1. No category

Contemporary Mathematics
Volume 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 ordinary homotopy groups equal the v1 -periodic homotopy groups of the space.
It is proved that the mapping telescopes of the spaces are homotopy equivalent.
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 S 3 -bundle over S 7 with
attaching map α1 . In [4], the 3-primary v1 -periodic homotopy groups v1−1 π∗ (K)
were computed, en route to the determination of v1−1 π∗ (F4 ). One can observe
that these groups satisfy
(1.2)
v1−1 π∗ (K) ≈ v1−1 π∗+d (S 25 ), where d = 8 · 310 + 2.
For any spherically resolved space Y , such as K or Ωd S 25 , there is a periodic
Ω-spectrum Tel1 (Y ), defined in [7], satisfying
π∗ (Tel1 (Y )) ≈ v1−1 π∗ (Y ).
Here a space Y is called spherically resolved if, for some k ≥ 0, Ωk Y can be
built from spheres or loops on spheres by fibrations. To see that K is spherically
resolved, we use (1.1) and the fibration
S 7 → ΩW → ΩS 23 ,
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 publication elsewhere.
c
0000
American Mathematical Society
0000-0000/00 $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 (Ωd S 25 ),
where d is as in (1.2).
11
This equivalence is induced by a map K → Ωd+4k3 S 25 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
S 3 -bundle over S 2p+1 with attaching map α1 . Then there is a p-local equivalence
0
Tel1 (B(3, 2p + 1)) ≈ Tel1 (Ωd S 2p+3 ),
where d0 = 2(p − 1)2 pp−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 → Sˆ2p+2 ,
where Sˆ2p+2 is the (p − 1)(2p + 2)-skeleton of ΩS 2p+3 . We remark that when
p = 3, Kp fibers over both Sˆ8 and W with the same fiber. The spaces Sˆ8 and
W are not 3-equivalent, but their loop spaces are. In Section 3, we will calculate
v1−1 π∗ (Kp ), and use this result to generalize Theorem 1.1 to the following result.
Theorem 1.3. There is an equivalence of p-local spectra,
00
2
Tel1 (Kp ) ≈ Tel1 (Ωd S 2p
where d00 = 2(p − 1)2 pp
2
+p−2
+2p+1
),
+ 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 show that both Tel1 (K) and Tel1 (Ωd S 25 )
10
are equivalent to the spectrum v1−1 M 4·3 +23 (312 ). Here M n (k) = S n−1 ∪k en
denotes a Moore spectrum, and, as in [9], v1−1 M n (pe ) is the telescope of
e−1
M n (pe ) → M n−qp
e−1
(pe ) → M n−2qp
(pe ) → · · · ,
with the maps all being suspensions of the same Adams map. Here and throughout the paper, q = 2(p − 1).
We begin by quoting from [4] the following result. Here and elsewhere νp (−)
denotes the exponent of p, with νp (0) = ∞.
Proposition 2.1. With d as in (1.2), and = 0 or 1,
(
10
Z/3min(12,ν3 (i−2·3 −11)+1)
−1
−1
d 25
v1 π2i− (K) ≈ v1 π2i− (Ω S ) ≈
0
i odd
i even.
Let Y = K or Ωd S 25 . For0 t = 4·3e with e ≥ 11, there is, for some nonnegative
A
integer m, a map Ωmt Y −→ Ω(m+1)t Y 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 Ωmt Y .
If T denotes the mapping telescope of
A0
Ωt A0
Ωmt Y −→ Ω(m+1)t Y −→ Ω(m+2)t Y → · · · ,
then Tel1 (Y ) is the Ω-spectrum with T in 00each index which is a multiple of t.
A
There is an obvious equivalence Tel1 (Y ) −→ Ωt Tel1 (Y ).
Let n = 4 · 310 + 23 throughout this section, and let S n−1 → Tel1 (Y ) represent
a generator of πn−1 (Tel1 (Y )) ≈ v1−1 πn−1 (Y ) ≈ Z/312 . This extends uniquely
to a map g : M n (312 ) → Tel1 (Y ), inducing an isomorphism in πn−1 (−). Let
g 0 : M n−t (312 ) → Ωt Tel1 (Y ) be adjoint to g. We can form a commutative
diagram
A
M n
(312 ) −→ M n−t(312 )
g
 0
yg
y
A00
t
Tel1 (Y )
−→ Ω Tel1 (Y )
with A an appropriate Adams map. The commutativity is a consequence of the
isomorphism
[M n (312 ), Ωt Tel1 (Y )] ≈ πn−1 (Ωt Tel1 (Y )) ≈ Z/312 .
This diagram induces a map of direct systems M n−mt (312 ) → Ωmt Tel1 (Y ),
yielding in the limit the map v1−1 M n (312 ) → Tel1 (Y ) which is to be our equivalence.
We now must show that this map induces an isomorphism in π∗ (−). When
Y = Ωd S 25 , the result is a special case of the equivalence
v1−1 M 0 (pk ) → Tel1 (Ω2k+1 S 2k+1 ),
4
DONALD M. DAVIS
which is proved by induction on k from the commutative diagram of fibrations
v1−1 M −1 (p)
↓
Tel1 (Ω2k−1 W (k))
→
→
v1−1 M 0 (pk−1 )
↓
Tel1 (Ω2k−1 S 2k−1 )
→
→
v1−1 M 0 (pk )
↓
Tel1 (Ω2k+1 S 2k+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
v1−1 M 25−d (312 ) ' v1−1 M n (312 ).
To prove the isomorphism when Y = K, we need to review from [4] the way
in which v1−1 π∗ (K) is obtained. Looping the map K → W of (1.1) and following by the map ΩW → ΩS 23 of [8] induces an epimorphism v1−1 π4i+1 (K) →
v1−1 π4i+1 (S 23 ), which is bijective unless v1−1 π4i+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.
v1−1 Mn (312 )

yh
→
v1−1 Mn (311 )
f
y
Tel1 (K)
→ Tel1 (W ) →
Tel1 (S 23 )
The preceding paragraph showed that f induces an iso in π4i+1 (−), and the
diagram and preceding statements then imply the same for h.
h
The map v1−1 M n (312 ) −→ Tel1 (K) induces an isomorphism in πi (−) whenever i ≡ n − 1 mod 4 · 311 , in which case both groups are Z/312 . We show that
h induces an iso in πi (−) whenever i ≡ 2 mod 4 by showing that we can choose
elements ai (resp. bi ) of order 3 in π4i+2 (v1−1 M n (312 )) (resp. π4i+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
= hai , 3, α1 i
(2.2)
bi+1
= hbi , 3, α1 i.
The case i ≡ n − 1 mod 4 · 311 starts the induction.
Equation (2.1) is well-known in M n (3), as this bracket defines the Adams
periodicity. The canonical map M n (3) → M n (312 ) induces an injection in
v1−1 π4i+2 (−), and so the bracket formula (2.1) in M n (312 ) follows from that
in M n (3).
In order to prove (2.2), it seems convenient to use the stable complexes introduced in [9]. There is a diagram of cofibrations of spectra
Σ3 P 4
↓
B0
↓
Σ7 P 12
Σ8 P 12
↓
→ K0 →
W0
↓
Σ23 P 44
EQUIVALENCES OF TELESCOPES
5
such that
v1−1 π∗ (K) ≈ v1−1 π∗s (K 0 ) ≈ v1−1 J∗ (K 0 ),
with a similar relationship for W and W 0 and for B and B 0 . Here J is the
fiber of ψ 2 − 1 : ` → Σ4 `, with ` a summand of bu(3) . We have adopted here
nonstandard notation P 4n for the 4n-skeleton of the 3-localization of BΣ∞ . This
space is usually called B 4n , but we are using B for sphere bundles. It is possible
that Σ23 P 44 or W 0 might have to be suspended more times in order that these
cofibrations exist, but, as in [6, 8.21], one can show that if this is the case then
0
the attaching map Σ23+L P 44 → v1−1 Σ9 P 12 or ΣL W 0 → v1−1 ΣB 0 has filtration
large enough that the analysis which follows remains valid.
The analysis of v1−1 π∗ (K) in [4] shows that, for any i > 15 (so that these
`∗ (−)-groups are isomorphic to the corresponding v1−1 `∗ (−)-groups), there are
isomorphisms and exact sequences as below.
`4i+2 (W 0 ) ≈ `4i+2 (Σ23 P 44 ) ≈ Z/311 ,
`4i+3 (W 0 ) ≈ `4i+3 (Σ8 P 12 ) ≈ Z/33 ,
0
→ `4i+2 (Σ3 P 4 ) → `4i+2 (B 0 ) → `4i+2 (Σ7 P 12 ) → 0
|
|
|
Z/3
Z/34
Z/33 ,
and
`4i+3 (K 0 ) → `4i+3 (W 0 ) → `4i+2 (B 0 ) → `4i+2 (K 0 ) → `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 `∗ (K 0 ), where •’s come from
Σ23 P 44 , ◦’s from Σ7 P 12 , ∗’s from Σ8 P 12 , and the × from Σ3 P 4 .
×
b ∗
×
b ∗
b ∗
b ∗
b ∗
b ∗
r
@@
@
@@
@
r
r
r
r
r
r
r
r
r
r
r
r
`∗ (K 0 )
r
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 (K 0 ) corresponds to the element b0i of order
3 in `4i+2 (K 0 ), which is the bottom ◦ in the chart. It is clear from the above
chart or exact sequences that these satisfy v1 b0i = b0i+1 , and, on elements of order
3, multiplication by v1 corresponds to the bracket in (2.2). This completes the
proof of Theorem 1.1.
The equivalence Tel1 (K) → Tel1 (Ωd S 25 ) is of course obtained as the composite
h−1
h0
Tel1 (K) −→ v1−1 M n (312 ) −→ Tel1 (Ωd S 25 ).
If this is preceded by the inclusion K → Tel1 (K), the composite must factor
11
through a map K → Ωd+4k3 S 25 for some nonnegative integer k. As explained
in [7], such a map will induce a map of telescopes, which will be the equivalence.
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]/(z p ),
with |x| = 3, y = P 1 x, and z = βy. 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
g
f
B(3, 2p + 1) −→ Kp −→ Sˆ2p+2
such that f ∗ sends E[x, y] isomorphically onto H ∗ (B(3, 2p + 1)), and g ∗ sends
H ∗ (Sˆ2p+2 ) isomorphically onto Zp [z]/(z p ).
The spaces B(3, 2p + 1) and Sˆ2p+2 were 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 v1−1 πj (Kp ) are
v1−1 πqi+2 (Kp ) ≈ v1−1 πqi+1 (Kp ) ≈ Z/pmin(p
2
+p,νp (i−p−2−pp
2 +p−2
)+1)
.
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 results from [11]. A space X is a power space if it has a self-map which induces
multiplication by on the indecomposables of H ∗ (X), where 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 = hx, P 1 x, βP 1 xi
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 category 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 ψX and ψY , and
H ∗ (X) ≈ U (N ) for some unstable A-module N . Let
e ∗ (Y ) = {y ∈ H
e ∗ (Y ) : ψY∗ y = y},
C H
and suppose
s
e∗
Exts+1
U (N, Σ C H (Y )) = 0
for s ≥ 1. Then any morphism g0 : H ∗ X → H ∗ Y of A-algebras satisfying
∗
g0 ψX
= ψY∗ g0 can be realized by a map g : Y → X.
Proposition 3.1 will be applied to
g0 : H ∗ (Sˆ2p+2 ) = Zp [z]/(z p ) → H ∗ (Kp )
defined by g0 (z) = z. We use that
H ∗ (Sˆ2p+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 Sˆ2p+2 is a power space, with power map defined
as a cellular approximation of the -power map of ΩS 2p+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 = hx, y, z, xyz p−2 , xz p−1 , yz p−1 i.
Then
ExtsU (S(2p + 2), Σs−1 L) = 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)
s,s−2p−3
(Zp , L);
ExtsU (S(2p + 2), Σs−1 L) ≈ ExtsA (Σ2p+2 Zp , Σs−1 L) ≈ ExtA
i.e., the unstable Ext group is actually stable.
There is a standard spectral sequence, described in [11, p. 10], converging to
d,s,t
Ext∗,∗
= Exts,t
A (Zp , L) with E1
A (Zp , Ld ). Here Ld denotes the component of
d,s,t
L in grading d. There are differentials dr : Erd,s,t → Erd+r,s+1,t , and E∞
is the
s,t
dth subquotient in a filtration of ExtA (Zp , L). We have Ld = 0 unless d ∈ SL ,
where SL = {3, 2p + 1, 2p + 2, 2p2, 2p2 + 1, 2p2 + 2p − 1}, with Ld ≈ Zp if d ∈ SL .
Then
(
Exts,t+d
(Zp , Zp ) if d ∈ SL
d,s,t
A
E1
=
0
if d 6∈ SL .
s,s−2p−3
To determine ExtA
(Zp , L) from this spectral sequence, we need to know
s,t
ExtA (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 if d = 2p2 + 1 and 2 ≤ s ≤ p
d,s,s−2p−3
E1
=
0
otherwise.
The nonzero classes here correspond to a0s−2 b1 ∈ Exts,s+pq−2
(Zp , Zp ).
A
We claim that
2
d1 6= 0 : E12p
,s−1,s−2p−3
2
→ E12p
+1,s,s−2p−3
for 3 ≤ s ≤ p, and
2
dq 6= 0 : Eq2p
+1,2,−2p−1
2
→ Eq2p
+2p−1,3,−2p−1
.
s,s−2p−3
d,s,s−2p−3
This will then leave E∞
= 0 for all s and d, so that ExtA
(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
s−1,s−2p−3
s,s−2p−3
ExtA
(Zp , L2p2 ) → ExtA
(Zp , L2p2 +1 ),
where these two nonzero groups of L are connected by the Bockstein β. The
first Ext group is generated by a0s−3 b1 , and the second by a0s−2 b1 . According to
[1, 2.6.1], this boundary morphism is given by multiplication by a0 because of
the β-action, and hence is nonzero here.
The dq -differential is
2,−2p−1
3,−2p−1
ExtA
(Zp , L2p2 +1 ) → ExtA
(Zp , L2p2 +2p−1 ),
where these two nonzero groups of L are connected by P 1 . The generators of
1,q−1
the Ext-groups are b1 and h0 b1 , where h0 ∈ ExtA
(Zp , Zp ) arises from P 1 , 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 S 3 ∪α1 e2p+1 ∪ e2p+4 . If the 3-cell is collapsed, the 2-cell complex
which remains has a trivial attaching map, and so its top cell can be collapsed
too, yielding a degree-1 map F → S 2p+1 with fiber S 3 .
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 determination
of v1−1 π∗ (Sˆ2p+2 ), which could be of some interest in its own right.
Proposition 3.3. Let p be an odd prime, Mi =
Mi0 = min(p2 + p − 1, νp (i − p − 2) + 1). Then


Z/pMi



Z/p1+max(Mi ,Mi0 )
−1
2p+2
ˆ
)≈
v1 πqi+ (S
0

Z/pMi




0
min(p, νp (i − 1) + 1), and
if = 3
if = 2
if = 1
if 4 ≤ ≤ q.
The entire proof of [4, 2.2] can be adapted directly to this case of an arbitrary
odd prime. This is primarily just an analysis of the exact sequences in v1−1 π∗ (−)
of the EHP fibrations
(3.2)
E
H
2
S 2p+1 −→ ΩSˆ2p+2 −→ ΩS 2p
+2p−1
and
(3.3)
2
E
H
Sˆ2p+2 −→ ΩS 2p+3 −→ ΩS 2p +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 v1−1 πqi+2 (Kp ) ≈ v1−1 πqi+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 S 2p+1 → ΩSˆ2p+2 → B(3, 2p + 1) → S 2p+1 has
degree p.
Finally we consider the case of v1−1 πqi+2 (Kp ) and v1−1 πqi+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
ρ
Lemma 3.1. Let B = B(3, 2p + 1) −→ S 2p+1 be the projection,
2
H
ΩSˆ2p+2 −→ ΩS 2p
+2p−1
from (3.2), and let φ : v1−1 πqi+2 (Sˆ2p+2 ) → v1−1 πqi+1 (B) be the boundary morphism from Theorem 3.1. Suppose y ∈ v1−1 πqi+1 (ΩSˆ2p+2 ) has H∗ (y) repre2
sented by A ⊗ ι2p2 +2p−1 ∈ E21,qi+2 (S 2p +2p−1 ). Then ρ∗ (φ(y)) is represented
p
by A ⊗ v1 h1 ι2p+1 mod terms that desuspend to S 2p−1 .
Proof. We need the following result.
2
Lemma 3.2. ΣΩSˆ2p+2 splits as X ∨ J, where X = S 2p+2 ∪u e2p +2p−1 and J
is a bouquet of spheres of dimension at least 2p2 + 4p. The attaching map u is
Σu0 , where u0 ∈ π2p2 +2p−3 (S 2p+1 ) is detected in the UNSS by d(hp+1
)ι2p+1 .
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 π2p2 +2p−2 (S 2p+2 ). That
the UNSS is 0 in other filtrations in this stem follows from a calculation similar
to that of [3, §6].
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+1
1
and qi + 2, respectively.
Now we can prove the following analogue of [4, Proposition 2.12].
Proposition 3.4. The morphism
0
Z/pMi +1 ≈ v1−1 πqi+2 (Sˆ2p+2 ) → v1−1 πqi+1 (B(3, 2p + 1)) ≈ Z/p
2
is nonzero unless i − p − 2 ≡ pp
+p−2
2
mod pp
+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 v1−1 πqi+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 generator of v1−1 πqi+2 (Sˆ2p+2 )
2
is mapped by H∗ to a generator of v1−1 πqi+2 (S 2p +2p−1 ), which is represented by
αm/e with e = min(p2 + p − 1, νp (m) + 1). By Lemma 3.1, it suffices to prove
Lemma 3.3. i.) If e < p2 + p − 1, then αm/e ⊗ v1 hp1 ι2p+1 = 0, and (ii.) if
2
νp (m) ≥ p2 +p−2, then αm/(p2 +p−1) ⊗v1 hp1 ι2p+1 = 0 if and only if m/pp +p−2 ≡ 1
mod p.
EQUIVALENCES OF TELESCOPES
11
Proof. Part (i) is immediate from [3, 5.3], just like the proof of [4, 2.17].
2
For part (ii), let s = m/pp +p−2 . Similarly to [4, 2.19], mod elements that
desuspend to S 2p−1 , we can write −αm/(p2 +p−1) ⊗ v1 hp1 ι2p+1 as
2
sh1 ⊗ v1m hp1 ι2p+1 + pv1m−p
(3.4)
−p p2 +p
h1
⊗ v1 hp1 ι2p+1 .
As in [4], H 0 of the first term of (3.4) is sv1m h1 , and so we will be done once we
show that H 0 of the second term is −v1m h1 . Similarly to [4, 2.20], the second
term of (3.4) can be rewritten as
(3.5)
2
v1m−p
−p+1 p2 +p−1
h1
2
⊗ v1 hp1 ι2p+1 − v1m−p
−p p2 +p−1
h1
⊗ v12 hp1 ι2p+1 ,
and the second term T of (3.5) has H 0 (T ) = −v1m h1 . We will be done once we
show that the first term of (3.5) desuspends to S 2p−1 .
Let Y denote the sum obtained by multiplying the right hand side of the
relation (of [5, 2.6iii])
(3.6)
0 = −v2 + ph2 + (1 − pp−1 )hp1 v1 + ηR (v2 ) − (p + 1)v1p h1 +
p
X
ai v1p+1−i pi hi1 ,
i=2
with ai ∈ Z, by v1m−p −p+1 h1p −1 ⊗ hp1 ι2p+1 . As in [4], the first, second, and
fourth terms of Y have H 0 (−) = 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 H 0 (−) = 0 for the
reason cited in [4]. Finally, we consider the next-to-last term of Y . Omitting
2
v1pwr and (p + 1), it becomes hp1 ⊗ hp1 ι2p+1 , which we show is d(hp2 ι2p+1 ) mod
terms that desuspend to S 2p−1 .
To see this, avoiding unnecessary units, we write d(hp2 ) as
2
2
(h2 ⊗ 1 + 1 ⊗ h2 + hp1 ⊗ h1 + h1 ⊗ hp−1
v1 + · · · + hp−1
⊗ h1 v1 )p − hp2 ⊗ 1 − 1 ⊗ hp2 .
1
1
A term in the expansion will be the product of p terms times a multinomial
coefficient, and will be of the form
(3.7)
chi1 hj2 ⊗ hk1 h`2 v1m .
We first consider the unstable condition for hk1 h`2 v1m being defined on S 2p−1 . The
excess k + ` − (p − 1)m must be less than p. Since this excess is no greater than
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 interested,
and mixed terms will all have a factor of p in the multinomial coefficient, which
can be used to remove one of the h1 ’s via the relation ph1 = v1 − ηR (h1 ). Thus
2
all terms, with the possible exception of hp1 ⊗ hp1 , desuspend to S 2p−1 .
A similar analysis applies to the unstable condition on the factor hi1 hj2 on the
left side of ⊗. Here the excess is the number of h’s on the left side of the ⊗
minus 12 times 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.
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15-24.
Lehigh University, Bethlehem, Pennsylvania 18015
E-mail address: [email protected]