa-2x = d(hN+(N+2)(p-1)+1)yN+2 + w0yN+1 + w1yN ;
with each term unstable. Now we can use (5.4), (2.6), 2.4, and 2.5 to write
pa-1x = d(hN+(N+2)(p-1)yN+2) + pw0yN+1 + pw1yN
i i j j
-hN+(N+2)(p-1) (N + (N + 1)(p - 1))hyN+1 - NphvyN + Np2h2yN :
We ignore the boundary term and apply Corollary 2.14 to the terms on yN+1. We
also write hv = vh - ph2 and ignore the p2w0yN term (with w0yN defined) which t*
*his
yields. We obtain
pa-1x = -(N+1)p-1_m-N-1d(hN+(N+1)(p-1)+1+ w)yN+1 - NphN+(N+2)(p-1) (vh + uh2)*
*yN
+pw1yN ;
32 D. DAVIS AND H. YANG
where wyN+1 is defined, and u is a unit. We multiply by another p and ignore te*
*rms
which are p2 times an unstable class on yN . We obtain
pax u0d(hN+(N+1)(p-1)+ pw)yN+1
i j
u0 d((hN+(N+1)(p-1)+ pw)yN+1) - Np(hN+(N+1)(p-1)+ pw) hyN
with u0a unit. Here we have applied (5.4) again. By Lemma 2.12, this is p times*
* a
generator of E22(XNN).
Case 2: N > ^mor N < i < ^mor (N < ^m i and i+(m-m^) < ^m(p-1)+2N).
In these cases, E2;j1= E2;j1for j N - 1, and so in the short exact sequence
(5.7) 0 ! coker(@) ! E22(Xi0) ! E22(XiN+1) ! 0;
the generator g of coker(@) lives on yN , and pg lives on yN-1. Hence (5.7) map*
*s to a
short exact sequence
OE02 i 2 i
0 ! E22(XNN-1)-! E2(XN-1) ! E2(XN+1) ! 0;
and it suffices to show OE0(pg) = pax in this latter sequence.
The analysis is quite similar to Case 1. We write
pa-2x = d(hN+(N+2)(p-1)+1)yN+2 + w0yN+1 + w1yN + w2yN-1
with each term unstable. Then, ignoring terms which are p times an unstable cla*
*ss
on yN-1, and letting u, u0, etc., denote units in Z(p), we obtain
pa-1x d(hN+(N+2)(p-1)yN+2) + pw0yN+1 + pw1yN
i i j
-hN+(N+2)(p-1) ((N + 1)p - 1)hyN+1 - NphvyN + Np2h2yN
j
+(uhv2 + u0h2v)yN-1 :
Now we ignore boundaries and use 2.14, 2.3(1), and 2.3(2) to obtain
pa-1x -(N+1)p-1_m-N-1d(hN+(N+1)(p-1)+1+ w)yN+1
+NphN+(N+1)(p-1) hyN + pw1yN + u00phN+(N+2)(p-1) h2yN
-hN+(N+2)(p-1) (uhv2 + u0h2v)yN-1:
Now we multiply by p again, again omit p times unstable classes on yN-1, and in*
*cor-
porate the two surrounding terms into w1, obtaining
pax u000d(hN+(N+1)(p-1)+ pw)yN+1 + p2w01yN :
FORMULAS FOR v1-PERIODIC HOMOTOPY GROUPS 33
Now we apply (5.4) again, and omit writing the d(hy)-term, obtaining
i
pax -u000(hN+(N+1)(p-1)+ pw) NphyN - ((N - 1)p + 1)hvyN-1
i j j!
+ (N-1)p+12h2yN-1 + p2w01yN :
We omit lots of terms which are p times an unstable class on yN-1. We omit the
-u000-factor, which can be considered as a factor of the entire expression. We*
* also
combine together several terms that are p2 times an unstable class on yN . Fina*
*lly,
we apply Lemma 2.3(1) to hv. This yields
pax Np(hN+(N+1)(p-1) h + pw001)yN - hN+(N+1)(p-1) vhyN-1:
Now we apply 2.14 to the first term and 2.3(2) to the second term, which effect*
*ively
cancels hp-1v. This yields
pax _N__m-Npd(hN+N(p-1)+1+ w3)yN - hN+N(p-1) hyN-1
_N__m-Nd(hN+N(p-1)+ pw3)yN - hN+N(p-1) hyN-1;
with w3yN defined. We apply (5.4) once again, and omit writing the term of the *
*form
d(hy), obtaining
pax -__N_m-N((N - 1)p + 1)(hN+N(p-1)+ pw3) hyN-1 - hN+N(p-1) hyN-1
-(__N_m-N+ 1)hN+N(p-1) hyN-1:
Since __N_m-N+ 1 = __m_m-N, this is nonzero by Lemma 2.12 since m 6 0 mod p by
assumption.
Case 3: N < ^m i and i+(m-m^) ^m(p-1)+2N. In this case, Theorem 3.7
(with Table 2 again recommended to provide insight) implies (1) E2;j1= 0 for j *
*< N,
and hence E22(XN0)= im(@) Z=p, and (2) E22(XiN+1) ! E22(Xi+1N+1) is multiplica*
*tion
by p of groups each isomorphic to Z=pa. To see (2), note first that the generat*
*or of
E22(Xi+1N+1) lives on yi+1, and second that E12(Xi+1i+1)-@!E22(XiN+1) hits the *
*elements of
order p. If i = p - 1, then we need to extend Theorem 3.7 to include the case i*
* = p.
This extension is straightforward; it is only Theorem 3.2 that must be modified*
* when
i = p.
34 D. DAVIS AND H. YANG
Now the splitting follows from the following commutative diagram of exact se-
quences.
0 ! E22(XN0)=?im(@) ! E22(Xi0)?! E22(XiN+1)?! 0
?? ? ?
y= ?yOE ?y.p
0 ! E22(XN0)= im(@) !E22(Xi+10)! E22(Xi+1N+1) ! 0
Indeed, suppose to the contrary that E22(Xi0) Z=pa+1 with generator G. Then
the left square implies paOE(G) 6= 0, and so we must have E22(Xi+10) Z=pa+1 wi*
*th
generator OE(G). Then the composition around the bottom of the right square is
surjective, but the composite around the top is not surjective. ||
Finally, we prove the third and fourth cases of Theorem 1.9, which we restate*
* as
follows. Again, we remind the reader that the conventions which were restated *
*at
the beginning of this section continue to be in effect.
Theorem 5.8. Suppose ^m= 0, i > N, and t = min(N; (m) + 1). Then
8
0 in
Theorem 3.2, or that in Table 1 the only elements hit by differentials have j =*
* 0.
Now we prove the last isomorphism of (5.8). We use the following commutative
diagram of short exact sequences.
0 ! E12(XN0)?! E12(Xi0)!?E12(XiN+1)?! 0
?? 0 ? ?
yOE ?yOE ?y=
0 ! E12(XN1)-! E12(Xi1)! E12(XiN+1) ! 0
By Theorem 3.2, illustrated in Table 1, the diagram is
0 ! Z=pN?! Z=pi? ! Z=pi-N?! 0
?? t ? ?
y.p ?yOE ?y=
0 ! Z=pN -! E ! Z=pi-N ! 0:
The key observation here is that OE0= .pt, which we now explain. By Theorem 3.2,
there are no differentials in the cellular spectral sequence for XN1, but in th*
*e spectral
sequence for XN0, there are differentials from E1;jto E2;0for N - t + 1 j N.
Thus the generator of E12(Xi0) sits on yN-t, which is where pttimes the generat*
*or of
E12(Xi1) lives.
Now it is a matter of simple algebra to determine the structure of the abelian
group E = E12(Xi1) in the diagram above. Let g denote a generator of E12(Xi0), *
*and
let h denote a generator of E12(XN1). Then pi-NOE(g) = pt(h). If t = N, this im*
*plies
pi-NOE(g) = 0, and so E Z=pN Z=pi-N with generators (h) and OE(g). If t < N
and t i - N, then E Z=pt Z=pi-twith generators (h) - pi-N-tOE(g) and
OE(g). If t < N and t i - N, then E Z=pN Z=pi-N with generators (h) and
OE(g) - pt-i+N(h). ||
We close with a lemma which was used in the preceding proof. Previous convent*
*ions
for N and ^mapply.
36 D. DAVIS AND H. YANG
Lemma 5.10. Suppose (L) N + m(p - 1) - 1, ^m= 0, and 1 j < p. Then
ffL;N+m(p-1) : E1;2N+1+qm2(S2N+1+qj) ! E2;2N+1+qm+qL2(S2N+1+qj)
is bijective.
Proof.Since both groups have order p, it suffices to show ffL=(N+m(p-1))ffm-j2N*
*+1+qj
is nonzero in E2. Let s = L-N -m(p-1). By Theorem 2.8(2), this class is equival*
*ent,
mod terms that desuspend, to
vshN+m(p-1) vm-j-1h2N+1+qj:
By Lemma 2.13, this is equivalent to
i j
_1_ s-1 N+m(p-1)+1m-j s-1 N+m(p-1)m-j
m-j d(v h v )2N+1+qj+ Lv h h v 2N+1+qj:
Since L is highly p-divisible, the second term is 0 in E2. The first term is si*
*mplified
by using Lemma 2.12(2) to replace (hpv)m-j by (vph)m-j. Thus the desired term i*
*s a
unit times d(vL-N+m-jp-1hN+j(p-1)+1)2N+1+qj, which is nonzero by Theorem 2.8(4).
||
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Lehigh University, Bethlehem, PA 18015
E-mail address: dmd1@lehigh.edu
Lehigh University, Bethlehem, PA 18015
E-mail address: hy02@lehigh.edu
*