COMPOSITIONS IN THE v1-PERIODIC HOMOTOPY GROUPS OF
SPHERES
MARTIN BENDERSKY AND DONALD M. DAVIS
Abstract.Let aei2 ssn+8i-1(Sn) denote an element which sus-
pends to a generator of the image of the stable J- homomorphism.
We determine the image of the composite aejO aek in v1-periodic
homotopy v-11ssn+8i+8j-2(Sn). The method is to use Adams op-
erations to compute the 1-line of an unstable homotopy spectral
sequence constructed by Bendersky and Thompson.
1. Main theorem
The p-primary v1-periodic homotopy groups of a space X, denoted v-11ss*(X; p),
are a localization of the portion of ss*X(p)detected by K-theory.([14]) The v1-*
*periodic
homotopy groups of spheres contain the image of the J-homomorphism.([18]) Until
the last two sections of this paper, we will deal with 2-primary homotopy theor*
*y, let
(-) denote the exponent of 2 in an integer, and let v-11ss*(X) = v-11ss*(X; 2).
We need the following known result.
Proposition 1.1. i.) v-11ss2n+8i-1(S2n+1) Z=2min(n;(i)+4);
ii.) there are morphisms v-11: ss*(S2n+1) ! v-11ss*(S2n+1), * > 2n + 1, whic*
*h are
split surjections for * = 2n + 8i - 1 with 4i - (i) n + 8.
Proof.i.) See Theorem 2.1 and the accompanying diagrams, 2.2 and 2.3. (ii.) In *
*[14,
1.7] and [12, 2.4] it is shown that if pe : nX ! nX is null homotopic, then the*
*re
is a natural morphism v-11: ssj(X) ! v-11ssj(X) for j > n. In [22], it is noted*
* that
23n=2+1is null homotopic on 2nS2n+1<2n + 1>. Thus v-11is defined on ss*(S2n+1) *
*for
* > 2n + 1. The split surjection follows from [18, 1.3,1.5]. ||
__________
Date: March 7, 2000.
1991 Mathematics Subject Classification. 55Q40,55Q50.
Key words and phrases. homotopy groups of spheres.
1
2 BENDERSKY AND DAVIS
In this paper, we determine the v1-periodic homotopy class of the composition
aek N+8j-1 aej N
SN+8j+8k-2-! S -! S ; (1.2)
where aei denotes a map (or its homotopy class) which suspends to a generator of
the image of the stable J-homomorphism in the 2-primary (8i - 1)-stem. That is,
we will determine v-11(aej O aek) in v-11ssN+8j+8k-2(SN ), and will sometimes s*
*ay that
aejO aek is a certain multiple of the generator in the v1-periodic summand to m*
*ean that
v-11(aejO aek) equals this multiple.
These compositions were first considered in [19] and were determined in [20] *
*when
(k) 4j - 5. Although our main theorem, 1.3, will be stated when N in (1.2) is *
*an
arbitrary integer, most of our discussion will, for simplicity, center around o*
*dd values
of N; the minor interpolation required for even values of N can be obtained by *
*the
same methods, using results from [18].
Let OE(n) denote the8number of positive integers i satisfying i n and i 0; *
*1; 2; 4
<3 if (j) + 4j > (k)
mod 8, and let ffi = : . Our main theorem is
2 if (j) + 4j (k)
Theorem 1.3. Let mj;k= max((j) + 4; (k) + 4 - 4j), and let N satisfy
OE(N + ffi) mj;k+ 3: (1.4)
Then1 v-11(aejOaek) is 2OE(N+3)-(mj;k+3)+(e(j;k))times a generator of v-11ssN+8*
*j+8k-2(SN ),
where 8
iP iij i jjj
<1_ 80i-1 j + (24j- 1) j+k if (k) = 4j + (j)
e(j; k) = :k i1 i i
0 if (k) 6= 4j + (j).
An integer N satisfies (1.4) if and only if the composite (1.2) is defined; a*
*n explicit
version is given by:
8
>><0if mj;k 0; 3 mod 4
N 2mj;k+ 4 - ffi + >1 if mj;k 2 mod 4 (1.5)
>:
2 if mj;k 1 mod 4.
Our theorem may be summarized by saying that
__________
1except in the (rare) cases specified in Proposition 2.14, in which cases we *
*can
only assert that v-11(aejO aek) is a multiple of the specified element
COMPOSITIONS IN HOMOTOPY OF SPHERES 3
Remark 1.6. aejO aek generates the v1-periodic summand the first time it is de*
*fined,
with two exceptions:
(i) If (k) > (j) + 4j and (k) 1; 2 mod 4, then aejO aek is 2 times a generator*
*. (ii)
If (k) = (j) + 4j, then aejO aek is e(j; k) or 2e(j; k) times a generator, and *
*e(j; k) is
even, since, when (k) = 4j + (j), the mod 8 value of e(j; k) is that of (24jj -*
* k)=k,
which is even. We hope that Remark 1.6 illuminates the technical content of The*
*orem
1.3. Once we know aej O aek on the smallest sphere on which it is defined, its*
* value
on larger spheres is easily determined from the well-known effect of the suspen*
*sion
homomorphism on the unstable v1-periodic homotopy groups of spheres.
In [20], the noncommutativity of these compositions when (j) 6= (k) is empha-
sized; here this is seen from the fact that mj;k6= mk;jin these cases. The resu*
*lts of
[20] are a subset of our cases in which (k) < 4j + (j); [20] does not see the c*
*hange
that occurs when (k) = (j) + 4j.
It is immediate to read off from Theorem 1.3 that (provided (k) 6= 4j + (j))
v-11(aejO aek) = 0 in v-11ssN+8j+8k-2(SN ) if and only if
OE(N + ffi) mj;k+ (j + k) + 7: (1.7)
However, we cannot conclude that aejOaek is 0 when (1.7) is satisfied, for it m*
*ight have
a nontrivial v1-torsion component. The composite ae1 O ae1 is an extreme exampl*
*e of
this, which will be discussed at the end of Section 2.
In [8] the first author and Thompson constructed a spectral sequence Es;tr(X)*
* con-
verging to the p-primary homotopy groups of the K-completion X^Kof a space X.
The second author wishes to call this the BTSS. Our theorems are proved by using
Adams operations in K-theory to compute what would be the 1-line of the BTSS of
B(aej), which is defined to be the total space of a fibration
S2n+1! B(aej) ! S2n+8j+1 (1.8)
with attaching map aej, if such a fibration existed. Our algebraic calculation *
*has the
desired homotopy-theoretic implication, even if the fibration does not exist.
The odd-primary analogue of the composition (1.2) was considered in [4] and [*
*17],
and results obtained in a large family of cases. In Section 6 we give complete *
*results
at the odd primes, following exactly the same methods that we employ at the pri*
*me
4 BENDERSKY AND DAVIS
2. The odd-primary case is slightly simpler because the d3-differentials in the*
* BTSS
which make the 2-primary case complicated to state are not present at the odd p*
*rimes.
In Section 7, we determine compositions of elements which suspend to elements*
* of
stable ImJ which are not necessarily generators. This generalizes our main theo*
*rem,
1.3, and is not implied by it, since the multiples of generators are defined on*
* smaller
spheres than are the generators.
In [12], homotopy-theoretic methods were used to determine the 2-primary v1-
periodic homotopy groups of F4=G2, which, by [15], is an S15-bundle over S23 wi*
*th
attaching map ae1. The results obtained there are not consistent with those obt*
*ained
in this paper for such a sphere bundle. The reason for the discrepancy is appar*
*ently
a mistake in an aspect of the proof in [12] which was also used in computing the
v1-periodic homotopy groups of G2 in [15]. The results for G2 in [15] are corre*
*ct, but
the proof requires some modification. These matters will be discussed in Sectio*
*n 5.
We would like to thank Pete Bousfield for a detailed reading of a portion of *
*this
paper, and for allowing us to incorporate work which has not yet been published.
Also, we would like to thank Johns Hopkins University and their JAMI Program for
supporting both authors during the development and writing of this paper.
2.Outline of proof
In this section, we outline the proof of Theorem 1.3. Indeed, in this sectio*
*n we
reduce the proof of this theorem to a certain calculation in the BTSS. This cal*
*cu-
lation is then performed in Section 4. The required relationship of the BTSS wi*
*th
compositions is established in Section 3.
We begin by discussing the relevant features of the spectral sequence for odd-
dimensional spheres.
Theorem 2.1. Let s = 1 or 2, j 1, and n > 2.
1. Es;2n+8j+12(S2n+1) Z=2min(n;(j)+4).
2. The double suspension
Es;2n+8j-12(S2n-1) ! Es;2n+8j+12(S2n+1)
is multiplication by 2 of isomorphic groups if s = 2 and n >
(j) + 4, and is injective otherwise.
COMPOSITIONS IN HOMOTOPY OF SPHERES 5
3. There is an isomorphism
v-11ss*(S2n+1) v-11ss*((S2n+1)^K)
and an exact sequence
0 ! K1 K2 ! v-11ss2n+8j+1-s(S2n+1) ! Es;2n+8j+12(S2n+1) ! C ! 0;
with 8
(j) + 4
K1 = :
0 otherwise
8
2n + 1.
The reader may find the following charts of the BTSS of S2n+1 helpful in unde*
*r-
standing this theorem, and also, perhaps, the proof of Theorem 1.3 presented la*
*ter
in this section. These charts are similar to those presented in [7, p.488] and *
*[3, p.58].
The primary difference is that those presented E2, while these present E1 when
n 0; 3 mod 4, and a subquotient of E3 after most of the d3-differentials have *
*been
taken into account when n 1; 2 mod 4. That the charts of [7] and [3] were of t*
*he
v1-periodic UNSS while the ones here are of the BTSS is inconsequential, since *
*the
two spectral sequences are isomorphic in dimension > 2n + 1.
The horizontal coordinate t - s corresponds to the homotopy group, dots are Z*
*=2,
integers are cyclic groups of the indicated order, diagonal lines of slope 1 ar*
*e multi-
plication by j, and dotted vertical lines are extensions (multiplication by 2).*
* In these
charts = min(n; (j) + 4), and the dotted d3-differential occurs iff = n.
6 BENDERSKY AND DAVIS
Diagram 2.2.
|S2n+1;| n |0; 3 |mod4 | | | | |
|______|_____|______|_____|______|______|_____|______|
| | | | | | | | |
| | r | | | | | | |
4 | | .. | | | | | | r |
| | .. | | | | | | |
|______|_____|______|_____|______|______|_____|______|..
| | .. | | | | | | |
| r | .. | | | | | | |
3 | r | .. | r | | | | r | |
| | .. | .. | | | | | |
|______|_____|______|_____|______|______|_____|______|...
| | .. | .. | | | | | |
| | 4 | .. | | | | | |
| | r | .. | | | 2 | | r |
2 | | | .. | | | | | r |
| | | .. | | | | | |
|______|_____|______|_____|______|______|_____|______|..
| | | . | | | | | |
| r | | .. | | | | | |
s = 1 | | | 4 | | | | 2 | |
| | | | | | | | |
|______|_____|______|_____|______|______|_____|______|
t - s = 2n + 8j - 7+1 2 3 4 5 6 7 8
Diagram 2.3. r
B
|S2n+1;| n |1; 2 |mod4 | B | | | |
|______|_____|______|_____|______|______|_____|______|
| | | | | B | | | |
| | | | r | B | r | | |
4 | | | | | B| .. | | |
| | | | | | ... | | |
|______|_____|______|_____|______|______|_____|______|..B..
| | | | | |B ... | | |
| | | r | | |B ... | | |
3 | r | | r | | r | ... | | |
| | | .. | | |B .. .|. | |
|______|_____|______|_____|______|______|_____|______|....B.
| | | .. | | | B.. |. | |
| | 8 | .. | | | .B |. | |
| | r | .. | r | | 2 |.. | |
2 | | | .. | | | |. | r |
| | | .. | | | |. | |
|______|_____|______|_____|______|______|_____|______|....
| | | . | | | | . | |
| r | | .. | | | | | |
s = 1 | | | 4 | | | | 2 | |
| | | | | | | | |
|______|_____|______|_____|______|______|_____|______|
t - s = 2n + 8j - 7+1 2 3 4 5 6 7 8
Proof of Theorem 2.1.There is also a v1-periodic BTSS, denoted v-11Es;tr(X), *
*with the
localization performed as in [3]; it is isomorphic to the (unlocalized) BTSS *
*in t - s >
dim(X). In [8, 5.2] it is proved that, for any prime p, the mod p v1-periodic*
* BTSS
of S2n+1is isomorphic to the mod p v1-periodic unstable Novikov spectral sequ*
*ence
COMPOSITIONS IN HOMOTOPY OF SPHERES 7
(UNSS). (The BTSS is an unstable Adams spectral sequence based on periodic K-
theory, while the UNSS is an unstable Adams spectral sequence based on BP -theo*
*ry.)
A Bockstein spectral sequence argument implies that the two v1-periodic spectral
sequences are isomorphic (integrally). Note that [8, 5.2] just deals with E2, b*
*ut since
there is a morphism of spectral sequences which is an isomorphism of E2-terms, *
*it is
an isomorphism of spectral sequences. Thus parts 1 and 2 follow from computatio*
*ns
made in [3] of the 2-primary v1-periodic UNSS.
The first part of part 3 follows since the periodic UNSS converges to v-11ss**
*(S2n+1),
while the periodic BTSS converges to v-11ss*((S2n+1)^K), and the spectral seque*
*nces are
isomorphic. The second part of part 3 follows from results about the periodic U*
*NSS
in [3], as depicted in our charts here. Cases of C = Z=2 correspond to a nonze*
*ro
d3-differential on Es2, and cases of K1 = Z=2 correspond to a nontrivial extens*
*ion into
Es+22. The K2 = Z=2 when n 0; 3 is the element in filtration 3. Part 4 follows*
* from
the isomorphism of the BTSS and periodic BTSS above dim(X). ||
If K*X is a free commutative algebra, then Es;t2(X) Ext sU(P K*St; P K*X),
where U is an abelian category of unstable K*K-comodules.([8, 4.9]) We abbrevi-
ate ExtU(P K*St; M) as Exts;tU(M). Let ffi=e= -d(vi1)=2e 2 K2i(K) be as in [8,
x5].
Definition 2.4.Assume n (j) + 4 and e 0. Let OEj = ff4j=(j)+4. Let Mn;j;ebe
the U-object which is a free K*-module on generators gn and gn+4jwith |gi| = 2i*
* + 1
and coaction
(gn) = 1 gn; (gn+4j) = 1 gn+4j+ 2eOEj gn:
Proposition 2.5. With the notation and hypotheses of Definition 2.4, then
1. OEj2n+1generates E1;2n+1+8j2(S2n+1), and
2. If the fibration (1.8) exists, then P K1(B(aej)) Mn;j;0.
Proof.1. In the BP -based UNSS, the generator of E1;2n+1+8j2(S2n+1) described i*
*n [5,
9.10]2 is d(v4j1+ 2(j)+3v4j-31v2)=2(j)+4. In the K-based UNSS, the correction t*
*erm
__________
2Actually, [5, 9.10] contains a misprint in the exponent; the result here is *
*what
it should have said.
8 BENDERSKY AND DAVIS
2(j)+3v4j-31v2 is not needed to obtain maximal divisibility. To see this, we no*
*te that
K*K v-11BP*BP=(vi; jRvi: i > 1):
The relation jRv2 implies that v21h1+ v1h21is divisible by 2. On the other hand,
d(v4j1)=2(j)+3= ((v1-2h1)4j-v4j1)=2(j)+3 v4j-31(v21h1+v1h21) mod 2;
and hence this is divisible by 2 in K*K.
2. If e is as in (2.8), then 2eOEjsurvives the BTSS to an element 2eaej 2 v-1*
*1ss2n+8j(S2n+1).
To see this, note that e = 1 iff C = Z=2 in Theorem 2.1 when s = 1 iff there is*
* a
nonzero differential in the BTSS emanating from E1;2n+1+8j3(S2n+1). It is immed*
*iate
that if (1.8) exists, then P K1(B(aej)) Mn;j;0because the K*K-coaction detects*
* the
attaching map. ||
Regardless of whether or not the fibration exists, algebraic analysis yields *
*informa-
tion about topological compositions by the following result, whose proof appear*
*s in
Section 3.
Theorem 2.6. If n (j) + 4 and e 0, there are exact sequences
0 ! E1;t2(S2n+1) ! Ext1;tU(Mn;j;e) ! E1;t2(S2n+1+8j) -@e!E2;t2(S2n+1);
(2.7)
where @e(x2n+1+8j) = x 2eOEj2n+1. Let i : ss*(SN ) ! ss*((SN )^K) denote the c*
*omple-
tion. If x2n+1+8jrepresents i(2o) with o 2 sst-3(S2n+8j-1), and if
8
<1 if n 1; 2 mod 4, and n = (j) + 4
e = : (2.8)
0 otherwise,
then @e(x2n+1+8j) represents the composite i(2eaejOo) 2 sst-2(S2n+1)^K v-11sst-*
*2(S2n+1).
The compositions we have to compute lie in filtration 2 in the BTSS. Theorem *
*2.6
implies that the order of Ext1;tU(Mn;j;e) determines the order of the compositi*
*on class.
The following result is an algebraic analogue of [6, 1.1] and [6, 2.2] which re*
*duces the
calculation of the 1-line of the BTSS to a calculation involving Adams operatio*
*ns.
Theorem 2.9. For any positive integers n, j, and k, there is an isomorphism
Ext1;2n+8j+8k+1U(Mn;j;0) #;
(2.10)
COMPOSITIONS IN HOMOTOPY OF SPHERES 9
where
Rt = (tn+4j+4k- tn)G + tn(t4j- 1)=2-(j)-4G0
R0t = (tn+4j+4k- tn+4j)G0;
and (-)# denotes the Pontryagin dual.
Proof.Because the argument is so similar to that of [6, 1.1], we merely sketch.*
* We
begin with a result similar to [6, 2.2] for a K*K-module M which is free as a K*
**-
module. The proof is essentially the same as that of [6, 2.2], and is omitted.
P
We say that sy 2 K*KM is unstable if s is in the K*-span of {hI : 2 ij < |y*
*|}.
Proposition 2.11. An element s y 2 K*K M2d+1 Q is (integral and ) unstable
if and only if td< t; s> 2 Z(2)for all integers t.
We define U(M) K*K M to be the span of unstable elements. As in [6], we
have
Ext1;2t+1U(M) ker(d : M2t+1 Q=Z ! K*K M Q=U(M));
where d is induced by the boundary d. We apply 2.11 to determine ker(d) when
M = Mn;j;0and t = n + 4j + 4k. A basis for M in this grading is {v4j+4k1gn; v4k*
*1gn+4j}
with
d(v4j+4k1gn)= ((jRv1)4j+4k- v4j+4k1) gn
d(v4k1gn+4j)= ((jRv1)4k- v4k1) gn+4j+ v4k1((jRv1)4j- v4j1)2-(j)-4 gn:
By [9, p.676] < t; jR(vi1)> = tivi1and < t; vi1> = vi1, and hence, by 2.11, k*
*er(d) is the
intersection over all integers t of the kernel of the morphism of free Q=Z(2)-m*
*odules
with matrix !
tn(t4j+4k- 1)tn(t4j- 1)2-(j)-4
At= n+4j 4k
0 t (t - 1)
By [6, 2.5], this intersection is Pontryagin dual to the abelian group presente*
*d by
all matrices At stacked. By an argument, presented in the next paragraph, simil*
*ar
to that of [6, 3.9], the relations from A-1, A2 and A3 imply all other relation*
*s, but
A-1 = 0, completing the proof of Theorem 2.9.
Although the rows of At may not be tn+4j+4k- t acting on P K1(X) for a topo-
logical space X, they behave as if they were:
10 BENDERSKY AND DAVIS
i:Analogous to [2, 5.1], we verify directly that At At+me mod
me=2(j)+4;
ii:Transformations e tdefined by e t(x1) = tnx1-tn(t4j-1)2-(j)-4x2,
e(tx2) = tn+4jx2 satisfy e se=te st.
Let s be odd. Use [1, 2.9] to write s = (-1)ffl3`+ c2N with ffl = 0 or 1, c 2 Z*
*, and
N sufficiently large. By (i), mod 2N-(j)-4, As A(-1)ffl3`= (-1)fflnA3`. The re*
*lations
`
in A3 say that e 3= 3n+4j+4kon the group presented by A3. Thus by (ii), e3 =
(3n+4j+4k)` on this group. Thus the rows of A3`, and hence of As, are consequen*
*ces
of those of A3. Similarly, using (ii), the relations in A2sare a consequence of*
* those in
A2 and those in As. ||
In Section 4, we shall calculate the right hand side of (2.10), obtaining the*
* following
result.
Theorem 2.12. Let m = mj;k= max((j) + 4; (k) + 4 - 4j). Then
Ext1;2m+8j+8k+1U(Mm;j;0) Z=2(k)+4+f(j;k);
where 8
<0 if (k) 6= 4j + (j)
f(j; k) = :
min((e(j; k)); (j) + 4)if (k) = 4j + (j),
with e(j; k) as in Theorem 1.3.
Proof of Theorem 1.3.We first assume (k) < 4j + (j). We will first prove that,
under this assumption, v-11(aejO aek) generates v-11ssN+8j+8k-2(SN ) when
8
>><1if mj;k 0; 3 mod 4
N = 2mj;k+ >2 if mj;k 2 mod 4 (2.13)
>:
3 if mj;k 1 mod 4
is the smallest integer such that OE(N + 3) mj;k+ 3.
The exact sequence (2.7) with n = mj;k, e = 0, and t = 2n + 8j + 8k + 1 is
0 ! Z=2(j+k)+4! Z=2(k)+4! Z=2(k)+4-@0!Z=2(j+k)+4
unless (k) = (j), in which case (j + k) should be replaced by (j) twice in the
exact sequence. This easily implies that @0 is surjective (e.g., by Euler chara*
*cteristic).
We emphasize that it is here that the calculation of ExtU(Mn;j;0) is being used.
COMPOSITIONS IN HOMOTOPY OF SPHERES 11
Now let e be as in (2.8), with n and t as in the previous paragraph. By the p*
*revious
paragraph, @e in (2.7) is multiplication by 2e. We apply Theorem 2.6 with x2n+1*
*+8j
representing 2a times a generator of E1;t2(S2n+1+8j), subject to the condition *
*that this
element is the double suspension of a homotopy class which suspends to 2aaek. T*
*his is
possible since n + 4j (k) + 4, which implies that E1;2n+8j+8k+12(S2n+1+8j) sur*
*jects
under stabilization. Using Theorem 2.1, we find that this can be done (i.e., 2a*
* times
a generator of E1;2n+8j+8k+12(S2n+1+8j) is the double suspension of a permanent*
* cycle)
if 8
>>>2if (j) + 4j (k) and mj;k 2; 3 mod 4
>><
1 if (j) + 4j (k) and mj;k 0; 1 mod 4
a = >
>>>1if (j) + 4j = (k) + 1 and mj;k 2; 3 mod 4
>:
0 otherwise.
We deduce from 2.6 that v-11(2eaejO 2aaek) is represented by 2e+atimes a genera*
*tor of
v-11E2;2mj;k+8j+8k+12(S2mj;k+1).
If mj;k 0; 3 mod 4, then (by 2.1.3) the E22-generator represents the generator
of v-11ss2mj;k+8j+8k-1(S2mj;k+1), and so, since we can divide by 2e+a in this g*
*roup of
order divisible by 24, v-11(aej O aek) generates the homotopy group in SN with *
*N =
2mj;k+ 1 in this case, as desired. If mj;k 1 mod 4, then the E2-generator does *
*not
yield a homotopy class, but by 2.1 (or see [7, p.483]) its double suspension ge*
*nerates
v-11ssN+8j+8k-2(SN ) with N = 2mj;k+ 3, and so we deduce that v-11(aejO aek) ge*
*nerates
this group in this case, as desired. Similar considerations apply when mj;k 2 m*
*od
4, except that only a single suspension is required in order that the E2 genera*
*tor yield
a homotopy generator (This can be deduced from [16]), and so aejO aek generates*
* in
SN with N = 2mj;k+ 2 in this case.
The case of an arbitrary value of N in Theorem 1.3 now follows from the fact *
*that
v-11ssN+8j+8k-3(SN-1) -! v-11ssN+8j+8k-2(SN ) is surjective if N + 3 6 0; 1; 2*
*; 4 mod 8,
and hits the multiples of 2 if N +3 0; 1; 2; 4 mod 8. This is probably best se*
*en in the
table at the top of page 483 of [7], with an obvious intermediate tower interpo*
*lated
when N 6 mod 8.
Now assume (k) > 4j + (j). The entire argument above goes through, except
aek N+8j-1
that SN+8j+8k-2-! S is not defined. One more suspension is required. That
is why, in this case, N in (1.5) must be 1 greater than its value in (2.13). T*
*he
12 BENDERSKY AND DAVIS
statement in the previous paragraph about the effect on one suspension on the v*
*1-
periodic (8i - 2)-stem implies the minor modifications required in the proof in*
* this
case.
The situation when (k) = (j) + 4j is exactly the same except that @0 in (2.7)*
* is
multiplied by e(j; k) because of Theorem 2.12 and the Euler characteristic argu*
*ment.
There is, however, a problem that if e + a is as in the preceding paragraphs an*
*d if
2e+ae(j; k) is 0 in Z=2(j)+4, we cannot divide by 2e+ato make an assertion abou*
*t the
image of aejO aek. This causes the exceptional case in Theorem 1.3.
Proposition 2.14. Let
8
>><1if (j) 0 mod 4
E = >2 if (j) 1; 3 mod 4
>:
3 if (j) 2 mod 4.
If (k) = (j) + 4j and (e(j; k)) (j) + 4 - E, we can only assert in Theorem 1.3
that v-11(aejO aek) is a multiple of 2OE(N+3)-(mj;k+3)+(e(j;k))(and not that it*
* equals this
2-power times a generator).
We close this section by discussing the special case ae1O ae1. Theorem 1.3 an*
*d (1.7)
say v-11(ae1 O ae1) is a generator of v-11ss23(S9) and is 0 in v-11ssN+14(SN ) *
*for N 21.
Yet ae1O ae1 is nonzero in ssN+14(SN ) for all N 9.
An explanation for this is given by noting that the v1-localization for S2n+1*
*can be
obtained as the composite (see [18] or [12])
ss2n+8i-1(S2n+1) -s*!J2n+8i-1(2n+1P 2n) -v!v-11J2n+8i-1(2n+1P 2n) v-11ss2n+8i-*
*1(S2n+1):
By [18, 1.5], s* is surjective for 4i - (i) n + 8. The morphism v is bijectiv*
*e if
4i-2 n; for smaller values of i, v-11J2n+8i-1(2n+1P 2n) involves elements of n*
*egative
filtration, which are not present in J2n+8i-1(2n+1P 2n). The class ae1 O ae1 is*
* present
and nonzero in filtration 2 in J2n+15(2n+1P 2n) for n 4, but for n 10 it is k*
*illed
by a d3-differential from filtration -1 in v-11J2n+8i(2n+1P 2n).
This is the only case of aejO aek for which this happens_that aejO aek is non*
*zero in
J2n+8j+8k-1(2n+1P 2n) while 0 in v-11J2n+8j+8k-1(2n+1P 2n). To avoid such a sit*
*ua-
tion, we need (roughly)
4(j + k) - 2 mj;k+ (j + k) + 5;
COMPOSITIONS IN HOMOTOPY OF SPHERES 13
which is satisfied unless j = k = 1.
3. Compositions in the BTSS
In this section, we prove Theorem 2.6. We begin with some background on pairi*
*ngs
in the BTSS.
Let {Es;trX } be the homotopy spectral sequence of an augmented cosimplicial *
*space
X . ([11]) For a space X, a K-based cosimplicial space KX is constructed in [8*
*]. The
BTSS of X, {Es;tr(X)}, is the homotopy spectral sequence of KX .
We wish to show that the Yoneda pairing of E2-terms of the BTSS induces a
pairing of spectral sequences which survives to the composition pairing of homo*
*topy
groups. One problem is that the BTSS converges to the homotopy groups of the
K-completion of X rather than those of X itself. More importantly, the cosimpli*
*cial
space used to construct the spectral sequence in [8] is not termwise abelian. *
*The
required generalization of the construction in [11] will appear in [10]. In or*
*der to
understand why one needs this generalization, we first review the constructions*
* and
results of [11].
If X is a simply-connected space and R is a commutative ring, then RX is the
cosimplicial space obtained by resolving X with respect to R. The R-completion *
*of
X, X^R, is defined to be Tot(RX). The Bousfield-Kan spectral sequence, {ErRX},
converges to the homotopy groups of X^R.
In [11, x10], an action
0;t0m * s+s0;t+t0
Es;t+mrRX Esr RS - ! Er RX
is constructed for t - s 1. This action survives to a pairing
sst+m-sX^R sst0-s0(Sm )^R-*! sst+t0-s-s0X^R
for t - s 1. In [11, 10.2], it is shown that this pairing is compatible with *
*the
composition pairing in homotopy, in the sense that, with i : X ! X^Rthe natural
map, there is a commutative diagram
sst+mX sst0Sm --O-! sst+t0X
?? ?
?yi*i* ??yi*
sst+mX^R sst0(Sm )^R*---!sst+t0X^R
14 BENDERSKY AND DAVIS
In [11, x18], this pairing is shown to correspond to the Yoneda preoduct.
The results of [11] do not immediately apply to our situation. The pairing * *
*depends
on a map c, constructed in [11, 9.1], between cosimplicial spaces. That c is a *
*map of
cosimplicial spaces uses the fact that RX is termwise abelian. Since the cosimp*
*licial
space in [8] is not termwise abelian, we cannot use the results of [11]. Howev*
*er,
Bousfield has shown in [10] that there is a pairing
0;t0m * s+s0;t+t0
Es;t+mrX Esr S - ! Er X
(with t - s 1 and r 2) which generalizes the pairing in [11] to the K-theory
situation. Specializing to odd spheres, we have the following result.
Theorem 3.1. The *-pairing is compatible with the O-pairing in the sense that *
*the
following diagram commutes.
v-11 -1 2n+1
sst+2m+1S2n+1 sst0S2m+1 --O-! sst+t0S2n+1---! v1 sst+t0S
?? ?? ??
ii?y i?y i?y
sst+2m+1(S2n+1)^K sst0(S2m+1)^K*---!sst+t0(S2n+1)^K---!v-11sst+t0(S2n+1)^K
We apply this theorem to the element 2eaej o of Theorem 2.6, and deduce that
the element i(2eaejO o) in the conclusion of that theorem equals
i(2eaej) * i(o): (3.2)
Now we interpret the pairing in the BTSS in terms of the usual pairing on Ext.
This follows since K*S2n+1is isomorphic to a free commutative algebra, so that,*
* as
noted prior to 2.4, Es;t2(S2n+1) ExtsU(K*St; K*S2n+1):
Lemma 3.3. The * pairing corresponds to the Yoneda product
0 t0 2m+1
ExtsU(K*St+2m+1; K*S2n+1) ExtsU(K*S ; K*S )
0 t+t0 t+2m+1
! ExtsU(K*St+2m+1; K*S2n+1) ExtsU(K*S ; K*S )
0 t+t0 2n+1
! Exts+sU(K*S ; K*S ):
Proof.This follows as in [11, x18], using the cosimplicial pairings constructed*
* in [10].
||
COMPOSITIONS IN HOMOTOPY OF SPHERES 15
In the notation of Theorem 2.6, the homotopy pair (3.2) comes from the E2 pair
2eOEj * x. The value of e given in (2.8) is required in order that 2eOEj is a p*
*ermanent
cycle in the BTSS. By Lemma 3.3, this * product of E2 classes is the Yoneda pro*
*duct
of Ext elements. The relationship of this Yoneda product with the boundary in t*
*he
exact sequence (2.7) is standard, and is formalized and proved in [21, 2.3.4].
Indeed the sequence (2.7) is induced from the short exact sequence in U
0 ! K*S2n+1! Mn;j;e! K*S2n+1+8j! 0; (3.4)
and [21, 2.3.4] asserts that the boundary morphism in (2.7) is given by Yoneda *
*product
with the element of Ext1corresponding to the extension (3.4). Our expression of*
* this
boundary as x 2eOEj is easily seen by the usual way of seeing such a boundary
morphism by diagram-chasing.
4. The calculation
In this section we prove Theorem 2.12 by computing the right hand side of (2.*
*10)
with n = mj;k. We begin by letting n be arbitrary. We remove terms which are
divisible by 2n+4j+4k, for, as we shall see, the groups will be of order smalle*
*r than
this. The four relations, after multiplying by unit factors, are
2nG - 2n-4-(j)(24j- 1)G0
2n+4jG0
(34(j+k)- 1)G + (34j- 1)2-4-(j)G0
(34k- 1)G0
We use that (3i- 1) = (i) + 2 if i is even, and note that the coefficient of *
*G0
in the third relation is a unit. We use the third relation to eliminate G0. Wit*
*h the
second and fourth relations combined, we have two relations (on G)
2n + 2n(24j- 1)(34j+4k- 1)=(34j- 1) (4.1)
2(j+k)+4+min(n+4j;(k)+4): (4.2)
Multiplying (4.1) by the unit (34j- 1)=2(4j)+4yields
2n-4-(j)(34j- 1 + (24j- 1)(34j+4k- 1)):
16 BENDERSKY AND DAVIS
Expanding 34i- 1 = (80 + 1)i- 1 brings this to the form
X iijj ij+kjj
5 . 2n-(j) 80i-1 i + (24j- 1) i : (4.3)
i1
The 2-exponent of our desired group is the smaller of the exponents of (4.2) *
*and
(4.3) when n = mj;k. Considering several cases, one verifies that if (k) 6= 4j *
*+ (j),
then the exponent of (4.3) is (k)+4, while that of (4.2) is larger. If (k) = 4j*
* +(j),
then the exponent of (4.2) is (j)+(k)+8, while that of (4.3) is (e(j; k))+(k)+4,
as claimed. ||
5.Discussion of previous work on G2 and F4=G2
In an attempt to determine v-11ss*(F4=G2), the second author proposed a proof*
* in
[12, 8.15] that v-11(ae1Oaek) is 8 times a generator of v-11ss8k+21(S15) Z=16.*
* According
to our Theorem 1.3, its actual coefficient is
8
>><8 if (k) < 4
>>:0 if (k) = 4
27-(k) if 5 (k) 6.
The composite is not defined on S15if (k) > 6.
The proof of [12, 8.15] relied on the existence of a certain map
1 (2n+1P 2n^ J) ! (S2n+1)K (5.1)
when n = 7. Such a map had been produced when n = 2 in [15, 4.6]. It appears
that the maps (5.1) must not exist, because of the contradiction that they impl*
*y for
ae1O aek when (k) 4, and many other similar composites. The method of [12] was
similar to that employed by Mahowald and Thompson in [20], except that [20] did*
* not
use maps (5.1). If maps (5.1) exist, this method applied to many composites aej*
*O aek
disagrees with Theorem 1.3. The results of [20] about aejO aek did not deal wit*
*h cases
where existence of (5.1) yields results that contradict our Theorem 1.3.
One aspect of the proof of [15, 4.6] seems incomplete. This is apparently the*
* cause
of the problem. A diagram
1 (2n+1P ^ J) ---! 1 (2n+1P2n+1^ J)
?? ?
?y ??y
1 (1 S2n+1)K ---! 1 (2n+1P2n+1)K
COMPOSITIONS IN HOMOTOPY OF SPHERES 17
is constructed. Its commutativity was tacitly assumed. The bottom map was induc*
*ed
from the K-localization of the Snaith map QS2n+1! Q2n+1P2n+1. It is probably
not an infinite loop map. So there seems no clear way to establish commutativit*
*y of
this diagram.
The (apparently false) result [15, 4.6] was used in computing a family of hig*
*her
differentials in an Adams-type spectral sequence converging to v-11ss*(G2). Th*
*ere
were two possibilities for this family of differentials, yielding two possible *
*results for
v-11ss*(G2). Another approach to v-11ss*(G2) is via the BTSS. Our expertise wit*
*h the
BTSS is not yet at the level where we can determine, strictly within BTSS techn*
*ology,
the entire BTSS for G2. However, we can use [6, 1.1] to calculate the 1-line, a*
*nd the
result of this calculation is consistent with the picture of v-11ss*(G2) claime*
*d in [15],
and is not consistent with the alternate picture. Thus we assert that [15, 1.3]*
*, the
explicit listing of the groups v-11ssi(G2), is correct, even though the argumen*
*t there
apparently had a gap.
Indeed, we have that
E1;2m+12(G2) (P K1(G2)= im( 2; 3 - 3m ))#:
Using software LiE as in [13], we find that P K1(G2)(2)has basis {w1; w2} satis*
*fying
k(w1) = kw1+ 1_2(k - k5)w2 and k(w2) = k5w2. This agrees with results of [23,*
* 2.5].
From this, we obtain
E1;4i+32(G2) Z=23+min((i-2);3):
These groups are isomorphic to the groups v-11ss4i+2(G2) asserted in [15, 1.3].*
* A
picture of v-11ss*(G2) is presented on [15, p.666]. The two possibilities ther*
*e are
whether a differential d is nonzero for even values of k or for odd values of k*
*. In
either case, the fact that the element in the top of the tower is divisible by *
*j2 implies
that in the BTSS there must be an extension from the 1-line group to a Z=2 on t*
*he
3-line. In order to make the E2-group and the homotopy group have the same orde*
*r,
there must be a compensating d3-differential on the generator of the 1-line gro*
*up. On
the other hand, there is no way to create a BTSS picture with 1-line as computed
which yields the picture of [15, p.666] with the opposite pattern for d.
18 BENDERSKY AND DAVIS
6. The odd-primary analogue
The exact same methods which were applied at the prime 2 yield similar results
when applied at an odd prime p. In this section, p is an odd prime, q = 2(p - 1*
*),
and (-) is the exponent of p in an integer. Let aei denote a map (or its homoto*
*py
class) which suspends to a generator of the image of the stable J-homomorphism *
*in
the p-primary (qi - 1)-stem.
It was shown in [4] that v-11ss2n+qi-1(S2n+1) Z=pmin(n;(i)+1), and
v-11 -1 2n+1
ss2n+qi-1(S2n+1) -! v1 ss2n+qi-1(S )
is split surjective if i - (i) n + 1. Our theorem at the odd prime p, analogou*
*s to
Theorem 1.3 is
Theorem 6.1. Let nj;k= max((j) + 1; (k) + 1 - (p - 1)j). Then, by [5], S2nj;k+1
is the smallest odd sphere on which aejO aek is defined. If n nj;k, then v-11(*
*aejO aek) is
2n-nj;k+(e(j;k))times a generator of v-11ss2n+qj+qk-1(S2n+1), where
8
<1 if (k) 6= (p - 1)j + (j)
e(j; k) = :1_P i-1iijj (p-1)j ij+kjj
k i1 (ffp) i + (p - 1) i if (k) = (p - 1)j + (j),
where ff 6 0 mod p satisfies ffp + 1 = rp-1 with r a generator of (Z=p2)x.
Again it is true that aejO aek generates the v1-periodic summand the first ti*
*me it is
defined, provided (k) 6= (p - 1)j + (j). Also note that one can easily read off*
* when
the v1-periodic component of aejO aek is 0, but cannot usually deduce that aejO*
* aek is in
fact 0. The proof of Theorem 6.1 is a direct adaptation of our proof in the 2-p*
*rimary
case, and is omitted.
In [4, 4.1], the first author used the UNSS to prove Theorem 6.1 when (k) <
(p - 1)j + (j).
7. Compositions of non-generators
In this section, we show how our main theorem generalizes to compositions of *
*any
elements which suspend to elements in the image of J, not necessarily the gener*
*a-
tors. Results for multiples of the generators are not usually implied by result*
*s for the
COMPOSITIONS IN HOMOTOPY OF SPHERES 19
generators because the multiples are defined on smaller spheres than are the ge*
*nera-
tors, and the double suspension homomorphism is often not injective on the unst*
*able
homotopy groups where these compositions lie.
For simplicity, we just treat the odd-primary case here, and we do not deal w*
*ith
borderline cases which cause coefficients such as e(j; k) of Theorem 6.1 to ari*
*se. We
just write what the composite is on the smallest sphere on which it is defined.*
* Results
for these composites on larger spheres are, of course, then determined since th*
*e double
suspension induces multiplication by 2.
We use the notation that ffj=eis an element of order pe in the (qj -1)-stem. *
*This is
defined provided e (j) + 1. Then aej of Section 6 is ffj=(j)+1. The following *
*result
generalizes Theorem 6.1. Composites of this sort were considered in a limited r*
*ange
of cases in [4] and [17].
Theorem 7.1. The smallest odd sphere S2n0+1on which ffj=eO ffk=fis defined has
n0 = min(e; f - (p - 1)j + 1). The composite v-11(ffj=eO ffk=f) equals pc1+c2ti*
*mes a
generator of v-11ssqj+qk+2n0-1(S2n0+1), where
c1 = max (0; min((k) - (p - 1)j; (j)) + 1 - e)
c2 = max (0; min((k) + 1; e + (p - 1)j - 1) - f);
provided (k) 6= (j) + (p - 1)j.
Note that if ffj=e= aej, then c1 = 0, and if ffk=f= aek, then c2 = 0, and so *
*the result
agrees with Theorem 6.1, which states that the composite of elements which susp*
*end
to generators of ImJ is a generator the first time it is defined.
The proof is similar to that of Theorem 6.1, which was similar to that of The*
*orem
1.3. We define an unstable K*K-comodule M which depends on n0, j, and e, and fi*
*ts
into an exact sequence
0 ! E1;t2(S2n0+1) ! Ext1;tU(M) ! E1;t2(S2n0+1+qj) -@! E2;t2(S2n0+1):
We work with t0 = 2n0+1+qj +qk. Then ffk=fis represented by pc2times a generator
of E1;t02(S2n0+1+qj). Similarly to Theorem 2.12, we compute the order of Ext1;t*
*0U(M).
The group is not always cyclic. It turns out that
| Ext1;t0U(M)|=|E1;t02(S2n0+1+qj)| = pc1:
20 BENDERSKY AND DAVIS
It is elementary to conclude that @ sends a generator to pc1times a generator. *
*By an
analogue of Theorem 2.6, @ sends ffk=fto ffj=eO ffk=f, yielding Theorem 7.1.
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Hunter College, CUNY, NY, NY 10021
E-mail address: mbenders@shiva.hunter.cuny.edu
Lehigh University, Bethlehem, PA 18015
E-mail address: dmd1@lehigh.edu