ContemporaryVMathematicsolume 00, 0000
Equivalences of some v1-telescopes
DONALD M. DAVIS
Abstract.Certain naturally occurring spaces have isomorphic v1-periodic
homotopy groups. To each is associated a mapping telescope whose ordi-
nary homotopy groups equal the v1-periodic homotopy groups of the space.
It is proved that the mapping telescopes of the spaces are homotopy equi*
*v-
alent.
1. Main theorem
In [10 ], Harper constructed a 3-primary finite complex K which is a direct
factor of the exceptional Lie group F4 localized at 3. This complex fits into a
3-local fibration
(1.1) B(3; 7) ! K ! W;
where W is the Cayley plane, and B(3; 7) is a 3-local S3-bundle over S7 with
attaching map ff1. In [4], the 3-primary v1-periodic homotopy groups v-11ss*(K)
were computed, en route to the determination of v-11ss*(F4). One can observe
that these groups satisfy
(1.2) v-11ss*(K) v-11ss*+d(S25); where d = 8 . 310+ 2:
For any spherically resolved space Y , such as K or dS25, there is a periodic
-spectrum Tel1(Y ), defined in [7], satisfying
ss*(Tel1(Y )) v-11ss*(Y ):
Here a space Y is called spherically resolved if, for some k 0, kY can be
built from spheres or loops on spheres by fibrations. To see that K is spherica*
*lly
resolved, we use (1.1) and the fibration
S7 ! W ! S23;
____________
1991 Mathematics Subject Classification. 55P15.
Key words and phrases. v1-telescopes, homotopy equivalence.
This paper is in final form and no version of it will be submitted for publi*
*cation elsewhere.
cO00000American0Mathematical0Societ*
*y0-0000/00 $1.00 + $.25 per page
1
2 DONALD M. DAVIS
established in [8].
The isomorphism (1.2) leads one to conjecture the following result, which is
our main theorem.
Theorem 1.1. There is an equivalence of 3-local spectra,
Tel1(K) Tel1(dS25);
where d is as in (1:2).
This equivalence is induced by a map K ! d+4k311S25 for some nonnegative
integer k.
A result similar to Theorem 1.1 holds for the space B(3; 7) which appears in
(1.1). We have chosen to emphasize the result for K because it seems a bit more
surprising. The result for B(3; 7) can be generalized to an arbitrary odd prime
as follows.
Theorem 1.2. Let p be an odd prime, and let B(3; 2p + 1) be the p-local
S3-bundle over S2p+1 with attaching map ff1. Then there is a p-local equivalence
0 2p+3
Tel1(B(3; 2p + 1)) Tel1(d S );
where d0= 2(p - 1)2pp-1 + 2.
The analogue of Harper's complex K can also be constructed at an arbitrary
odd prime p, and indeed he performed this construction in [11 ]. In Section 3,
we present Harper's unpublished proof that this complex Kp fits into a p-local
fibration
B(3; 2p + 1) ! Kp ! ^S2p+2;
where ^S2p+2is the (p - 1)(2p + 2)-skeleton of S2p+3. We remark that when
p = 3, Kp fibers over both ^S8and W with the same fiber. The spaces ^S8and
W are not 3-equivalent, but their loop spaces are. In Section 3, we will calcul*
*ate
v-11ss*(Kp), and use this result to generalize Theorem 1.1 to the following res*
*ult.
Theorem 1.3. There is an equivalence of p-local spectra,
002p2+2p+1
Tel1(Kp) Tel1(d S );
where d00= 2(p - 1)2pp2+p-2+ 2.
These somewhat surprising equivalences lead one to wonder to what extent
v1-telescopes are determined by their homotopy groups. We would like to thank
Mark Mahowald for pointing out the possibility of such equivalences, and John
Harper for answering with a proof a question which became Theorem 3.1.
EQUIVALENCES OF TELESCOPES 3
2. Proof of Theorem 1.1
In this section, we prove Theorem 1.1, leaving the similar and easier proof
of Theorem 1.2 for the reader. We show1that0both Tel1(K) and Tel1(dS25)
are equivalent to the spectrum v-11M4.3 +23(312). Here Mn (k) = Sn-1 [k en
denotes a Moore spectrum, and, as in [9], v-11Mn (pe) is the telescope of
e-1 e n-2qpe-1 e
Mn (pe) ! Mn-qp (p ) ! M (p ) ! . .;.
with the maps all being suspensions of the same Adams map. Here and through-
out the paper, q = 2(p - 1).
We begin by quoting from [4] the following result. Here and elsewhere p(-)
denotes the exponent of p, with p(0) = 1.
Proposition 2.1.With d as in (1:2), and ffl = 0 or 1,
( 10
Z=3min(12;3(i-2.3 -11)+1)i odd
v-11ss2i-ffl(K) v-11ss2i-ffl(dS25)
0 i even.
Let Y = K or dS25. For0t = 4.3e with e 11, there is, for some nonnegative
integer m, a map mtY -A! (m+1)tY constructed as in [7] using the period-t
Adams map of the mod 3e+1 Moore space and a null-homotopy of 3e+1 on mtY .
If T denotes the mapping telescope of
0 tA0
mtY -A! (m+1)tY -! (m+2)tY ! . .;.
then Tel1(Y ) is the -spectrum with T in each0index0which is a multiple of t.
There is an obvious equivalence Tel1(Y ) A-! tTel1(Y ).
Let n = 4.310+23 throughout this section, and let Sn-1 ! Tel1(Y ) represent
a generator of ssn-1(Tel1(Y )) v-11ssn-1(Y ) Z=312. This extends uniquely
to a map g : Mn (312) ! Tel1(Y ), inducing an isomorphism in ssn-1(-). Let
g0 : Mn-t(312) ! tTel1(Y ) be adjoint to g. We can form a commutative
diagram
Mn (312)? -A! Mn-t(312)?
?yg ?yg0
00
Tel1(Y ) -A! tTel1(Y )
with A an appropriate Adams map. The commutativity is a consequence of the
isomorphism
[Mn (312); tTel1(Y )] ssn-1(tTel1(Y )) Z=312:
This diagram induces a map of direct systems Mn-mt (312) ! mt Tel1(Y ),
yielding in the limit the map v-11Mn (312) ! Tel1(Y ) which is to be our equiva-
lence.
We now must show that this map induces an isomorphism in ss*(-). When
Y = dS25, the result is a special case of the equivalence
v-11M0(pk) ! Tel1(2k+1S2k+1);
4 DONALD M. DAVIS
which is proved by induction on k from the commutative diagram of fibrations
v-11M-1 (p) ! v-11M0(pk-1) ! v-11M0(pk)
# # #
Tel1(2k-1W (k)) ! Tel1(2k-1S2k-1) ! Tel1(2k+1S2k+1):
Here W (k) is the usual fiber of the double suspension; that the left vertical
arrow is an equivalence was noted in [9]. We have also used the equivalence
v-11M25-d(312) ' v-11Mn (312).
To prove the isomorphism when Y = K, we need to review from [4] the way
in which v-11ss*(K) is obtained. Looping the map K ! W of (1.1) and follow-
ing by the map W ! S23 of [8] induces an epimorphism v-11ss4i+1(K) !
v-11ss4i+1(S23), which is bijective unless v-11ss4i+1(K) Z=312, in which case *
*it
has kernel Z=3. Obstruction theory shows that, after looping sufficiently, there
is a commutative diagram as below. We continue to let n = 4 . 310+ 23.
v-11Mn?(312) ! v-11Mn?(311)
?yh ?yf
Tel1(K) ! Tel1(W ) ! Tel1(S23)
The preceding paragraph showed that f induces an iso in ss4i+1(-), and the
diagram and preceding statements then imply the same for h.
The map v-11Mn (312) -h! Tel1(K) induces an isomorphism in ssi(-) when-
ever i n - 1 mod 4 . 311, in which case both groups are Z=312. We show that
h induces an iso in ssi(-) whenever i 2 mod 4 by showing that we can choose
elements ai (resp. bi) of order 3 in ss4i+2(v-11Mn (312)) (resp. ss4i+2(Tel1(K)*
*))
so that h*(ai) = bi. This is accomplished by induction on i once we observe that
the classes are related by the following Toda brackets.
(2.1) ai+1 =
(2.2) bi+1 = :
The case i n - 1 mod 4 . 311 starts the induction.
Equation (2.1) is well-known in Mn (3), as this bracket defines the Adams
periodicity. The canonical map Mn (3) ! Mn (312) induces an injection in
v-11ss4i+2(-), and so the bracket formula (2.1) in Mn (312) follows from that
in Mn (3).
In order to prove (2.2), it seems convenient to use the stable complexes int*
*ro-
duced in [9]. There is a diagram of cofibrations of spectra
3P 4 8P 12
# #
B0 ! K0 ! W 0
# #
7P 12 23P 44
EQUIVALENCES OF TELESCOPES 5
such that
v-11ss*(K) v-11sss*(K0) v-11J*(K0);
with a similar relationship for W and W 0and for B and B0. Here J is the
fiber of 2 - 1 : ` ! 4`, with ` a summand of bu(3). We have adopted here
nonstandard notation P 4nfor the 4n-skeleton of the 3-localization of B1 . This
space is usually called B4n, but we are using B for sphere bundles. It is possi*
*ble
that 23P 44or W 0might have to be suspended more times in order that these
cofibrations exist, but, as in [6, 8.21],0one can show that if this is the case*
* then
the attaching map 23+LP 44! v-119P 12or L W 0! v-11B0 has filtration
large enough that the analysis which follows remains valid.
The analysis of v-11ss*(K) in [4] shows that, for any i > 15 (so that these
`*(-)-groups are isomorphic to the corresponding v-11`*(-)-groups), there are
isomorphisms and exact sequences as below.
`4i+2(W 0) `4i+2(23P 44) Z=311;
`4i+3(W 0) `4i+3(8P 12) Z=33;
0 ! `4i+2(3P 4) ! `4i+2(B0) ! `4i+2(7P 12) ! 0
| | |
Z=3 Z=34 Z=33;
and
`4i+3(K0) ! `4i+3(W 0) ! `4i+2(B0) ! `4i+2(K0) ! `4i+2(W 0) ! 0
| | | | |
0 Z=33 Z=34 Z=312 Z=311:
We prefer to picture this as in the chart below for `*(K0), where o's come from
23P 44, O's from 7P 12, *'s from 8P 12, and the x from 3P 4.
x
x b|*@
b|*@ b|*@
b|*@ b|*@
b|*@ r||
r|| r|
r| r||
r|| r|
r| r|
r| r||
r|| r|
r|`*(K0)r|
r| r||
r|| r|
r| r|
r|
4i + 2 4i + 6
6 DONALD M. DAVIS
The element bi of order 3 in J4i+2(K0) corresponds to the element b0iof order
3 in `4i+2(K0), which is the bottom O in the chart. It is clear from the above
chart or exact sequences that these satisfy v1b0i= b0i+1, and, on elements of o*
*rder
3, multiplication by v1 corresponds to the bracket in (2.2). This completes the
proof of Theorem 1.1. ||
The equivalence Tel1(K) ! Tel1(dS25) is of course obtained as the compos-
ite -1 0
Tel1(K) h-! v-11Mn (312) -h! Tel1(dS25):
If this is preceded by the1inclusion1K ! Tel1(K), the composite must factor
through a map K ! d+4k3 S25 for some nonnegative integer k. As explained
in [7], such a map will induce a map of telescopes, which will be the equivalen*
*ce.
This explains the sentence which follows Theorem 1.1.
3. Harper's complex at an arbitrary odd prime
In [11 ], Harper constructed a finite p-local H-space Kp for any odd prime p,
which he calls the "torsion molecule." It satisfies
H*(Kp) E[x; y] Zp[z]=(zp);
with |x| = 3, y = P1x, and z = fiy. We use Zp = Z=p, all cohomology groups
have coefficients in Zp, and E denotes an exterior algebra.
In this section, we present several results about these spaces Kp. First is *
*an
unpublished proof of Harper of the following result, which generalizes (1.1) to
an arbitrary odd prime.
Theorem 3.1. There is a p-local fibration
B(3; 2p + 1) -f! Kp -g! ^S2p+2
such that f* sends E[x; y] isomorphically onto H*(B(3; 2p + 1)), and g* sends
H*(S^2p+2) isomorphically onto Zp[z]=(zp).
The spaces B(3; 2p + 1) and ^S2p+2were defined in Section 1.
Next we prove the following result, which generalizes to any odd prime the
3-primary calculation in [4, 1.3].
Theorem 3.2. The only nonzero groups v-11ssj(Kp) are
2+p; (i-p-2-pp2+p-2)+1)
v-11ssqi+2(Kp) v-11ssqi+1(Kp) Z=pmin(p p :
The proof of Theorem 1.1 then generalizes directly to yield Theorem 1.3.
In order to construct the map g of Theorem 3.1, we need the following re-
sults from [11 ]. A space X is a power space if it has a self-map which induces
multiplication by ffl on the indecomposables of H*(X), where ffl is a generator*
* of
EQUIVALENCES OF TELESCOPES 7
(Z=p)*. An H-space is a power space. En route to proving in [11 , Thm B] that
Kp is an H-space, Harper proves that it is a power space, and this weaker result
is all that we will need here.
We will also need the observation that H*(Kp) = U(M), where
M =
with |x| = 3. Here U(M) denotes the unstable A-algebra generated by M. We
will need the following general result, [11 , 2.3.6], in which U is the categor*
*y of
unstable A-modules, and A is the mod p Steenrod algebra.
Proposition 3.1.[11 , 2.3.6] Suppose X and Y are simply-connected p-local
finite complexes which are power spaces with power maps X and Y , and
H*(X) U(N) for some unstable A-module N. Let
CffleH*(Y ) = {y 2 eH*(Y ) : *Yy = ffly};
and suppose
Exts+1U(N; sCffleH*(Y )) = 0
for s 1. Then any morphism g0 : H*X ! H*Y of A-algebras satisfying
g0 *X= *Yg0 can be realized by a map g : Y ! X.
Proposition 3.1 will be applied to
g0 : H*(S^2p+2) = Zp[__z]=(__zp) ! H*(Kp)
defined by g0(__z) = z. We use that
H*(S^2p+2) U(S(2p + 2));
where S(2p + 2) is the unstable A-module with Zp in grading 2p + 2 as its only
nonzero component, and that ^S2p+2is a power space, with power map defined
as a cellular approximation of the ffl-power map of S2p+3.
Existence of the map g in Theorem 3.1 now follows from Proposition 3.1
together with the following result.
Proposition 3.2.Let L H*K be given by
L = :
Then
Ext sU(S(2p + 2); s-1L) = 0
for s 1.
8 DONALD M. DAVIS
Proof. By [11 , 1.3.2], sharpened slightly using [11 , 1.1.11b,1.3.1],
(3.1)
Ext sU(S(2p + 2); s-1L) ExtsA(2p+2Zp; s-1L) Exts;s-2p-3A(Zp; L);
i.e., the unstable Ext group is actually stable.
There is a standard spectral sequence, described in [11 , p. 10], converging*
* to
Ext*;*A(Zp; L) with Ed;s;t1= Exts;tA(Zp; Ld). Here Ld denotes the component of
L in grading d. There are differentials dr : Ed;s;tr! Ed+r;s+1;tr, and Ed;s;t1i*
*s the
dth subquotient in a filtration of Exts;tA(Zp; L). We have Ld = 0 unless d 2 SL,
where SL = {3; 2p + 1; 2p + 2; 2p2; 2p2+ 1; 2p2+ 2p - 1}, with Ld Zp if d 2 SL.
Then (
Ext s;t+dA(Zp; Zp)ifd 2 SL
Ed;s;t1=
0 ifd 62 SL:
To determine Exts;s-2p-3A(Zp; L) from this spectral sequence, we need to know
Exts;tA(Zp; Zp) for t-s 2p2-3. The groups ExtA(Zp; Zp) are tabulated through
this range in [11 , 1.4.1]. One verifies from this that
(
Zp ifd = 2p2 + 1 and 2 s p
Ed;s;s-2p-31=
0 otherwise.
The nonzero classes here correspond to as-20b1 2 Exts;s+pq-2A(Zp; Zp).
We claim that
2;s-1;s-2p-3 2p2+1;s;s-2p-3
d1 6= 0 : E2p1 ! E1
for 3 s p, and
2+1;2;-2p-1 2p2+2p-1;3;-2p-1
dq 6= 0 : E2pq ! Eq :
This will then leave Ed;s;s-2p-31= 0 for all s and d, so that Exts;s-2p-3A(Zp; *
*L) =
0 for all s, which implies the proposition.
Both of these nonzero differentials are a consequence of [1, Lemma 2.6.1]. In
the first case, d1 is
Exts-1;s-2p-3A(Zp; L2p2) ! Exts;s-2p-3A(Zp; L2p2+1);
where these two nonzero groups of L are connected by the Bockstein fi. The
first Ext group is generated by as-30b1, and the second by as-20b1. According to
[1, 2.6.1], this boundary morphism is given by multiplication by a0 because of
the fi-action, and hence is nonzero here.
The dq-differential is
Ext 2;-2p-1A(Zp; L2p2+1) ! Ext3;-2p-1A(Zp; L2p2+2p-1);
where these two nonzero groups of L are connected by P1. The generators of
the Ext-groups are b1 and h0b1, where h0 2 Ext1;q-1A(Zp; Zp) arises from P1, so
that [1, 2.6.1] again implies the nonzero boundary. ||
EQUIVALENCES OF TELESCOPES 9
We complete the proof of Theorem 3.1 by showing that the fiber F of the
map g just constructed is B(3; 2p + 1). The Serre spectral sequence shows that
F has the correct cohomology, and since it is simply-connected, F can be given
a cell structure S3 [ff1e2p+1[ e2p+4. If the 3-cell is collapsed, the 2-cell co*
*mplex
which remains has a trivial attaching map, and so its top cell can be collapsed
too, yielding a degree-1 map F ! S2p+1 with fiber S3. ||
The proof of Theorem 3.2 is a direct adaptation of the 3-primary case proved*
* in
[4]. We will sketch the steps in the argument. The first step is the determinat*
*ion
of v-11ss*(S^2p+2), which could be of some interest in its own right.
Proposition 3.3.Let p be an odd prime, Mi = min(p; p(i - 1) + 1), and
M0i= min(p2 + p - 1; p(i - p - 2) + 1). Then
8
>>>Z=pMi ifffl = 3
>< 0
Z=p1+max(Mi;Mi) ifffl = 2
v-11ssqi+ffl(S^2p+2) > 0
>>>Z=pMi ifffl = 1
: 0 if4 ffl q:
The entire proof of [4, 2.2] can be adapted directly to this case of an arbi*
*trary
odd prime. This is primarily just an analysis of the exact sequences in v-11ss**
*(-)
of the EHP fibrations
2+2p-1
(3.2) S2p+1 -E! S^2p+2-H! S2p
and
(3.3) ^S2p+2-E! S2p+3 -H! S2p2+2p+1:
Next the exact sequence of the fibration of Theorem 3.1 can be used exactly
as in [4, 2.10] (and the material which preceded it) to establish the 0-groups *
*in
our Theorem 3.2. The same exact sequence can be used as in [4] to show that
if i 1 mod p, then v-11ssqi+2(Kp) v-11ssqi+1(Kp) Z=p. As in [4], one must
divide into cases depending upon whether or not i pp-1 + 1 mod pp, and the
main tool is that the composite S2p+1 ! S^2p+2! B(3; 2p + 1) ! S2p+1 has
degree p.
Finally we consider the case of v-11ssqi+2(Kp) and v-11ssqi+1(Kp) with i 6 1
mod p. In [4], this required the key technical result, Lemma 2.16. This also
adapts to an arbitrary odd prime, as follows, but in this case we will comment
more extensively on the proof. We now need the unstable Novikov spectral
sequence (UNSS). Since the reader must be referring to [4] already, we refer to
the material preceding Lemma 2.16 of that paper for UNSS notation. Noting
that elements in the v1-localized UNSS can be represented by elements in the
unlocalized UNSS (perhaps after multiplying by a power of v1), we use E2 to
refer to elements in either spectral sequence.
10 DONALD M. DAVIS
Lemma 3.1. Let B = B(3; 2p + 1) -ae!S2p+1 be the projection,
2+2p-1
S^2p+2-H! S2p
from (3:2), and let OE : v-11ssqi+2(S^2p+2) ! v-11ssqi+1(B) be the boundary mor-
phism from Theorem 3.1. Suppose y 22v-11ssqi+1(S^2p+2) has H*(y) repre-
sented by A 2p2+2p-1 2 E1;qi+22(S2p +2p-1). Then ae*(OE(y)) is represented
by A v1hp12p+1 mod terms that desuspend to S2p-1.
Proof. We need the following result.
Lemma 3.2. S^2p+2splits as X _ J, where X = S2p+2[u e2p2+2p-1and J
is a bouquet of spheres of dimension at least 2p2 + 4p. The attaching map u is
u0, where u02 ss2p2+2p-3(S2p+1) is detected in the UNSS by d(hp+11)2p+1.
Proof. The splitting was proved in [12 , Thm. 6]. We have used [2, 3.7] to
give the UNSS name of the nonzero attaching map in ss2p2+2p-2(S2p+2). That
the UNSS is 0 in other filtrations in this stem follows from a calculation simi*
*lar
to that of [3, x6]. ||
Now the proof of Lemma 3.1 is a direct adaptation of the proof of [4, Lemma
2.16]. This was the hardest part of [4], but the adaptation really just amounts
to replacing 7, 8, 23, h31, h41, and 4i + 2 by 2p + 1, 2p + 2, 2p2+ 2p - 1, hp1*
*, hp+11,
and qi + 2, respectively. ||
Now we can prove the following analogue of [4, Proposition 2.12].
Proposition 3.4.The morphism
0+1 -1 2p+2 -1
Z=pMi v1 ssqi+2(S^ ) ! v1 ssqi+1(B(3; 2p + 1)) Z=p
is nonzero unless i - p - 2 pp2+p-2 mod pp2+p-1, in which case it is zero.
This result, together with the exact sequence of (3.1) and the argument of [*
*4,
2.13] for cyclicity of v-11ssqi+1(Kp), completes the proof of Theorem 3.2. Again
we provide a road map for translating the proof of [4, 2.12] to the prime p.
Proof of Proposition 3.4. Let m = i-p-2. The generator2of v-11ssqi+2(S^2p+2)
is mapped by H* to a generator of v-11ssqi+2(S2p +2p-1), which is represented by
ffm=e with e = min(p2 + p - 1; p(m) + 1). By Lemma 3.1, it suffices to prove
Lemma 3.3. i.) If e < p2 + p - 1, then ffm=e v1hp12p+1 = 0, and (ii.) if
p(m) p2+p-2, then ffm=(p2+p-1)v1hp12p+1 = 0 if and only if m=pp2+p-2 1
mod p.
EQUIVALENCES OF TELESCOPES 11
Proof. Part (i) is immediate2from [3, 5.3], just like the proof of [4, 2.17].
For part (ii), let s = m=pp +p-2. Similarly to [4, 2.19], mod elements that
desuspend to S2p-1, we can write -ffm=(p2+p-1) v1hp12p+1 as
2-p p2+p p
(3.4) sh1 vm1hp12p+1+ pvm-p1 h1 v1h12p+1:
As in [4], H0 of the first term of (3.4) is svm1h1, and so we will be done once*
* we
show that H0 of the second term is -vm1h1. Similarly to [4, 2.20], the second
term of (3.4) can be rewritten as
2-p+1 p2+p-1 p m-p2-p p2+p-1 2 p
(3.5) vm-p1 h1 v1h12p+1- v1 h1 v1h12p+1;
and the second term T of (3.5) has H0(T ) = -vm1h1. We will be done once we
show that the first term of (3.5) desuspends to S2p-1.
Let Y denote the sum obtained by multiplying the right hand side of the
relation (of [5, 2.6iii])
(3.6)
Xp
0 = -v2 + ph2 + (1 - pp-1)hp1v1 + jR (v2) - (p + 1)vp1h1 + aivp+1-i1pihi1;
i=2
2-p+1 p2-1 p
with ai 2 Z, by vm-p1 h1 h12p+1. As in [4], the first, second, and
fourth terms of Y have H0(-) = 0, and the third term of Y is a unit times our
desired term. Each term in the sum at the end of Y has H0(-) = 0 for the
reason cited in [4]. Finally, we2consider the next-to-last term of Y . Omitti*
*ng
vpwr1and (p + 1), it becomes hp1 hp12p+1, which we show is d(hp22p+1) mod
terms that desuspend to S2p-1.
To see this, avoiding unnecessary units, we write d(hp2) as
(h2 1 + 1 h2+ hp1 h1+ h1 hp-11v1+ . .+.hp-11 h1v1)p - hp2 1 - 1 hp2:
A term in the expansion will be the product of p terms times a multinomial
coefficient, and will be of the form
(3.7) chi1hj2 hk1h`2vm1:
We first consider the unstable condition for hk1h`2vm1being defined on S2p-1. T*
*he
excess k + ` - (p - 1)m must be less than p. Since this excess is no greater th*
*an
1 for any term in the sum whose pth power is being taken, the only problem
can occur when all factors are 1 h2 or hp1 h1. The pth power of the first is
subtracted off, the pth power of the second is the term in which we are interes*
*ted,
and mixed terms will all have a factor of p in the multinomial coefficient, whi*
*ch
can be used to remove one of the h1's via the2relation ph1 = v1 - jR (h1). Thus
all terms, with the possible exception of hp1 hp1, desuspend to S2p-1.
A similar analysis applies to the unstable condition on the factor hi1hj2on *
*the
left side of . Here the excess is the number of h's on the left side of the
minus 1_2times the degree of the terms on the right side of the . This is 1 for
12 DONALD M. DAVIS
the terms h2 1 and hp1 h1, and less than 1 for the others, and so the same
argument as above applies. ||
References
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*f Math 72
(1960) 20-104.
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3._________, The derived functors of the primitives for BP*(S2n+1), Trans Amer*
* Math
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*oc
Adams Symposium, London Math Soc Lecture Note Series 176 (1992) 55-72.
8._________, Three contributions to the homotopy theory of the exceptional Lie*
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*ed spaces, Topol-
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15-24.
Lehigh University, Bethlehem, Pennsylvania 18015
E-mail address: dmd1@lehigh.edu