CHAPTER 1
Computing v1periodic homotopy groups
of spheres and some compact Lie groups
Donald M. Davis
Lehigh University
Bethlehem, PA 18015
dmd1@lehigh.edu
Contents
1.Introduction:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: *
* 3
2.Definition of v1periodic homotopy groups::::::::::::::::::::::::::::::::::::*
*::: 5
3.The isomorphism v11ss*(S2n+1) v11sss*2n1(Bqn):::::::::::::::::::::::::::*
*:: 7
4.Jhomology :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: *
*12
5.The v1periodic homotopy groups of spectra ::::::::::::::::::::::::::::::::::*
*::: 23
6.The v1periodic UNSS for spheres:::::::::::::::::::::::::::::::::::::::::::::*
*: 26
7.v1periodic homotopy groups of SU(n) ::::::::::::::::::::::::::::::::::::::::*
*:: 36
8.v1periodic homotopy groups of some Lie groups ::::::::::::::::::::::::::::::*
*::: 44
References:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: *
* 55
HANDBOOK OF ALGEBRAIC TOPOLOGY
Edited by I.M. James
cO1995 Elsevier Science B.V. All rights reserved
1
2 Donald M. Davis Chapter 1
Section 1 Computing periodic homotopy groups 3
1.Introduction
In this paper, we present an account of the principal methods which have been u*
*sed
to compute the v1periodic homotopy groups of spheres and many compact simple
Lie groups. The two main tools have been Jhomology and the unstable Novikov
spectral sequence (UNSS), and we shall strive to present all requisite backgrou*
*nd
on both of these.
The v1periodic homotopy groups of a space X, denoted v11ss*(X; p), are a cer
tain localization of the actual plocal homotopy groups ss*(X)(p). We shall dro*
*p the
p from the notation except where it seems necessary. Very roughly, v11ss*(X) i*
*s a
periodic version of the portion of ss*(X) detectable by real and complex Ktheo*
*ry
and their operations. It is the first of a hierarchy of theories, v1nss*(X), w*
*hich
should account for all of ss*(X). Each group v11ssi(X) is a direct summand of *
*some
actual group ssi+L(X), at least if X has an Hspace exponent and v11ssi(X) is a
finitelygenerated abelian group, which is the case in all examples discussed h*
*ere.
The v1periodic homotopy groups are important because for spaces such as
spheres and compact simple Lie groups they give a significant portion of the ac*
*tual
homotopy groups, and yet are often completely calculable. The goal of this pape*
*r is
to explain how those calculations can be made. One might hope that these methods
can be adapted to learning about vnperiodic homotopy groups for n > 1.
One application of v1periodic homotopy groups is to obtain lower bounds for t*
*he
exponents of spaces. The pexponent of X is the largest e such that some homoto*
*py
group of X contains an element of order pe. We can frequently determine the lar*
*gest
ptorsion summand in v11ss*(X). Such a summand must also exist in some ssi(X),
although we cannot usually specify which ssi(X). Thus we obtain lower bounds for
the pexponents of spaces, which we conjecture to be sharp in many of the cases
studied here. It is known to be sharp for S2n+1 if p is odd. See Corollaries 7.*
*8 and
8.9 for estimates of the pexponent of SU (n) when p is odd.
Another application, which we will not discuss in this paper, is to James numb*
*ers,
which are an outgrowth of work on vector fields in the 1950's. It is proved in *
*[23]
that, for sufficiently large values of the parameters, the unstable James numbe*
*rs
equal the stable James numbers, and these equal a certain value which had been
conjectured by a number of workers.
In Section 2, we present the definition and basic properties of the v1periodic
homotopy groups. In Section 3, we describe the reduction of the calculation of
the unstable groups v11ss*(S2n+1) to the calculation of stable groups v11sss**
*(Bqn).
Here Bqn is a space which will be defined in Section 3; when p = 2, it is the r*
*eal
projective space P 2n. Here we have begun the use, continued throughout the pap*
*er,
of q as 2p  2. The original proof of this result, 3.1, appeared in [40] and [5*
*6]; it
involved delicate arguments involving the lambda algebra. We present a new proo*
*f,
due to Langsetmo and Thompson, which involves completely different techniques,
primarily Ktheoretic.
In Section 4, we explain how to compute J*(Bqn), while in Section 5, we sketch
the proof that if X is a spectrum, then v11ss*(X) v11J*(X). Combining the
results of Sections 3, 4, and 5 yields a nice complete result for v11ss*(S2n+1*
*), which
4 Donald M. Davis Chapter 1
can be summarized as
v11ss2n+1+i(S2n+1) v11sssi(Bqn) v11Ji(Bqn)
and, if p is odd,
ae min(n;p(a)+1)
Z=p if i = qa  2 or qa  1
0 if i 6 1 or 2 mod q.
We will use Z=n and Zn interchangeably, and let p(n) denote the exponent of p
in n. The subscript p of will sometimes be omitted if it is clear from the con*
*text.
The final result for v11ss*(S2n+1) when p = 2 is more complicated; see Theorem
4.2.
In Section 6, we sketch the formation of the UNSS and its v1localization, and*
* for
S2n+1 we compute the entire v1localized UNSS and part of the unlocalized UNSS.
In Section 7, we discuss the computation of the v1periodic UNSS and v1periodic
homotopy groups in general for spherically resolved spaces and specifically for*
* the
special unitary groups SU(n). This is considerably easier at the odd primes tha*
*n at
the prime 2. The following key result of Bendersky ([4]) will be proved by obse*
*rving
that the homotopytheoretic calculation and UNSS calculation agree for S2n+1.
If p is odd, and X is built by fibrations from finitely many odddimensional sp*
*heres,
then v11Es;t2(X) = 0 in the v1periodic UNSS unless s = 1 or 2 and t is odd, in
which case
ae1 1;i+1
v11ssi(X; p) v1E212;(X)i+2if i is even
v1 E2 (X) if i is odd.
In Section 7 we also review the computation of E12(SU (n)) in [6] and combine *
*it
with Theorem 1.1 to obtain the following result, which was the main result of [*
*23].
Let p(m) denote the exponent of p in m, and define integers a(k; j) and ep(k; n)
by
X xk
(ex  1)j = a(k; j)__ ;
kj k!
and
ep(k; n) = min{p(a(k; j)) : n j k}:
If p is odd, then v11ss2k(SU (n)) Z=pep(k;n), and v11ss2k1(SU (n)) is an ab*
*elian
group of order pep(k;n), although not always cyclic.
In Section 8, we illustrate the two principal methods used in computing v1
periodic homotopy groups of the exceptional Lie groups, focusing on v11ss*(G2;*
* 5)
Section 2 Computing periodic homotopy groups 5
for UNSS methods, and on v11ss*(F4=G2; 2) for homotopy (Jhomology) methods.
We also discuss the recent thesis of Yang ([58]), which gives formulas more tra*
*ctable
than that of Definition 1.2 for the numbers ep(k; n) which appear in Theorem 1.*
*3,
provided n p2 p.
2.Definition of v1periodic homotopy groups
In this section, we present the definition and basic properties of the v1perio*
*dic
homotopy groups. We work toward the definition of v11ss*(X) by recalling the
definition of v11ss*(X; Z=pe). Let Mn(k) denote the Moore space Sn1 [k en. The
mod k homotopy group ssn(X; Z=k) is defined to be the set of homotopy classes
[Mn(k); X]. With the prime p implicit, and q = 2(p  1), let
aee1
s(e) = pmaxq(8; 2e1if)piisfoddp = 2. (2.1)
Let A : Mn+s(e)(pe) ! Mn(pe) denote a map, as introduced by Adams in [1],
which induces an isomorphism in Ktheory. Such a map exists provided n 2e+3.
([28, 2.11]) Then v11ssi(X; Z=pe) is defined to be dirlimN[Mi+Ns(e)(pe); X], w*
*here
the maps A are used to define the direct system. The map is what Hopkins and
Smith would call a v1map, and they showed in [35] that any two vnmaps of a
finite complex which admits such maps become homotopic after a finite number
of iterations (of suspensions of the same map), and hence v11ss*(X; Z=pe) does
not depend upon the choice of the map A. Note that although v11ss*(X; Z=pe)
is a theory yielding information about the unstable homotopy groups of X, the
maps A which define the direct system may be assumed to be stable maps, since
the direct limit only cares about large values of i + Ns(e). Note also that the
groups v11ssi(X; Z=pe) are defined for all integers i and satisfy v11ssi(X; Z*
*=pe)
v11ssi+s(e)(X; Z=pe).
There is a canonical map ae : Mn(pe+1) ! Mn(pe) which has degree p on the top
cell, and degree 1 on the bottom cell. It satisfies the following compatibility*
* with
Adams maps.
[34, p. 633] If A : Mn+s(e)(pe) ! Mn(pe) and
A0: Mn+s(e+1)(pe+1) ! Mn(pe+1)
are v1maps, then there exists k so that the following diagram commutes.
Mn+ks(e+1)(pe+1)ae!Mn+ks(e+1)(pe)??
?yA0k ?y 0
Akp
Mn(pe+1) ae! Mn(pe)
Here p0= p unless p = 2 and e < 4, in which case p0= 1.
6 Donald M. Davis Chapter 1
Thus, after sufficient iteration of the Adams maps, there are morphisms ae* be
tween the direct systems used in defining v11ss*(X; Z=pe) for varying e, and p*
*assing
to direct limits, we obtain a direct system
* ae*
v11ss*(X; Z=pe) ae!v11ss*(X; Z=pe+1) ! . .:. (2.2)
The following definition was given in [28], following less satisfactory definit*
*ions in
[30] and [23].
For any space X and any integer i,
v11ssi(X) = dirlimev11ssi+1(X; Z=pe);
using the direct system in (2.2).
The reason for the use of the (i+1)st modpe periodic homotopy groups in defini*
*ng
the ith (integral) periodic groups is that the maps ae of Moore spaces have deg*
*ree
1 on the bottom cells, but modpe homotopy groups are indexed by the dimension
of the top cell.
The modpe periodic homotopy groups have received more attention in the liter
ature, especially when e = 1. For spaces with Hspace exponents, there is a clo*
*se
relationship between the integral periodic groups and the modpe groups, which *
*we
recall after giving the relevant definition.
A space X has Hspace exponent pe if for some positive integer L the pepower
map ffiLX ! ffiLX is nullhomotopic.
By [20] and [36], spheres and compact Lie groups have Hspace exponents.
(i)[28, 1.7] If X has Hspace exponent pe, then there is a split short exact
sequence
0 ! v11ssi(X) ! v11ssi(X; Z=pe) ! v11ssi1(X) ! 0:
(ii)On the category of spaces with Hspace exponents, there is a natural tran*
*s
formation ss*()(p)! v11ss*(; p).
(iii)If X has an Hspace exponent, and v11ssi(X) is a finitely generated abe*
*lian
group, then v11ssi(X) is a direct summand of ssi+L(X) for some nonnegative in
teger L.
The proof of part ii utilizes the fibration
e
map *(Mn+1(pe); X) ! ffinX p! ffinX;
where map *(; ) denotes the space of pointed maps. If the second map is null
homotopic, then the first map admits a section s. The natural transformation is
induced by
ffinX s! map*(Mn+1(pe); X) ! dirlimkmap*(Mn+1+ks(e)(pe); X):
Section 3 Computing periodic homotopy groups 7
Techniques of [28] imply naturality of this construction. To prove part iii, we*
* note
that the map s allows v11ssi(X) to be written as dirlimkssi+ks(e)(X), which is*
* a
direct summand of one of the groups in the direct system, provided the direct l*
*imit
is finitely generated.
If X is a spectrum, then v11ss*(X) is defined in exactly the same way as for *
*spaces,
that is, as in Definition 2.2. If X is a space, then stable groups, v11sss*(X)*
*, can be
defined either as v11ss*(21 X), where 21 X denotes the suspension spectrum of
X, or as v11ss*(QX), where QX = ffi1 21 X is the associated infinite loop spac*
*e.
3.The isomorphism v11ss*(S2n+1) v11sss*2n1(Bqn)
In this section, we sketch a new proof, due to Thompson and Langsetmo ([39, 4.2*
*]),
of the following crucial result.
There is a map
ffi2n+1S2n+1! QBqn (3.1)
which induces an isomorphism in v11ss*().
Here QX = ffi1 21 X, and Bqn is the qnskeleton of the plocalization of the cl*
*as
sifying space B 2p of the symmetric group 2p on p letters. Note that if p = 2, *
*then
Bqn is the real projective space RP 2n. The original proof, from [40] when p = 2
and [56] when p is odd, involved delicate arguments involving the lambda algebra
and unstable Adams spectral sequences. We feel that the following argument, pri
marily Ktheoretic, will speak to a broader crosssection of readers. The follo*
*wing
elementary result shows that it is enough to show that the map (3.1) induces an
iso in v1periodic mod p homotopy.
If a map X ! Y induces an isomorphism in v11ss*(; Z=p), then it induces an
isomorphism in v11ss*(; p).
Proof. There are cofibrations of Moore spaces which induce natural exact se
quences
*
! v11ssn(X; Z=pe)ae!v11ssn(X; Z=pe+1) !
v11ssn(X; Z=p)! v11ssn1(X; Z=pe) ! :
Induction on e using the 5lemma implies that there are isomorphisms
v11ss*(X; Z=pe) ! v11ss*(Y ; Z=pe)
for all e, compatible with the maps ae* which define the direct system (2.2). T*
*he
desired isomorphism of the direct limits is immediate. 
The construction of the map (3.1) takes us far afield, and is not used elsewhe*
*re in
the computations. For completeness, we wish to say something about it, but we w*
*ill
8 Donald M. Davis Chapter 1
be extremely sketchy. The map is due to Snaith. Work of many mathematicians,
especially Peter May, is important in the construction. However, we shall just *
*refer
the reader to [37], where the proof of naturality of these maps is given, along*
* with
references to the earlier work.
i. There are maps
sn : ffi2n+1S2n+1! QBqn
which are compatible with respect to inclusion maps as n increases, and such th*
*at
the adjoint map
21 ffi2n+1S2n+1!21 Bqn
is the projection onto a summand in a decomposition of 21 ffi2n+1S2n+1 as a wed*
*ge
of spectra. g
ii. There is a map QS2n+1 ! Q 22n+1B(n+1)q1whose fiber, F, satisfies
v11ss*(ffi2n+1F; Z=p) v11ss*(QBqn; Z=p):
Sketch of proof. Let CN (k) denote the space of ordered ktuples of disjoint l*
*ittle
cubes in IN . If X is a based space, let CN X denote the space of finite collec*
*tions
of disjoint little cubes of IN labeled with points of X. More formally,
a
CN X = CN (k) x2k Xk= ~;
k1
where
[(c1; : :;:ck); (x1; : :;:xk1; *)] ~ [(c1; : :;:ck1); (x1; : :;:xk1)]:
There are natural maps, due to May,
CN X ! ffiN 2N X; (3.2)
which are weak equivalences if X is connected.
The space CN X is filtered by defining Fm (CN X) to be the subspace of m or
fewer little labeled cubes. The successive quotients are defined by
DN;mX = Fm (CN X)=Fm1 (CN X) CN (m)+ ^2m X[m];
where X[m]denotes the mfold smash product. Snaith proved that if X is path
connected, there is a weak equivalence of suspension spectra
_
21 CN X ' 21 DN;mX: (3.3)
m1
Section 3 Computing periodic homotopy groups 9
Let N = 2n + 1 and X = S0. Stabilize the equivalence of (3.2), and project onto
the summand of (3.3) with m = p to get
21 ffi2n+1S2n+1!21 C2n+1(p)= 2p :
The identification of C2n+1(p)= 2p as the nqskeleton of B 2p after localizatio*
*n at
p was obtained by Fred Cohen in [19, p. 246].
Note that (3.2) and (3.3) yield a map ffiN 2N X ! QDN;mX. The map g of ii)
is obtained from the case N = 1 and X = S2n+1 as the composite
QS2n+1! QD1;pS2n+1= Q(B 2+p^2p(S2n+1)[p]) ' Q 22n+1B(n+1)q1:
Here we have used results of [45] for the last equivalence.
The compatible maps sn of 3.3(i) combine to yield a map s0: QS0 ! QB1 , and
one can show that the right square commutes in the diagram of fibrations below.
2n+1g
ffi2n+1F!?QS0?ffi! QB(n+1)q1?
?yt ?ys0 ?y=
QBqn ! QB1 ! QB(n+1)q1
The map t then follows, and will induce an isomorphism in v11ss*(; Z=p), as
asserted in Theorem 3.3(ii), once we know that s0does. Kahn and Priddy showed
that there is an infinite loop map : QB1 ! QS0 such that O s0 induces an
isomorphism in ssj() for j > 0. It is easily verified using methods of the next
two sections that the associated stable map B1 ! S0 induces an isomorphism in
v11ss*(; Z=p). Hence so does s0. 
Throughout this section, let K*() denote mod p Khomology, and Mk denote
the mod p Moore space Mk(p). The following two theorems, whose proofs occupy
most of the rest of this section, imply that j : S2n+1 ! F induces an iso in
v11ss*(; Z=p).
[16, 14.4] If k 2, and OE : X ! Y is a map of kconnected spaces such that ffi*
*kOE
is a K*equivalence, then OE induces an isomorphism in v11ss*(; Z=p).
[39] Let F be as in Theorem 3.3ii. The map S2n+1 ! QS2n+1 lifts to a map
j : S2n+1! F such that ffi2j is a K*equivalence.
The map j of Theorem 3.5 can be chosen so that t O ffi2n+1j = sn, where t is t*
*he
map in the proof of Theorem 3.3ii which induces the isomorphism in v11ss*(; Z*
*=p).
Theorem 3.1 is now proved by applying Lemma 3.2 to sn.
Bousfield localization is involved in the proof of Theorem 3.4 and several oth*
*er
topics later in the paper, and so we review the necessary material. If E is a s*
*pec
trum, then a space (or spectrum) X is E*local if every E*equivalence Y ! W
induces a bijection [W; X] ! [Y; X]. The E*localization of X is an E*local sp*
*ace
10 Donald M. Davis Chapter 1
(or spectrum) XE together with an E*equivalence X ! XE . The existence and
uniqueness of these localizations were established in [18] and [17]. We will ne*
*ed the
following result of Bousfield, the proof of which is easily obtained using the *
*ideas
at the beginning of Section 5.
[18, 4.8] A spectrum is K*local if and only if its mod p homotopy groups are
periodic under the action of the Adams map.
We will omit the proof of Theorem 3.4, as it requires many peripheral ideas.
Instead, we will sketch a proof of the following result, which, although weaker*
* than
3.4, has the same flavor, and predated it. An alternate proof of Theorem 3.1 ca*
*n be
given by using Theorem 3.7 and strengthening Theorem 3.5 to show that K*(ffi3j)
is bijective. As the calculations for ffi3 seem significantly more difficult th*
*an for ffi2,
we omit that approach.
[57] Let p be an odd prime, and let X and Y be 3connected spaces. Suppose that
f : X ! Y is a map such that K*(ffikf) is an isomorphism for k = 0, 1, 2, and 3.
Then f induces an isomorphism in v11ss*(; Z=p).
Sketch of proof. If F ! E ! B is a principal fibration, there is a bar spec
tral sequence converging to K*(B) with E2s;t T orK*Fs;t(K*E; K*). (See [55] for*
* a
discussion of this spectral sequence.)
Applied to the commutative diagram of principal fiber sequences
ffi3Xp!ffi3X?!map*(M3;?X)?
?yffi3f?yffi3f ?yf0
ffi3Yp!ffi3Ym!ap*(M3; Y );
the spectral sequence and the hypothesis of the theorem imply that f0 induces an
isomorphism in K*(). Hence there is an equivalence of the K*localizations
0
(map *(M3; X))K fK!(map *(M3; Y ))K :
Let V (X) denote the mapping telescope of
* A*
map *(M3; X) A!map *(M3+q; X) ! . .:.
Then V (X) is K*local, since it is ffi1 of a periodic spectrum which is K*lo*
*cal by
Theorem 3.6. This implies that there are maps i0 making the following diagram
commute.
0
map *(M3;?X)! (map *(M3;?X))K i!V?(X)
?yf0 ?yf0 ?
K 0y V (f)
map *(M3; Y!)(map *(M3; Y ))K i!V (Y )
Section 3 Computing periodic homotopy groups 11
Since v11ss*(X; Z=p) ss*(V (X); Z=p), the desired isomorphism is a consequence
of the following construction of an inverse to
V (f)* : ss*(V (X); Z=p) ! ss*(V (Y ); Z=p):
An element ff 2 ssk(V (Y ); Z=p) can be represented by a map
Mk ! map*(M3+qpj; Y );
or, adjointing, by a map Mk+qpj! map*(M3; Y ). Here some care is required to
see that we can switch the Moore space factor on which the map A is performed.
The element which corresponds to ff is the composite
Mk+qpj ! map *(M3; Y ) ! (map *(M3; Y ))K
(f0K)1!(map 3 i0
*(M ; X))K ! V (X):

The proof of Theorem 3.5 involves a good bit of delicate computation. The hard
est part is the determination of K*(ffi2F) as a Hopf algebra. In order to conve*
*niently
obtain the coalgebra structure of K*(ffi2F), we proceed in two steps. We first *
*calcu
late the algebra K*(ffi3F), using the bar spectral sequence associated to the p*
*rincipal
fibration
ffi4QS2n+1! ffi4Q 22n+1Bq(n+1)1! ffi3F:
This spectral sequence is calculated in [38], obtaining an algebra isomorphism
K*(ffi3F) P [y1; y2] E[z]: (3.4)
Here yi (resp. z) has bidegree (1; 1) (resp. (0; 1)) in the spectral sequence, *
*and
hence even (resp. odd) degree in K*(ffi3F). The calculation of this spectral se*
*quence
requires some preliminary computation regarding the algebra structure of K*(ffi*
*2F),
and this requires major input from [44].
Now we calculate the bar spectral sequence associated to the principal fibrati*
*on
ffi3F ! * ! ffi2F:
This spectral sequence, with E2 T orP[y1;y2]E[z](K*; K*), collapses to yield an
isomorphism of Hopf algebras
K*(ffi2F) E[a1; a2] .[b]; (3.5)
where . denotes the divided polynomial algebraPover Zp. The coproduct has a1, a*
*2,
and fl1(b) as the primitives, and (fli(b)) = flj(b) flij(b).
12 Donald M. Davis Chapter 1
Dualizing eq. (3.5) yields an isomorphism of algebras
K*(ffi2F) E[ff1; ff2] P [fi];
with ffi odd and fi even. This matches nicely with the following result fro*
*m [52,
3.8].
For any prime p, there is an isomorphism of algebras
K*(ffi2S2n+1) E[u0; u1] P [w];
where E and P denote exterior and polynomial algebras over K*.
We will use the AtiyahHirzebruch spectral sequence to show that the map
2j
ffi2S2n+1 ffi!ffi2F
of Theorem 3.5 sends the generators of the isomorphic K*()algebras across. Th*
*is
will imply the second half of Theorem 3.5.
In [52], it is shown how the generators u0, w, and u1 of K*(ffi2S2n+1) arise i*
*n the
AtiyahHirzebruch spectral sequence whose E2term is H*(ffi2S2n+1; K*). Indeed,
they arise from the bottom three cohomology classes, of grading 2n  1, 2pn  2,
and 2pn  1, respectively. The map ffi2j induces an isomorphism in Hi(; Zp) for
i < 2pn1+min(q; 2n2), and so it maps onto the three generators of K*(ffi2S2n+*
*1).
4.Jhomology
In this section, we show how to compute J*(Bqn). When combined with Theorems
3.1 and 5.1, this gives an explicit computation of v11ss*(S2n+1), which we sta*
*te at
the end of this section as Theorem 4.2. This will be extremely important in our
calculation of v11ss*(Y ) for other spaces Y . We begin with the case p odd, w*
*here
the results are somewhat simpler to state. Historically it worked in the other *
*order,
with Mahowald's 2primary results in [40] preceding Thompson's oddprimary work
in [56].
Let p be an odd prime. We follow quite closely the exposition in [24] and [56,*
* x3].
The spectrum bu(p)splits as a wedge of spectra 22i` satisfying H*(`; Zp) A==E,
where A is the mod p Steenrod algebra, and E is the exterior subalgebra generat*
*ed
by Q0 = fi and Q1 = P 1fi  fiP 1. The spectrum ` is sometimes written BP <1>.
Then `* = ss*(`) is calculated from the Adams spectral sequence (ASS) with
Es;t2 Exts;tA(H*`; Zp) Exts;tE(Zp; Zp) Zp[a0; a1]; (4.1)
where ai has bigrading (1; iq + 1). Here we have used the changeofrings theor*
*em
in the middle step. There are no possible differentials in the spectral sequenc*
*e, and
since multiplication by a0 corresponds to multiplication by p in homotopy, we f*
*ind
Section 4 Computing periodic homotopy groups 13
that ss*(`) is a polynomial algebra over Z(p)on a class of grading q. Using the*
* ring
structure of `, one easily sees that there is a cofibration
2q ` ! ` ! HZ(p):
Let k be a (p  1)st root of unity mod p but not mod p2, and let k denote the
Adams operation. The map k1 : ` ! ` lifts to a map : ` !2q `. The connective
Jspectrum, J, is defined to be the fiber of . The homotopy exact sequence of
easily implies that
( Z
(p) if i = 0
ssi(J) Z=pp(j)+1 if i = qj  1 with j > 0
0 otherwise.
The image of the classical Jhomomorphism is mapped isomorphically onto these
groups by the map S0 ! J; this is the reason for the name of the spectrum.
We now proceed toward the calculation of J*(Bqn). We let B be the plocalizati*
*on
of the suspension spectrum of B 2p. Then, with coefficients always in Zp, the o*
*nly
nonzero groups Hi(B) occur when i 0 or 1 mod q, and i > 0. These groups are
cyclic of order p with generator xisatisfying Q0xaq1= xaqand Q1xaq1= x(a+1)q.
We will work with the skeleta Bqn and the quotients Bq(n+1)1= B=Bqn; these
are suspension spectra of the spaces which appeared in Theorem 3.3.
There is a plocal map B ! S0 constructed by Kahn and Priddy. If R de
notes its cofiber, there is a filtration of the Emodule H*R with subquotients
2qiE==E0 for i 0. Here E0 is the exterior subalgebra of E generated by Q0. Sin*
*ce
ExtE(E==E0) ExtE0(Zp) Zp[a0], we find that ExtE(H*R) has a "spike," con
sisting of the powers of a0, for each nonnegative value of t  s which is a mul*
*tiple
of q. Here we have begun a practice of omitting Zp from the second variable of
ExtB(; ) if B is a subalgebra of the Steenrod algebra. The action of ExtE(Zp)
on ExtE(H*R) has a1 always acting nontrivially.
We draw ASS pictures with coordinates (t  s; s), so that horizontal component
refers to homotopy group. A chart for the ASS of R ^ ` is given in fig. 1.
The short exact sequence
0 ! H*(2 B) ! H*(R) ! H*(S0) ! 0
induces an exact sequence
! Exts;tE(H*S0) i*!Exts;tE(H*R) ! Exts;tE(H*(2 B)) ! Exts+1;tE(H*S0) ! :
(4.2)
These morphisms are ExtE(Zp)module maps, and the action of a1 implies that i*
14 Donald M. Davis Chapter 1
Figure 1. ASS for `*(R)
6 6 6
  
  
  
  
r r r
  
r r r. . .
  
_____________________________________________rrr
t  s =0 q 2q
is injective. Thus there are elements xiq12 Ext0;iq1E(H*B) for i > 0 such that
ae
Exts;tE(H*B) = Zp0 ifotthse=riqwi1,sie>,0, 0 s < i
with generators as0xiq1. Hence
ae (i+1)=q
`i(B) Z=p if i 1 mod q, and i > 0
0 otherwise.
There is an isomorphism of Emodules H*(Bq(n+1)1) 2qnH*(B), and so
`*(Bq(n+1)1) `*(2qnB):
The Ext calculation easily implies that the morphism `*(B) ! `*(Bq(n+1)1) in
duced by the collapse map is surjective, and so the exact sequence of the cofib*
*ration
Bqn ! B ! Bq(n+1)1implies that
ae min((i+1)=q;n)
`i(Bqn) Z=p if i 1 mod q, and i > 0
0 otherwise.
The ASS chart for B4q is illustrated in fig. 2.
The map S0 ! R implies that * : `qj(X) ! `(q1)j(X) is multiplication by the
same number for X = R as it was for X = S0. Thus it is multiplication by pp(j)+*
*1.
Now the map R !2 B implies that * : `qj1(B) ! `(q1)j1(B) is multiplication
by pp(j)+1. The maps Bqn ! B ! Bq(n+1)1imply that the same is true in Bqn
and Bq(n+1)1. We obtain
8 min(n; (j)+1)
> 0
Ji(Bqn) > Z=pmin(n1p; (jif)i)= jq  2, j > n (4.3)
: Z=p p if i = jq  2, 0 < j n
0 otherwise.
Section 4 Computing periodic homotopy groups 15
Figure 2. ASS for `*(B4q) for t  s < 6q
r

r r
 
r r r
  
r r r r
  
r r r r
  
____________________________________________________________rrrr
q  1 2q  1 3q  1 4q  1 5q  1 6q  1
This is illustrated in fig. 3, which is not quite an ASS chart. It is a combina*
*tion of
the charts for `*(Bqn) and `*q+1(Bqn) and the homomorphism * between them,
which is represented by lines of negative slope. The exact sequence
0 ! coker(*+1) ! J*(Bqn) ! ker(*) ! 0
says that elements which are not involved in these boundary morphisms com
prise J*(Bqn). There are several reasons for our having elevated the filtration*
*s of
`*q+1(Bqn) by 1 in this chart. One is that it makes all the boundary morphisms*
* go
up, so that it looks like an ASS chart. Another is that (by [40]) there is a re*
*solution
of Bqn^J (which is not an Adams resolution) for which the homotopy exact couple
is depicted by this chart. A third is that if J1 is defined to be the fiber of *
*J ! HZ2,
then the ASS chart for Bqn^ J1 will agree with this chart in filtration greater*
* than
1. See [13, x6] for an elaboration on this.
Figure 3. Beginning of chart for J*(B4q)
r r
A 
r r r r
@  AA
r r r@r rAr
@ @  A 
r r rr@ r@r rAr
A @ @ 
r r rr rr@ r@r
@  A @ 
____________________________________________________________rrrr@A@
q  1 2q  1 3q  1 4q  1 5q  1 6q  1
If X is a space or spectrum, then v11Ji(X) is defined analogously to Definiti*
*on 2.2
to be dirlime;k[Mi+1+ks(e)(pe); X ^J]. Since J is a stable object, we can Sdua*
*lize the
Moore space, obtaining dirlime;kJi+1+ks(e)(X ^ M(pe)), where the Moore spectrum
M(n) has cells of degree 0 and 1. The "+1" in this Jgroup is present due to the
maps M(pe) ! M(pe+1) having degree 1 on the 1cell.
16 Donald M. Davis Chapter 1
One can often compute v11J*(X) directly from J*(X) without having to worry
about the "^M(pe)". This can be done by extending the periodic behavior which
occurs in positive filtration down into negative filtrations and negative stems*
*. For
example (cf. fig. 3), a chart for v11J*(Bnq) has, for all integers a, adjacent*
* towers
of height n in aq  2 and aq  1 with d(a)+1differential. If (a) + 1 n, then *
*the
differential is 0. This interpretation of v11J*() can be justified using the *
*following
result.
Let p be an odd prime, and let KU be the spectrum for periodic Ktheory localiz*
*ed
at p. Let k be ak(p11)st root of unity mod p but not mod p2, and let Ad denot*
*e the
fiber of KU  ! KU. Then v11J*() Ad*().
This follows from the fact that if v11`*() is defined as dirlime;k`i+1+ks(e)(*
*X ^
M(pe)), then v11`*() KU*(), which is a consequence of the fact that A ^ =
v ^ 1M :2q M ! M ^ `, where A :2q M ! M and v : Sq ! `.
When p = 2, the results are a bit messier to state and picture. If bsp denotes*
* the
2local connected ffispectrum whose (8k)th space is BSp[8k], then 24 bsp ' bo[*
*4],
the spectrum formed from bo by killing ssi() for i < 4. The map 3  1 : bo ! *
*bo
lifts to a map : bo !24 bsp, and J is defined to be the fiber of .
Let A1 denote the subalgebra of the mod 2 Steenrod algebra A generated by Sq1
and Sq2. Then H*bo A==A1 and H*bsp A A1 N, where N = <1; Sq2; Sq3>.
Hence, using the change of rings theorem, the E2term of the ASS converging to
ss*(bo) is ExtA1(Z2), while that for bsp is ExtA1(N). These are easily computed
to begin as in fig. 4, with each chart acted on freely by an element in (t  s;*
* s) =
(8; 4). Positively sloping diagonal lines indicate the action of h1 2 Ext1;2A1(*
*Z2). It
corresponds to the Hopf map j in homotopy.
Figure 4. Part of ASS for bo and bsp
bo bsp 
6 6
 
 
r 
6  6 
 r  r
   
r r r r r
  
r r r rr
  
r r r r
 
_________________r _______________________r
0 4 0 4
Section 4 Computing periodic homotopy groups 17
There are no possible differentials in these ASS's, and so we obtain
( Z
(2) if i 0 mod 4, i 0
ssi(bo) Z2 if i 1; 2 mod 8, i > 0
0 otherwise,
in accordance with Bott periodicity.
From Adams' work, we have * : ss4j(bo) ! ss4j(24 bsp) hitting all multiples of
22(j)+3, while * is 0 on the Z2's. This yields ssi(J) = 0 if i < 0, while for i*
* 0
8
>>>Z(2) if i = 0
< Z=22(i)+1 if i 3 mod 4
ssi(J) > Z2 if i 0; 2 mod 8; i > 0 (4.4)
>>:Z2 Z2 if i 1 mod 8; i > 1
0 if i 4; 5; 6 mod 8.
From Adams' work and the confirmation of the Adams Conjecture, it is known that
ss*(S0) ! ss*(J) sends the image of the classical Jhomomorphism plus Adams'
elements j and jj isomorphically onto ss*(J). A chart for ssi(J) with i 18 is
given in fig. 5. Here the elements coming from bo are indicated by o's, while t*
*hose
from 24 bsp are indicated by O's.
Figure 5. 2primary ssi(J), i 18
6
b
BB
6 bBB
 b bBB r
 6b b@6r@ b6rrBB
6 6  B
 b bA b@r@ brBb
 6 A 6 
 b@r@ bArA b@r@ bb
  A  pp
 b@r@ bArA r b b
  A 
r b@r@ bArAr b
  A 
r b@r@ bAr b
 
r b@r@ b b
 pp
r r b b

r r b

__________________________________________________r
0 3 7 11 15
The first dotted jextension can be deduced from the fact that * : H4(24 bsp) !
H4(bo) hits Sq4, together with the relation j3 = 4. This jaction is then pushed
18 Donald M. Davis Chapter 1
along by periodicity. Another argument which is frequently useful for deducing
jextensions such as this involves Toda brackets. The generator of ss8i+4(bo) is
obtained from the element ff 2 ss8i+2(bo) as . Clearly ff pulls back *
*to ss*(J),
and if ffj were 0 here, then the bracket could also be formed in J. However, the
boundary morphism on ss8i+4(bo) implies that this bracket cannot be formed, and
so ffj must be nonzero in J. Moreover, it must be the element fi such that 2fi *
*is
ffi().
We will rename Bqn as P 2nwhen p = 2, with P denoting the suspension spectrum
of RP 1. As in the odd primary case, there is a map : P ! S0 with nontriv
ial cohomology operations in its mapping cone R. This 2primary map can be
viewed more geometrically than its oddprimary analogue, as an amalgamation of
composites
P n! SO(n + 1) J!ffinSn:
With A0 denoting the exterior subalgebra of A generated by Sq1, H*R can
be filtered as an A1module with subquotients 24i A1==A0 for i 0, and so
ExtA(H*(R ^ bo)) consists of h0spikes rising from each position (t  s; s) = (*
*4i; 0)
for i 0. Here h0 is the element of Ext1;1A(Z2) or Ext1;1A1(Z2) corresponding t*
*o Sq1
and to multiplication by 2 in homotopy. Also, we begin a practice of using with*
*out
comment the relation
ExtA(H*(X ^ bo)) ExtA(H*X A==A1) ExtA1(H*X):
Thus the nonzero groups boi(R) occur only when i 0 mod 4 and i 0, and
these groups are Z(2). We also need
( Z
(2) if i 0 mod 4, i 0
bspi(R) boi(R ^ (S0 [j e2[2e3)) Z2 if i 2 mod 4, i > 0
0 otherwise.
The Z2's are obtained from the exact sequence in bo*() associated to the cofib*
*ra
tion
R ^ S2 ! R ^ (S2 [2e3) ! R ^ S3 2!:
Analogous to (4.2) is an exact sequence which allows us to compute ExtA1(H*P )
from ExtA1(H*S0) and ExtA1(H*R). This is most easily seen in the chart of fig. *
*6,
in which o's are from ExtA1(H*S0), and O's are from ExtA1(H*R).
The groups are read off from this as
8 4j+3
> Z=2 i = 8j  1, j > 0
: Z2 i = 8j + 1 or 8j + 2, j 0
0 otherwise.
Section 4 Computing periodic homotopy groups 19
Figure 6. ExtA1(H*P), from ExtA1(H*S0) and ExtA1(H*R), t  s < 15
6
b
@r@6
6 b 
b br@@
b66 b@r6@ ppb
6   r
b b@r@ b@r@r b
@ 6 @  @ 
b@r b@r b@r b
@  @ 
b@r b@r b b
@  pp
b@r r b b b
@ 
_______________________________________br@rbbb
@ 
@ r 3 7 11
Next on the agenda is bsp*(P ), which is computed from bsp*(S0) (in o) and
bsp*(R) (in O) as in fig. 7.
Figure 7. bsp*(P), from bsp*(S0) and bsp*(R), * 11
6 6
 
66 b 
  b@r@
6 b  
  br@@ b@r@
6 6 b  
b  br@@ br@@ b
@  @  pp
b@r br@ r b b
@  @ 
b@r br@r b b
@  @ 
_______________________________________br@bbbbr@bb
@ 
@ r 1 3 7 11
Note that in positive filtration bsp*(P ) looks like bo*(24 P ) pushed up by 1
filtration. The explanation for this is the short exact sequence of A1modules
0 !25 Z2 ! A1==A0 ! N ! 0; (4.5)
20 Donald M. Davis Chapter 1
where N = <1; Sq2; Sq3>, as before. If this is tensored with any A0free A1mod*
*ule
M, such as P , then the exact ExtA1sequence reduces to isomorphisms
Exts1;tA1(25 M) ! Exts;tA1(N M)
when s > 1. When M = P , an iso is also obtained when s = 1.
The isomorphism of A1modules H*(P4n+1) H*(24n P ) allows one to imme
diately obtain bo*(P4n+1) and bsp*(P4n+1) from the above calculations. One way
of determining bo*(P4n+3) and bsp*(P4n+3) is from the short exact sequence of
A1modules
0 ! H*(24n+4Z2) ! H*(P4n+3) ! H*(24n+3R) ! 0:
This yields as bsp*(P4n+3) a chart which begins as in fig. 8, while bo*(P4n+7)
bo*(24 P4n+3) is obtained from this chart by deleting all classes in filtration*
* 0.
Figure 8. bsp*(P4n+3), * 15
r

r r
 
r r r
r r r
 
r r r
  
_______________________________________rrrrrrr
4n+ 3 7 11 15
Next we compute bo*(P 2m) and bsp*(P 2m) using the exact ExtA1sequence cor
responding to the short exact sequence
0 ! H*(P2m+1) ! H*(P ) ! H*(P 2m) ! 0:
For example, this yields the calculation of bo*(P 8n) indicated in fig. 9, wher*
*e bo*(P )
is in o's, while bo*(P8n+1) is in O's.
Next we form J*(P 2m) from bo*(P 2m) and (23 bsp)*(P 2m), with filtrations of *
*the
latter pushed up by 1, similarly to the odd primary case. The boundary morphism
bo4i1(P 2m) ! (24 bsp)4i1(P 2m) is pictured by a differential in the chart, a*
*nd, for
the same reason as in the oddprimary case, its value is the same as in bo4i1(*
*S0) !
(24 bsp)4i1(S0), namely a nonzero d(i)+1wherever possible. If m k, then the
charts for J*(P 2m) and J*(P 2k) are isomorphic through dimension 2k  1. This *
*is
illustrated in fig. 10 for k = 8.
For * 2m  1, the form of the chart for J*(P 2m) depends upon the mod 4
value of m. The last of the filtration1 Z2's occurs in * = 2m or 2m + 2. The c*
*hart
near * = 2m is indicated in fig. 11. Note how the bsppart is like the bopart *
*shifted
Section 4 Computing periodic homotopy groups 21
Figure 9. bo*(P8n), from bo*(P) and bo*(P8n+1)


 
 
 
r  
 
r  
  
  
  
  
  
  
  
  
  
  
  
r   
   
r r   r
   
r r  r b r
   @ 
r r  b r b@r
  p pp  @  @ 
r r r  b b@r b@r
    @  @ 
r r r r  b b@r b@r
   pp @  @ 
_______________________________________________________________________rrrr*
*@@rr
1 3 7 11 8n  1 8n + 7
Figure 10. J*(P2m) in * 15, provided m 8
r
r r
 
r r b r
r b r brA
@  A
r b b@r brAA
 @  A
r r b b@r brAA
  @  A
r r b b r b b@r b bbrAA
  @  A
________________________________________rrrr@rA
1 3 7 11 15
22 Donald M. Davis Chapter 1
one unit left and two units down. Differentials dr with r > 1 are omitted from *
*this
chart; they occur on the towers in 8n  1 and 8n + 7.
Figure 11. J*(P2m) where it starts to ascend
r r
P 8n P 8n+2 P 8n+4 r brrr P 8n+6brrrr

r r b br@ r b br@
r r r r rbbr@ rb br@
r r r r b r r r b r br@ b r br@
4n _ _ r b r r b r br rb r brbr@ br br@
r b br@ b b@r br bbr@ br br@ br @pp
rb br@ b b@r pp b br@ br @pp br p p
b r br@ b@r p p br@ pp p p r pp @br
pp p@p @pp br @pp p p brr@ p p br@
p p p p p p br p p br brb@ br br@
br br@ br@rb @br br bb br r b
br r b br@r b br@r b b brr b
brrb b b brb@ b br@br b brb
_____________r_______________ ______________________________________r@
8n 61 8n +63 8n +63 8n +611 8n +67
To obtain v11J*(P 2m) from J*(P 2m), one removes the filtration1 Z2's, and
extends into negative filtration the periodic behavior which is present in the *
*towers
to the right of dimension 2m. The justification for this is similar to that in *
*the odd
primary case, namely Proposition 4.1. For example, if i is any integer, v11J*(*
*P 8n+4)
for 8i + 6 * 8i + 13 looks like the portion of fig. 11 for P 8n+4between 8n +*
* 6
and 8n + 13, with a d(4i+4)differential on the tower in 8i + 7. One might find
it easier to compute v11J*(P 2m) directly without bothering to first compute t*
*he
nonperiodic J; however, one sometimes needs the nonperiodic Jgroups.
We combine the results of this section with Theorem 3.1 and Theorem 5.1 to
obtain the following extremely important result, Theorem 4.2.
If p is odd, then
ae min(n;p(a)+1)
v11ss2n+1+i(S2n+1; p) Z=p if i = qa  2 or qa  1
0 if i 6 1 or 2 mod q.
If n 1 or 2 mod 4, then
8
> Z2 Z2 min(3;n+1) if i 1; 4 mod 8
: Z2 Z=2 if i 2; 3 mod 8
Z=2min(n1;2(j)+4)if i = 8j  2 or 8j  1.
Section 5 Computing periodic homotopy groups 23
If n 0 or 3 mod 4, then
8
>>>Z2 Z2 Z2 if i 0; 1 mod 8
>> Z8 if i 3 mod 8
>>>0 if i 4; 5 mod 8
>:Z=2min(n;2(j)+4) if i = 8j  2
Z2 Z=2min(n;2(j)+4)if i = 8j  1.
5.The v1periodic homotopy groups of spectra
In this section, we sketch three proofs of the following central result.
If X is a spectrum, then v11ss*(X) v11J*(X).
This result was first stated, at least for mod p v1periodic homotopy groups, i*
*n [56].
Theorem 5.1 is a consequence of the following result, which is the special case
where X is the mod p Moore spectrum M = S0 [p e1.
Let v11M denote the mapping telescope of
M !2s M !22sM ! . .;.
where s = 8 if p = 2 and s = q if p is odd, and the maps are all suspensions of*
* an
Adams map A. Then the Hurewicz morphism
ss*(v11M) ! J*(v11M)
is an isomorphism.
This theorem implies that for any spectrum X, the map
X ^ v11M ! X ^ v11M ^ J
is an equivalence, which, after dualizing the Moore spectra, implies that Theor*
*em
5.1 is true with mod p coefficients. The general case of the theorem then follo*
*ws
from Lemma 3.2.
As an aside, we note that these results are equivalent to the validity of Rave*
*nel's
Telescope Conjecture ([53]) when n = 1. This result, for which the analogue with
n = 2 has been shown to be false, can be stated in the following way.
The v1telescope equals the K*localization, i.e., v11M = MK .
Proof. Since the Adams maps induce isomorphisms in K*(), the inclusion M !
v11M is a K*equivalence. Since v11M ' v11M ^ J ' M ^ v11J, and, similarly
to Proposition 4.1, there is a cofibration
v11J ! KO ! KO;
24 Donald M. Davis Chapter 1
it follows readily that v11M is K*local. 
The remainder of this section is concerned with proofs of Theorem 5.2. Three
distinct proofs have been given, although each is too complicated to present in
detail here. We sketch each, relegating details to the original papers.
The first proof, when p = 2, was given by Mahowald in [40], although he was
offering sketches of this proof as early as 1970. The oddprimary analogue was *
*given
in [24]. A sketch of Mahowald's proof, involving boresolutions, follows.
Using selfduality of M, it suffices to show that
dirlimi[2k+8iM; S0] ! dirlimi[2k+8iM; J] (5.1)
is an isomorphism. The target groups are easily determined by the methods of the
preceding section to be given by two sequences of "lightning flashes" as in fig*
*. 12.
It is easily seen, for example from the upper edge of Adams spectral sequence, *
*that
these elements all come from actual stable homotopy classes, i.e., the morphism
(5.1) is surjective.
Figure 12. dirlimi[2k+8iM; J]
r
p
r r p p

r r r
r r r

r r
r
k 2 0 2 3 mod 8
Thefinjectivity of the morphism (5.1) will be proved by showing thatiif 2kf
M ! S0 ! J is trivial, then for i sufficiently large, 2k+8iM A! 2k M ! S0
is trivial._This will be done using boresolutions.
Let bo denote the cofiber of the inclusion S0 ! bo. There is a tower of
(co)fibrations
__ __ __
S0 21 bo 22 bo^ bo
# #_ __# __
bo 21 bo^ bo 22 bo^ bo^ bo
The homotopy exact couple of this tower gives the boASS for S0. It was proved
in [25], following [41], that the E2term of this spectral sequence vanishes ab*
*ove a
Section 5 Computing periodic homotopy groups 25
line_of slope 1/5. That is, Es;t2(S0) = 0 if s > 1_5(t  s) + 3. One can show t*
*hat
bo^ bo '24 bsp _ W , where W can be written explicitly, and the map
__ 3
21 bo ffi!21 bo^ bo !2 bsp
may be used as the map whose fiber is J. Here ffi induces the lowest d1 in the
boASS, and the_second_map collapses W .
Let Es = (21 bo)^s, the sth stage of the tower. By explicit calculation it can
be shown that, if s > 1 or if s = 1 and the map is detected entirely in the W *
*part,
a map X ! Es of Adams (HZ=2) filtration greater than 1 can be varied so that
its projection to Es1 is unchanged, while the new map lifts to Es+1. Originall*
*y it
was thought that this was true for maps of Adams filtration greater than 0, but*
* a
complication was noted inf[27].
Now suppose that 2k M ! S0 ! J is trivial. Then f lifts to a map into E1
whose projection into 23 bsp is trivial. The Adams map A can be written as the
composite of two maps, each of HZ=2Adams filtration greater than 1. Thus, by
the result of the previous paragraph, f OAilifts to E2i+1. If i is chosen large*
* enough
that 2i + 1 > 1_5(k + 8i + 1) + 3, then this map 2k+8iM ! S0 will have bofiltr*
*ation
so large that all such maps are trivial by the vanishing line result, completin*
*g the
proof.
The first proof of Theorem 5.2 for p odd was given by Haynes Miller. A proof
analogous to his for p = 2 has not been achieved; [32] was a step in that direc*
*tion.
Miller's work did not involve the spectrum J. Instead, in [46], he defined a lo*
*calized
ASS for M, and computed its E2term. Then, in [47], using a clever comparison
with the BP based Novikov spectral sequence, he computed the differentials in *
*the
ASS, obtaining the following result.
If p is odd, then ss*(v11M) is free over Zp[v11] on two classes, namely [S0 ,!
M ! v11M] and [Sq1 ff1!M ! v11M].
The methods of Section 4 show easily that these map isomorphically to v11J*(M).
We provide a little more detail about Miller's calculations. In [46] he obtain*
*ed as
an E2term for the localized ASS
v11E[hi;0: i 1] P [bi;0: i 1]; (5.2)
wherePhi;0corresponds to [i] and has bigrading (1; 2(pi1)), while bi;0correspo*
*nds
to 1_ppj[jipji] and has bigrading (2; 2p(pi1)). The first step in obtainin*
*g this
is to use a changeofrings theorem to write the E2term as v11CotorA(1)*(Zp; *
*Zp),
where A(1)* is the quotient A*=(o0). This is then shown to be isomorphic to
Zp[v11] CotorP(1)(Zp; Zp);
where P (1) = Zp[1; 2; : :]:=(p1; p2; : :):, and this yields eq. (5.2).
In [47], the differential d2(hi;0) = v1bi1;0is established in the localized A*
*SS.
This leaves Zp[v11] E[h1;0] as E3 = E1 , and this is easily translated into Pr*
*opo
26 Donald M. Davis Chapter 1
sition 5.4. Miller first established this differential in an algebraic spectral*
* sequence
converging to the E2term of the BP based Novikov spectral sequence, and then
showed that this implies the desired differential in the ASS by a comparison th*
*eo
rem.
Somewhat later, Crabb and Knapp ([21]) gave a proof of Theorem 5.1 for finite
spectra X which was much less computational than those just discussed. Their
proof utilized the solution of the Adams conjecture, and some refinements there*
*of.
They let Ad*() be the generalized cohomology theory corresponding to the fiber
of k  1 : KO ! KO. By Proposition 4.1, this is just our v11J*. They prove
the following result about stable cohomotopy, which by Sduality is equivalent *
*to
Theorem 5.1 for finite spectra.
If X is a finite spectrum, then the Hurewicz morphism
v11ss*s(X; Z=pe) h! Ad*(X; Z=pe)
is bijective.
Their main weapon is a result of May and Tornehave which says that if A*()
is the connective theory associated to Ad*(), and j is the morphism given by a
solution of the Adams conjecture, then the composite
A0(X) j!ss0s(X) h!A0(X)
is bijective for a connected space X. This is used to show that, for k sufficie*
*ntly
large, there is a stable Adams map 2ks(e)M(pe) Ae!M(pe) which is in the image
under j from Aks(e)(M(pe); Z=pe). This is then used to show that for any eleme*
*nt
x of ssns(X; Z=pe), for L sufficiently large, ALex is in the image of the morph*
*ism j,
and this easily implies injectivity in Theorem 5.5. Care is required throughout*
* in
distinguishing stable maps from actual maps.
6.The v1periodic unstable Novikov spectral sequence for spheres
In this section, we review the basic properties of the unstable Novikov spectral
sequence (UNSS) based on the BrownPeterson spectrum BP , and sketch the de
termination of the 1 and 2lines of this spectral sequence when applied to S2n*
*+1.
Then we show how the v1periodic UNSS is defined, and compute it completely for
S2n+1.
The spectrum BP associated to the prime p is a commutative ring spectrum
satisfying BP* = ss*(BP ) = Z(p)[v1; v2; : :]:and BP*(BP ) = BP*[h1; h2; : :]:,*
* with
vi = hi = 2pi2. The generators viare those of Hazewinkel, while hiis conju*
*gate
to Quillen's generator ti. We shall often abbreviate BP*BP as ..
We will make frequent use of the right unit jR : BP* ! BP*BP .
Section 6 Computing periodic homotopy groups 27
jR(v1) = v1 ph1, and
Xp
jR(v2) = v2 ph2+ (pp1 1)hp1v1+ (p + 1)vp1h1+ aivp+1i1pihi1;
i=2
where ai2 Z.
In writing hp1v1 here, we have begun the practice of writing jR(v)h as hv. Thus
hp1v1 6= v1hp1. Proposition 6.1 is easily derived from formulas relating vi to *
*mi,
and for jR(mi). See [14, 2.6], where the following formula for the comultiplica*
*tion
#: BP*BP ! BP*BP BP*BP is also computed. All tensor products in this and
subsequent sections are over BP*.
# (h1) = h1 1 + 1 h1, and
p1X
# (h2) = h2 1 + 1 h2+ 1_ppihi1 hpi1v1+ hp1 h1:
i=1
Let BP n denote the nth space in the ffispectrum for BP . If X is a space, th*
*en
a space BP (X) is defined as limnffin(BP n ^ X). Define D1(X) to be the fiber
of the unit map X ! BP (X), and inductively define Ds(X) to be the fiber of
Ds1(X) ! Ds1(BP (X)). This gives rise to a tower of fibrations
. .!.D2(X) ! D1(X) ! X:
The homotopy exact couple of this tower is the UNSS of X; if X is simply connec*
*ted,
it converges to the localization at p of ss*(X).
In general, computing this spectral sequence can be extremely difficult, but if
BP*X is free as a BP*module, and cofree as a coalgebra, then it becomes somewh*
*at
tractable. Indeed, in such a case
Es;t2(X) ExtsU(At; P (BP*X)); (6.1)
where At denotes a free BP*module on a generator of degree t, P () denotes the
primitives in a coalgebra, and U denotes the category of unstable .comodules. *
*We
sketch a definition of the category U and the proof of eq. (6.1), referring the*
* reader
to [6, p. 744] or [8, x7] for more details. If M is a free BP*module, then U(M)
is defined to be the BP*submodule of . M spanned by all elements of the form
hI m satisfying the unstable condition
2(i1+ i2+ . .).< m; (6.2)
where hI = hi11hi22...I.f M is not BP*free, then U(M) is defined as coker(U(F1*
*) !
U(F0)), where F0 and F1 are free BP*modules with M = coker(F1 ! F0). We
define Us(M) by iterating U(). The category U consists of BP*modules equipped
28 Donald M. Davis Chapter 1
with morphisms M ! U(M), U(M) ffi!U2(M), and U(M) ffl!M satisfying
certain properties. The unstable condition (6.2) is analogous to the one for un*
*stable
right modules over the Steenrod algebra, but its proof relies on deep work of R*
*avenel
and Wilson in [54].
The category U is abelian. We abbreviate ExtsU(At; N) to Exts;tU(N). These
groups_may be calculated_as the homology groups of the_unstable_cobar complex
C*;*(N), defined by Cs;t(N) = Us(N)t, with boundary Cs d!C s+1defined by
d[fl1 . ..fls]m=[1fl1 . ..fls]m
X
+ (1)j[fl1 . ..fl0jfl00j . ..fls]m
X
+ (1)s+1[fl1 . ..flsfl0]m00;
P P
where # (flj) = fl0j fl00jand (m) = fl0 m00. __
We will use a reduced complex C*;*(N), which is chain equivalent to C*;*(N).
This is obtained from eU(N) = ker(U(N) ffl!N) and its iterates eUsby Cs;t(N) =
eUs(N)t. Finally, we illustrate how d(v) = jR(v)  v for v 2 BP* comes into pla*
*y.
P
Suppose # (h) = h 1 + 1 h + h0 h00and (I) = 1 I, and let v; v02 BP*.
Then
X
d([vh]v0I)=[1vh]v0I  [vh1]v0I  [vh]v0I  [vh0h00]v0I + [vhv0]I
X
= [jR(v)  vh]v0I  [vh0h00]v0I  [vhjR(v0)  v0]I
We abbreviate C*;*(BP*(X)) to C*;*(X).
The first result about the UNSS, both historically and pedagogically, is the f*
*ol
lowing, which appeared in [8, 9.12]. We repeat their proof because it gives a g*
*ood
first example of working with the unstable cobar complex. Recall that q = 2(p *
* 1).
Let p be an odd prime. If k > 0, then
E1;2n+1+kq2(S2n+1) Z=pmin(n;p(k)+1):
If t 6 2n + 1 mod q, or if t < 2n + 1, then Es;t2(S2n+1) = 0.
Proof. Since vi and hi are divisible by q, all nonzero elements in BP*(S2n+*
*1)
have degree congruent to 2n + 1 mod q, and so the only possible nonzero elements
in Es;t2(S2n+1) occur when t 2n + 1 mod q, and t 2n + 1. There is an injective
chain map
C*;*(S2n1) ! C*;*+2(S2n+1)
defined by A2n17! A2n+1, corresponding to the double suspension homomor
phism of homotopy groups. Since the boundaries in C1(S2n1) are sent bijectively
to those in C1(S2n+1), the morphism E1;t2(S2n1) ! E1;t+22(S2n+1) is injective.
Section 6 Computing periodic homotopy groups 29
We quote a result, originally due to Novikov (but see [51, x5.3] for the proof*
*),
about the stable groups: if n is sufficiently large, then
E1;2n+1+kq2(S2n+1) Z=pp(k)+1
with generator d(vk1)2n+1=pp(k)+1. We will prove Theorem 6.3 by showing that if
n p(k) + 1, then d(vk1)=pn is defined on S2n+1, but not on S2n1.
We begin by observing
d(vk1)=pn=((jR(v1))k  vk1)=pn = ((v1 ph1)k  vk1)=pn
Xk
= (1)j kjpjnvkj1hj1: (6.3)
j=1
k k
Note that the coefficients jpjn have nonnegative powers of p, since p( j )+j
p(k) + 1 n for j 1. Now we work mod terms that are defined on S2n1.
This allows us to ignore terms in the sum (6.3) for j < n. For the other terms,
we write pjnhj1as (v1 jRv1)jnhn1, and note that when this is expanded by the
binomial theorem, all terms except vjn1hn1may be ignored, since (jRv1)ihn12n1=
hn1vi12n1 satisfies eq. (6.2) when i > 0. Thus the sum (6.3) reduces to
Xk n1X
(1)j kjvkn1hn1=  (1)j kjvkn1hn1;
j=n j=0
P k k k
since j=0(1)j j = 0. If j > 0 in the righthand sum, then j is divisible by
p, and then phn1can be written as v1hn11 hn11v1, so that the term is defined*
* on
S2n1. Thus, mod S2n1, (6.3) reduces to vkn1hn1. This class is not defined on
S2n1. 
If p = 2, a similar argument establishes the following result.
If p = 2, then, for u > 0,
8
><0 u odd
E1;2n+1+u2(S2n+1) > Z=2 2(u) = 1
: Z=4 u = 4
Z=2min(n;2(u)+1)u 0 mod 4, and u > 4.
If u = 2k in the three nonzero cases, then the generators are, respectively, d(*
*vk1)=2,
d(v21)=4, and d(vk1+ 22(k)+1vk31v2)=22(k)+2.
When p is odd, the element (d(vk1)=pj)2n+1 2 E1;2n+1+kq2(S2n+1) is denoted
ffk=j. If j = 1, this will frequently be shortened to ffk. We note the followin*
*g from
the proof of Theorem 6.3.
If n j, then ffk=j2n+1= vkj1hj12n+1 mod terms defined on S2j1.
30 Donald M. Davis Chapter 1
Next we cull from [5] information about unstable elements in E2;*2(S2n+1), whi*
*ch
form a subgroup which we shall denote by eE2;*2(S2n+1). By "unstable," we mean
an element in the kernel of the iterated suspension. The main theorem of [5] is*
* the
following.
Let p be an odd prime, and let t = p(a). Then
8
< Z=pn if n t + 1
eE2;qa+2n+12(S2n+1) Z=pt+1 if t + 1 n < a  t
: Z=pan if a  t  1 n < a.
2 2;qa+2n+1
The homomorphism eE2;qa+2n12(S2n1) 2! eE2 (S2n+1) is
( injective if n t + 1
.p if t + 1 < n < a  t
surjectiveif a  t n < a.
Let m = min(n; a  t  1). Then pj times the generator of eE2;qa+2n+12(S2n+1) is
h1 vam+j11hmj12n+1 mod terms defined on S2(mj)1.
This is illustrated in the chart below, where we list just leading terms, an el*
*ement
connected to one just below it by a vertical line is p times that element, and *
*el
ements at the same horizontal level are related by the iterated double suspensi*
*on
Section 6 Computing periodic homotopy groups 31
homomorphism. We omit the subscript from h1 and v1, and the .
__S3______S5_________________S2t+3______S2t+5___
hva2h hva2h . . . hva2h
 
hva3h2 . . . hva3h2 hva3h2
 
... .. .. ..
. . .
 
hvat2ht+1 hvat2ht+1

hvat3ht+2
...
_________________________________________________________
_S2(at)1__S2(at)+1______________S2a1__
...
hv2tha2t1

hv2t1ha2thv2t1ha2t
 
... .. .. ..
. . .
hvthat1 hvthat1 . . . hvthat1
The proof of Theorem 6.6 requires results about Hopf invariants which we will
address shortly. We begin by describing a plausibility argument for it using on*
*ly ele
mentary ideas about the unstable cobar complex. We continue to omit the subscri*
*pt
of h1 and v1.
(i)The lead term h van+j1hnj does not pull back to S2(nj)1because
hnj2(nj)1does not satisfy the unstable condition.
(ii)By Proposition 6.1, ph = v  jR(v), a fact which we will begin to use
frequently. It implies that
p . h van+j1hnj = h van+jhnj1 h van+j1hnj1v:
The second term desuspends below S2(nj)1, and the first term is the next term
up the unstable tower.
(iii)We show that 22 applied to the element of order p on S2n+1 is a boundary
when n t + 1. To do this, we give a more precise description of this element
of order p as d(van1hn+1)2n+1. This clearly double suspends to the boundary
32 Donald M. Davis Chapter 1
d(van1hn+12n+3). Note how we had to wait until S2n+3 in order to put the in
side the d(), since hn+12n+1does not satisfy the unstable condition. It remain*
*s to
show that the lead term is correct, which is the content of the following propo*
*sition.
If t = p(a), then
d(van1hn+1) h va+tn1hnt
mod terms defined on S2(nt1)+1.
Proof. Replacing v by ph + jR(v) implies
van1hn+1 pan1ha (mod S2n+1):
Since boundaries on S2n+1 desuspend to S2(nt)1, we obtain
d(van1hn+1) pan1d(ha)
P a
mod the indeterminacy stated in the proposition. Now d(ha) = j hj haj,
and, since p( aj) t + 2  j for j > 1, we find
pan1d(ha) span1+th ha1 sh va+tn1hnt
mod terms defined on S2(nt1)+1. Here a = sptwith s not a multiple of p, and we
have freely replaced ph by v  jR(v). 
The main detail in the proof of Theorem 6.6 which is lacking in the plausibili*
*ty
argument above is an argument for why these are the only unstable elements on
the 2line. For this, we need the following result, which will also be useful i*
*n other
contexts.
i. There is an unstable .comodule W (n) and an exact sequence
P2!Es;t1 2n1 22 s;t+1 2n+1 H2 s1;t1 P2 s+1;t1 2n1
2 (S ) ! E2 (S ) ! ExtU (W (n)) ! E2 (S ):
(6.4)
ii. W (n) is a free module over BP*=p on classes x2pin1for i > 0 with coaction
X npi
(x2pkn1) = pkihki x2pin1:
i
iii. Ext0U(W (n)) Zp[v1]x2pn1. P
iv. If z 2 ExtU(W (n)) is represented by flk x2pkn1, then
X
P2(z) = d( flk pk1hnk) 2n1:
Section 6 Computing periodic homotopy groups 33
v. Every element x 2 Es2(S2n+1)Pmay be represented, mod terms which desuspend to
S2n1, by a cyclePof the form flk pk1hnk 2n+1, with flk 2 C*(A2pkn1 Zp).
Then H2(x) = flk x2pkn1.
Recall in (v.) that At is the free BP*module on a generator of degree t, and t*
*hat
C*() denotes the reduced unstable cobar complex.
We provide a bare outline of the proof, beginning with the construction from [*
*9].
There is a nonabelian category G of unstable .coalgebras, and a notion of ExtG
such that if BP*X is a free BP*module of finite type, then E2(X) (of the UNSS)
is ExtG(BP*X). Letting PG() denote the primitives in G, one finds that if M is
an object of G, then PG(M) is in the category U. By considering an appropriate
double complex, one can construct a composite functor spectral sequence converg*
*ing
to ExtG(M) with
Ep;q2 ExtpU(RqPG(M)):
Here RqPG denotes the qth right derived functor of PG. If M satisfies RqPGM = 0
for q > 1, then the spectral sequence has only two nonzero columns, and reduces
to an exact sequence
! ExtsU(PGM) ! ExtsG(M) ! Exts1U(R1PGM) ! : (6.5)
This will be the case when M = BP*(ffiS2n+1). One verifies that PGBP*(ffiS2n+1)
A2n, and that R1PGBP*(ffiS2n+1) is the comodule W (n) described in Theorem
6.8, and the exact sequence (6.5) reduces to the sequence (6.4) in this case. T*
*he
descriptions of the morphisms in (iv.) and (v.) are obtained in [3] using an al*
*ternate
construction of the exact sequence. Part (iii.) is proved in [5, p. 535] by stu*
*dying
explicit cycles.
Now we complete our observations on the proof of Theorem 6.6. If x is any
nonzero unstable element on the 2line, then there must be k and n so that
2 2;*+2
22kx 6= 0 2 ker(E2;*2(S2n1) 2!E2 (S2n+1)):
Then by Theorem 6.8, there must be an element ve1x2pn12 Ext0U(W (n)) such that
P2(ve1x2pn1) = d(ve1hn1)2n1=22kx:
But this is exactly the description of the unstable elements on the 2line whic*
*h was
given in the third part of our plausibility argument for Theorem 6.6.
When p = 2, the discussion above about the unstable elements on the 2line goes
through almost without change, as described in [10, pp. 482484]. The result is*
* that
if n a  2(a)  2, then
ae
eE2;2n+1+2a2(S2n+1) Z=2 if a is odd
Z=2min(2(a)+2;n)if a is even.
34 Donald M. Davis Chapter 1
The orders when a is even are 1 larger than in the odd primary case because v2 *
*can
be used to obtain 1 additional desuspension.
Now we construct the v1periodic UNSS, following [4] for the most part. In [7]*
* a
UNSS converging to map*(Y; X) was constructed. If Y = Mn(pe), the E2term is
the homology of C*(P (BP*(X))) Z=pe. The Adams map A induces
*
UNSS (map *(Mn(pe); X)) A!UNSS (map *(Mn+s(e)(pe); X));
where s(e) is as in eq. (2.1). By [35], on E2 this is just multiplication by a *
*power
of v1 after iterating sufficiently. As in our definition of v11ss*(), we defi*
*ne the
v1periodic UNSS of X by
v11E*;*r(X) = dirlime;kE*;*+1r(map *(Mks(e)(pe); X)):
The direct system over e utilizes the maps ae : Mn(pe+1) ! Mn(pe) used in Secti*
*on
1, and the shift of one dimension is done for the same reason as in our definit*
*ion of
v11ss*(). Similarly to Proposition 2.4, we have
On the category of spaces with Hspace exponents, there is a natural transforma*
*tion
from the UNSS to the v1periodic UNSS.
One of the main theorems of [4] is the following determination of the v1perio*
*dic
UNSS of S2n+1.
Let p be an odd prime. The v1periodic UNSS of S2n+1 collapses from E2 and
satisfies
ae min(n; (a)+1)
v11Es;2n+1+u2(S2n+1) Z=p p if s = 1 or 2, and u = qa
0 otherwise.
The morphism Es;t2(S2n+1) ! v11Es;t2(S2n+1) is an isomorphism if s = 1 and
t > 2n + 1, while for s = 2 it sends the unstable towers injectively, and bijec*
*tively
unless n a  p(a)  1, where t = 2n + 1 + qa.
Proof. It is readily verified that the elements on the 1 and 2lines described*
* in
Theorems 6.3 and 6.6 form v1periodic families. The main content of this theorem
is that there is nothing else which is v1periodic.
In order to prove this, we use a v1periodic version of the double suspension
sequence (6.4). It is proved in [4] that the morphisms of (6.4) behave nicely w*
*ith
respect to v1action, yielding an exact sequence
P2!v1 s;t1 2n1 22 1 s;t+1 2n+1 H2 1 s1;t1 P2
1 E2 (S ) ! v1 E2 (S ) ! v1 Ext U (W (n)) ! :
(6.6)
By Theorem 6.8(ii.), there is a spectral sequence converging to v11ExtU(W (n))
Section 6 Computing periodic homotopy groups 35
with
M s;t
Es;t1 v11ExtU(A2pin1 Zp): (6.7)
i1
There is a short exact sequence given by the universal coefficient theorem
0 ! v11Es;t2(Sn) Zp ! v11Exts;tU(An Zp) ! T or(v11Es1;t2(Sn); Zp) ! 0:
(6.8)
We will use eqs. (6.6), (6.7), and (6.8) to show inductively that there are no
unexpected elements in v11E2(S2n+1). But first we show how the known elements
fit into this framework. By (6.8), each summand in v11E12(Sn) gives two Zp's, *
*called
stable, in v11ExtU(An Zp), and similarly each summand in v11E22(Sn) gives two
summands in v11ExtU(An Zp), called unstable. We claim that
n
v11Exts;tU(W (n)) Zp if s = 0 or 1, and t 2n  1 mod(q6.9)
0 otherwise.
In [4, p. 57], the relationship between (6.9) and (6.7) is discussed: in the sp*
*ectral
sequence (6.7), stable classes from the (i+1)summand hit unstable classes from*
* the
isummand, yielding in v11E1 only the stable classes from the 1summand. These
are the elements described in eq. (6.9). On the other hand, in the exact.sequen*
*cep
(6.6), let t = 2n + kq with e = p(k). If e < n  1, then 22 is Z=pe ! Z=pe
when s = 2, yielding the elements in v11Exts;t1U(W (n)) for s = 0 and 1, whil*
*e if
e n  1, then for s = 1 and 2, 22 is Z=pn1 ,! Z=pn, also yielding elements in
v11Exts;t1U(W (n)) for s = 0 and 1.
It seems useful for this proof to have one more bit of input, namely the resul*
*t for
the v1periodic stable Novikov spectral sequence, which can be defined as a dir*
*ect
limit over e of stable Novikov spectral sequences of M(pe).
[13, x2] There is a v1periodic stable Novikov spectral sequence for S0, satisf*
*ying
v11Es;tr(S0) = dirlimnv11Es;t+2n+1r(S2n+1);
and
ae (t)+1
v11Es;t2(S0) Z=pp if s = 1 and t 0 mod q
0 otherwise.
We prove by induction on s that, for all n, v11Es2(S2n+1), v11Es+12(S2n+1), *
*and
v11ExtsU(W (n)) contain only the elements described in Theorem 6.10 and eq. (6*
*.9).
This is easily seen to be true when s = 0, where we know all groups completely.*
* As
sume it is true for all s < oe. Then v11Eoe2(S2n+1) contains no unexpected ele*
*ments
36 Donald M. Davis Chapter 1
because oe = (oe  1) + 1. If v11Eoe+12(S2n+1) contains an unexpected element,*
* then
some v11Eoe+12(S2L+1) must contain one in ker(22) by Theorem 6.11. Such an ele
ment must be P2(x), where x is an unexpected element of v11Extoe1U(W (n)), bu*
*t no
such element exists by our induction hypothesis. Finally, v11ExtoeU(W (n)) con*
*tains
no unexpected elements by (6.7) and (6.8) since, as just established, v11Eoe2(*
*S2m+1)
and v11Eoe+12(S2m+1) contain no unexpected elements for any m. 
The UNSS and v1periodic UNSS are considerably more complicated at the prime
2 than at the odd primes, but the v1periodic UNSS of S2n+1 is still completely
understood. We shall not discuss it in detail because most of our applications *
*in
this paper will be at the odd primes. The reader desiring more detail is referr*
*ed to
[4], which gives a chart with UNSS names of the elements.
We reproduce in figs. 13 and 14 the charts from [10, p. 488] of the v1periodic
UNSS of S2n+1 at the prime 2. Here "3" means Z=23, and "" means Z=2 , where
= min(2(8k + 8) + 1; n):
Differentials emanating from a summand of order greater than 2 are nonzero only
on a generator of the summand. Note how Z=8 in periodic homotopy is obtained
as an extension by the Z2 in filtration 3 of the elements in the 1line group w*
*hich
are divisible by 2 in a Z=8.
Figure 14 applies to S2n+1 when n 1 or 2 mod 4, with n > 2. The reader is
referred to [10, p.487] for the minor changes required when n 2. In fig. 14, t*
*he
dotted differential is present if and only if = n. In both charts, the left j*
*action
on the Z=2 on line 1, which is usually indicated by positively sloping solid l*
*ines,
is indicated by the dotted line if n < (8k + 8) + 1.
The argument establishing these charts appears in [4]. Note that v11Es;*4(S2n*
*+1) =
0 if s > 4, and hence no higher differentials are possible.
7.v1periodic homotopy groups of SU (n)
In this section we show how the v1periodic UNSS determines the v1periodic ho
motopy groups of spherically resolved spaces. This relationship is particular n*
*ice
when localized at an odd prime, and it is this case on which we focus most of o*
*ur
attention. At the end of the section, we discuss the changes required when p = *
*2.
After proving the general result for spherically resolved spaces, we specialize*
* to
SU(n), where the result is, in some sense, explicit.
It is not clear that the v1periodic UNSS of a space X must converge to the
pprimary v1periodic homotopy groups of X. There might be periodic homotopy
classes which are not detected in the periodic UNSS because multiplication by v1
repeatedly increases BP filtration. It is also possible that a v1periodic fami*
*ly in
E2 might support arbitrarily long differentials in the unlocalized spectral seq*
*uence,
in which case it would exist in all v11Er, but would not represent an element
of periodic homotopy. We now show that neither of these anomalies can occur
Section 7 Computing periodic homotopy groups 37
Figure 13. v11Es;2n+1+j2(S2n+1), n 0, 3 mod 4, p = 2
rr BrrB BrrBBB rrBB
BBBB BB
BB B
s = 4 rr BB BrrB B BrrBBB rrBB
BB BB BB B
B BB B B
rr B BrrBBB B BrrBBBB rrBB rr
 B  BB B B
B B  BB B B
rr 3Br B  rrBBB rB prr
B B ppp
B B ppp
s = 1 r 3 Br p
j  s = 8k+ 1 3 5 7
Figure 14. v11Es;2n+1+j2(S2n+1), n 1, 2 mod 4, n > 2, p = 2
r rr rrBB BBrrBB
BB BBBB BB B
B BB B B
s = 4 r B rr BB rrBB B BBrrBB
BB B BB BB pp B
B B B BB pp B
r B rr B B rrBBBpB BBrrBBB BBrr
B  B B BBpp B
B  B B BBpp B
r 3 r B  rrB B rB p rrBp
B B pp ppp
B B ppppp
s = 1 r 3 Br p
j  s = 8k+ 1 3 5 7
38 Donald M. Davis Chapter 1
for a spherically resolved space, essentially because they cannot happen for an*
* odd
sphere, where the v1periodic homotopy groups are known to agree with the v11E*
*2
term.
A space X is spherically resolved if there are spaces X0; : :;:XL, with X0 = *,
XL = X, and fibrations
Xi1! Xi! Sni (7.1)
with ni odd, and algebra isomorphisms
H*(Xi) H*(Xi1) H*(Sni):
The following result was stated as Theorem 1.1. It was proved in [4].
If p is odd, and X is spherically resolved, then v11Es;t2(X) = 0 unless s = 1 *
*or 2,
and t is odd. The v1periodic UNSS collapses to the isomorphisms
ae1 1;i+1
v11ssi(X) v1E212;(X)i+2if i is even
v1 E2 (X) if i is odd.
Proof. Each algebra BP *(Xi) is free, and so eq. (6.1) applies to give E2(Xi)
ExtU(M(xn1; : :;:xni)), where M() denotes a free BP*module on the indicated
generators. There are short exact sequences in U
0 ! M(xn1; : :;:xni1) ! M(xn1; : :;:xni) ! M(xni) ! 0;
and hence long exact sequences
! Es;t2(Xi1) ! Es;t2(Xi) ! Es;t2(Sni) ! Es+1;t2(Xi1)(!7:.2)
These exact sequences are compatible with the direct system of v1maps whose li*
*mit
is the v1periodic groups. Thus there is a v1periodic version of (7.2), and he*
*nce by
Theorem 6.10 and induction on i, v11Es;t2(X) = 0 unless s = 1 or 2, and t is o*
*dd.
Thus v11E2(X) v11E1 (X).
If u > max{ni}, s = 1 or 2, and s + u is odd, there are natural edge morphisms
ssu(X) ! Es;s+u2(X), and these are compatible with the direct system of v1maps,
giving morphisms v11ssu(X) ! v11Es;s+u2(X). These yield a commutative diagram
of exact sequences
Section 7 Computing periodic homotopy groups 39
0 ! v11ss2k(Xi1)!? v11ss2k(Xi)? ! v11ss2k(Sni)!?
?yOEi1 ?yOEi ?y
0 !v11E1;2k+12(Xi1)! v11E1;2k+12(Xi)v1!1E1;2k+12(Sni)!
! v11ss2k1(Xi1)!? v11ss2k1(Xi)? v!11ss2k1(Sni)!?0
?yOE0i1 ?yOE0 ? 0
i y
! v11E2;2k+12(Xi1)! v11E2;2k+12(Xi)v1!1E2;2k+12(Sni)! 0:
(7.3)
The zeros at the ends of the v11E2sequence follow from the previous paragraph.
The zero morphism coming into v11ss2k(Xi1) follows from v11E02(Sni) = 0 and
OEi1being an isomorphism, which is inductively known. The zero morphism coming
out from v11ss2k1(Sni) follows since anything in the image must have filtrati*
*on
2, but, by induction, v11ss2k2(Xi1) is 0 above filtration 1.
Comparison of Theorems 4.2 and 6.10 shows that the groups related by and by
0are isomorphic, and it is easy to see that and 0induce the isomorphisms. (*
*See
[23] for reasons.) Since X1 is a sphere, this comparison also shows that OE1 an*
*d OE01
are isomorphisms, which starts the induction. Thus all OEiand OE0iare isomorphi*
*sms
by induction and the 5lemma. 
Theorem 7.2 is a nice result, but it still leaves the formidable task of calcu*
*lating
v11E2(X). In [6], Bendersky proved the following seminal result, whose proof we
will discuss throughout much of the remainder of this section.
If k n, then in the UNSS E1;2k+12(SU (n)) Z=pep(k;n), where ep(k; n) is as
defined in Definition 1.2, and p is any prime.
This allows us to easily deduce Theorem 1.3, now demoted to corollary status,
which we restate for the convenience of the reader.
If p is odd, then v11ss2k(SU (n)) Z=pep(k;n), and v11ss2k1(SU (n)) is an ab*
*elian
group of the same order.
Proof of corollary. The first part of the corollary is a straightforward applic*
*ation
of Theorems 7.2 and 7.3, once we know that the groups in Theorem 7.3 are v1
periodic. This can be seen by observing how they arise, from exact sequences bu*
*ilt
from spheres, where the classes are all v1periodic. In [23], a slightly differ*
*ent proof of
this part of the corollary was given, before the v1periodic UNSS had been hatc*
*hed.
The exact sequence like the top row of (7.3) for the fibration
SU (n  1) ! SU(n) ! S2n1 (7.4)
implies, by induction on n, that v11ss2k1(SU (n)) = v11ss2k(SU (n)). Ind*
*eed, the
orders are equal for S2n1 by Theorem 6.10, and so if they are equal for SU(n1*
*),
40 Donald M. Davis Chapter 1
then they will be equal for SU (n), since the alternating sum of the exponents *
*of p
in an exact sequence is 0. The fact that SU (2) = S3 starts the induction. 
In [23], an example of a noncyclic group v11ss2k1(SU (n); 3) was given, and *
*in
[13] it was shown that v11ss2k1(SU (n); 2) will often have many summands (in
addition to a regular pattern of Z2's).
In order to prove Theorem 7.3, it is convenient to work with the UNSS based
on MU, rather than BP . This allows us to work with the ordinary exponential
series, rather than its ptypical analogue. The facts about MU that we need are
summarized in the following result.
i. MU*(MU) is a polynomial algebra over MU* with generators Bi of grading 2i
for i > 0. There are elements fii2 MU2i(CP 1) which form a basis for MU*(CP 1)
as an MU*module. P
ii. Let B denote the formal sum 1 + i>0Bi. The coaction
MU*(CP 1) ! MU*MU MU* MU*(CP 1)
satisfies
X
(fin) = (Bj)nj fij:
j
Here (Bj)nj denotes the component in grading 2(n  j) of the jth power of the
formal sum B.
iii. There is a ring homomorphism _e: MU*(MU) Q ! Q satisfying
o_e(Bi) = 1=(i + 1).
o_e(jR(a)) = 0 if a 2 MUi with i > 0.
o_einduces an injection E1;2n+1+2k2(S2n+1) ! Q=Z.
iv. The BP based UNSS is the plocalization of the MUbased UNSS.
Proof. Part (i.) is standard (e.g., [2]), while part (ii.) is [2, 11.4]. Part (*
*iii.) is from
[6]. There are elements mi 2 MU2i Q such that MU*MU Q is a polynomial
algebra over Q on all miand jR(mi). One defines _eto be the ring homomorphism
which sends mito 1=(i + 1) and jR(mi) to 0. The second property in (iii.) is cl*
*ear,
and the first follows by conjugating [2, 9.4] to obtain
X
mn = jR(mi)(Bi+1)ni;
and then applying _eto obtain _e(mn) = _e(Bn).
One way to see the third property is to localize at p and pass to BP . Then mM*
*Upi1
passes_to mBPi, and so our _epasses to that of [6, 4.3]. It is shown on [6, p.7*
*51] that
eBP sends the plocal 1line injectively, and this works for all p. 
Now we can state a general theorem which incorporates most of the work in
proving Theorem 7.3. This theorem was stated without proof in [11, 3.10], where
Section 7 Computing periodic homotopy groups 41
it was applied to X = Sp(n), p = 2. We will outline the proof, which is a direct
generalization of [6], later in this section.
Suppose X is spherically resolved as in Definition 7.1 with n1 < n2 < . .,.and
L possibly infinite. Then MU*(Xk) is an exterior algebra over MU* on classes
y1; : :;:yk; with yi = ni. Let
__1;t 1;t 1;t
E (Xk) = ker(E2 (Xk) ! E2 (XL)):
Let flk;j2 MU*(MU) be defined in terms of the coaction in MU*(Xi) by
Xk
(yk) = flk;j yj; (7.5)
j=1
and let bk;j= _e(flk;j) 2 Q. Then the matrix B = (bk;j) is lower triangular wit*
*h 1's
on the diagonal. Let C = (ck;j) be the inverse of B, and let
!k(m) = l:c:m:{den(ck;j) : m j k}:
__1;nk __1;nk
Then coker(E (Xm1 ) ! E (Xk1)) is cyclic of order !k(m).
Now we specialize to SU (n), where we need the following result.
In the UNSS
(i)Es;t2(SU ) = 0 if s > 0.
(ii)E1;2k+12(SU (n)) = 0 if n > k.
(iii)E1;2k+12(SU (k)) Z=k!.
(iv)If i < j k, the inclusion SU (i) ! SU(j) induces an injection in E1;2k+1*
*2.
(v)SU is spherically resolved as in Theorem 7.6 with Xk = SU (k + 1), nk =
2k + 1, and if the MUcoaction on SU is as in eq. (7.5), then
X _
e(flk;j)xk = ( log(1  x))j: (7.6)
kj
Proof. Part i is a nontrivial consequence of the fact that SU is an Hspace with
torsionfree homology and homotopy. See [6, 3.1]. The coalgebra MU*(SU (n)) is
cofree with primitives isomorphic to MU*(2 CP n1). Hence by (6.1)
E2(SU (n)) ExtU(MU*(2 CP n1));
and so the fibrations (7.4) induce exact sequences in E2. Part ii follows from *
*part
i and the exact sequence, and part iv is also immediate from the exact sequence.
Let y2k+12 MU2k+1(SU (n)) be the generator corresponding to
2 fik 2 MU2k+1(2 CP n1)
42 Donald M. Davis Chapter 1
for k < n. Part iii is proved on [6, p.748] by showing that the generator of
E0;2k+12(SU ) is of the form k!y2k+1+ lower terms, so that
E0;2k+12(SU (k + 1)) ! E0;2k+12(S2k+1)
sends the generator of one Z to k! times the generator of the other Z. This imp*
*lies
part iii.
By Proposition 7.5(ii.) and the relationship of MU*(SU (n)) with MU*(2 CP n1)
noted above, we find
X
(y2k+1) = (Bj)kj y2j+1:
P _
By Proposition 7.5(iii.), we have e(Bi)xi+1=  log(1x). These facts yield pa*
*rt
v. 
Now we can prove Theorem 7.3. If f(x) is a power series with constant termP0,
let [f(x)] denote the infinite matrix whose entries ak;jsatisfy f(x)j = kak;j*
*xk.
One easily verifies that [g(x)][f(x)] = [f(g(x))]. Hence the inverse of the mat*
*rix
[ log(1  x)] is [1  ex]. This observation, with Theorem 7.6 and Proposition*
* 7.7,
implies that there is a short exact sequence
0 ! E1;2k+12(SU (n)) ! E1;2k+12(SU (k)) ! Z=!k(n) ! 0
with middle group Z=k! and
!k(n) = l:c:m:{den(coef(xk; (1  ex)j)) : n j k}:
Thus E1;2k+12(SU (n)) is cyclic of order
k!= l:c:m:{den(coef(xk; (ex  1)j)) : n j k}
k
= gcd{coef(x_k!; (ex  1)j) : n j k}:
Looking at exponents of p yields Theorem 7.3 by Proposition 7.5(iv.). 
It remains to prove Theorem 7.6, the notation of which we employ without com
ment. Define bk;j(m) for m k recursively by bk;j(k) = bk;jand
bk;j(m) = bk;j(m + 1)  bk;m(m + 1)bm;j: (7.7)
We begin by noting that if row reduction is performed on (BI) so as to get at *
*each
step one more diagonal of 0's below the main diagonal of B, we find that the en*
*tries
ck;jof B1 satisfy
8
< 0 if j > k
ck;j= : 1 if j = k
bk;j(j + 1)if j < k.
Section 7 Computing periodic homotopy groups 43
Then
!k(m) = max(ord(bk;j(j + 1)) : m j k): (7.8)
Here and throughout this proof,_ord() refers to_order in Q=Z.
Fix k, and let o(m) = coker(E 1;nk(Xm1 ) ! E 1;nk(Xm )). We drop the
subscript k from eqs. (7.7) and (7.8)._The fibration Xk1 ! Xk ! Snk im
P
plies that the generator g(k  1) of E 1;nk(Xk1) is d(yk) = j 2.
ov11E4(X) = v11E1 (X), and v11Es4(X) = 0 if s > 4.
oIf the groups v11E1;t2(X) are cyclic, then the v1periodic UNSS converges to
v11ss*(X).
8.v1periodic homotopy groups of some Lie groups
In this section we focus on two examples. One uses UNSS methods to determine
v11ss*(G2; 5), while the other uses ASS methods to determine v11ss*(F4=G2; 2).
Here G2 and F4 are the two simplest exceptional Lie groups. The first example is
just one of many discussed in [14]. We also discuss how these UNSS methods can *
*be
used to give tractable formulas for v11ss*(SU (n); p) when p is odd and n p2*
* p.
We close by summarizing the status of the program, initially proposed by Mimura,
of computing the v1periodic homotopy groups of all compact simple Lie groups.
Our first theorem concerns the v1periodic homotopy groups of certain sphere
bundles over spheres, which appear frequently as direct factors of compact simp*
*le
Lie groups localized at p, according to the decompositions given in [49].
Let p be an odd prime, and let B1(p) denote an S3bundle over S2p+1with attachi*
*ng
map ff1. Then the only nonzero v1periodic homotopy groups of B1(p) are
p1))
v11ss2p+qm1(B1(p)) v11ss2p+qm(B1(p)) Z=pmin(p+1;1+p(mp :
This is the case k = 1 of [14, 2.1]. The spaces B1(p) were called B(3; 2p+1) i*
*n [14].
The following result follows immediately from the 5local equivalence G2 ' B1(5*
*).
( min(6;1+ (i5009))
Z=5 5 if i 1 mod 8
v11ssi(G2; 5) Z=5min(6;1+5(i5010))if i 2 mod 8
0 otherwise
Section 8 Computing periodic homotopy groups 45
The following result is the central part of the proof of Theorem 8.1. Indeed, *
*this
theorem, 8.3, along with Theorem 6.3, gives the order of each group E1;t2(B1(p)*
*),
and Theorem 8.5 shows the group is cyclic. Then Theorem 7.2 shows that this giv*
*es
v11ss*(B1(p)) when * is even, and the proof of Corollary 7.4 shows that
v11ss2k1(B1(p)) = v11ss2k(B1(p)):
Finally, v11ss*(B1(p)) is shown to be cyclic when * is odd in Theorem 8.6.
In the exact sequence
0 ! E1;qm+2p+12(S3)i*!E1;qm+2p+12(B1(p))
j*!E1;qm+2p+1 2p+1 @ 2;qm+2p+1 3
2 (S ) ! E2 (S );
the morphism @ is a surjection to Z=p unless
p(m) p  1 and m=pp1 1 mod p;
in which case it is 0.
We will use the double suspension Hopf invariant H2 discussed in Theorem 6.8.
We denote by H0 the morphism
H0: E22(S2n+1) H2!Ext1U(W (n)) ! E12(M);
obtained by following H2 by the stabilization. Here
Es2(M) ExtsBP*BP(BP*; BP*=p)
denotes the E2term of the stable NSS for the mod p Moore spectrum M. We will
eventually need the following facts about E2(M).
(i)E2(M) is commutative.
(ii)vk1h1 6= 0 2 E12(M).
(iii)If x 2 E2(M), then v2x = xv2 = 0.
(iv)v1hk(p1)+11= vk(p1)+11h1 in E2(M).
e11
(v)If s 6 0 mod p, then ffspe1=e= svsp1 h1 in E2(M).
Proof. Part i is wellknown, part ii is [51, p 157], and part iii follows from *
*[48,
2.10]. Part iv follows from Proposition 6.1, part iii of thiselemma,and1the fa*
*ct that
ph1 = 0 2 E2(M). To prove part v, we use 6.1 to expand vsp1 , obtaining
e1 spe1
ffspe1=e=1_pe(jR(vsp1 )  (ph1+ jR(v1)) )
spe1Xspe1 e1
=  pjehj1vsp1 j:
j=1 j
46 Donald M. Davis Chapter 1
All terms except j = 1 are divisible by p, and hence are 0. To insure that term*
*s with
j large are p times an admissible element, write pjehj1as p(v1jR(v1))je1he+*
*1.

Now we begin the proof of Theorem 8.3. We begin with the case (m) < p  1.
In this case,
@(gen) = ffm=(m)+1 ff13 ffm=(m)+1 h13; (8.1)
mod terms that desuspend to S1. Here we have used [6, 4.9] and Proposition 6.5.
By Proposition 6.5, the assumption that (m) < p  1 implies that ffm=(m)+1 is
defined on S2p1, and hence Theorem 6.8.v implies that
H0(@(gen)) = ffm=(m)+1 6= 0;
where the last step uses parts v and ii of Lemma 8.4. Thus @ 6= 0 in this case,*
* as
claimed.
Now we complete the proof of Theorem 8.3 by considering the case (m) p1.
We let s = m=p(m) and
ae
ffl = 01 ifi(m)f>(pm)1= p  1.
We will establish the following string of equations in the next paragraph, and *
*then
we will further analyze whether these terms are 0 by studying their Hopf invari*
*ant.
The following string is valid mod terms which desuspend to S1.
@(gen)= ffm=p ff13
= pp(jR(vm1)  (ph1+ jR(v1))m ) ff13 (8.2)
mX m
=  pjphj1 vmj1ff13
j=1 j
= pmp hm1 ff13 fflsh1 vm11ff13 (8.3)
= vmp1hp1 ff13 vmp11hp1 v1h13+ fflsh1 vm11h13(8.4)
= A + B + C; (8.5)
where A, B, and C denote the three terms in the preceding line.
Line (8.2) follows from Propositions 6.5 and 6.1. Line (8.3) has been obtained
by observing that in the sum all terms desuspend to S1 except j = m and, if
(m) = p  1, j = 1. To see this, we observe that we need to have a p to make
ff13 desuspend. This factor will be present unless j = 1 and (m) = p  1. The
requirement that j be 1_2times the degree of the symbols following hj1will only
be a problem for large values of j. When j is large, write the term as
m p(v  j (v ))jp1hp+1 vmj ff :
j 1 R 1 1 1 13
Section 8 Computing periodic homotopy groups 47
Since p times anything which is defined on S3 desuspends to S1, this desuspends*
* to
S1 provided p + 1 (p  1)(m  j + 1) + 1, which simplifies to 1 (p  1)(m  j*
*),
i.e., j < m. To obtain (8.4), we have rewritten the first term of eq. (8.3) as
p(v1 jR(v1))mp1hp+11 ff13;
observed that when this is expanded, all terms except the first desuspend, and *
*in
that first term we write ph1 = v1 jR(v1).
We note first that, by Proposition 6.2, A is d(h2) mod S1, and so H0(A) = 0. We
can evaluate the Hopf invariant of B and C by Theorem 6.8.v; using Lemma 8.4,
we obtain
H0(B + C) = (1 + ffls)vm11h1:
Hence H0(@(gen)) = 0 if and only if 1 + ffls 0 mod p. Since H0 is injective on
E22(S3), this completes the proof of Theorem 8.3. 
Now we settle the extension in the exact sequence of Theorem 8.3.
The groups E1;qm+2p+12(B1(p)) in Theorem 8.3 are cyclic.
Proof. We will show that whenever ker(@) 6= 0 in the exact sequence of 8.3, the*
*re
is an element z 2 E1;qm+2p+12(B1(p)) such that j*(z) = ffm 2p+1, the element of
order p, and pz = i*(gen). Since @(ffm 2p+1) = 0 in these cases, there is w3 2
C1;qm+2p+1(S3) such that d(w3) = ffm ff13. Let
z = ffm 2p+1 w3:
Then z is a cycle, since d(z) = ffm ff13  ffm ff13, and clearly j*(z) is as
required. Since pffm = d(vm1), we have
pz  d(vm12p+1)=d(vm1)2p+1 pw3 d(vm1)2p+1+ vm1ff13
= i*((vm1ff1 pw)3)
We show (vm1ff1pw)3 6= 0 2 E1;qm+2p+12(S3) by noting from [7, x7] that the Hopf
invariant
H2 : E12(S3) ! Ext0(W (1))
factors through the mod p reduction of the unstable cobar complex. Thus
H2((vm1ff1 pw)3) = H2(vm1h13) = vm16= 0:
The second "=" uses Theorem 6.8v, and the "6=" uses 6.8iii. 
The following result completes the proof of Theorem 8.1 according to the outli*
*ne
given after Corollary 8.2.
In Theorem 8.1, the group v11ssqm+2p1(B1(p)) is cyclic.
48 Donald M. Davis Chapter 1
Proof. We use the exact sequence in v11ss*() of the fibration which defines
B1(p). The cyclicity follows from that of v11ssqm+2p1(S2p+1) unless
@ = 0 : v11ssqm+2p(S2p+1) ! v11ssqm+2p1(S3): (8.6)
If eq. (8.6) is satisfied, then
Off1 6= 0 : v11ssqm+2(S2p+1) ! v11ssqm+2p1(S2p+1)
by [23, 6.2], and
@ : v11ssqm+2(S2p+1) ! v11ssqm+1(S3)
is an isomorphism of Z=p's by Theorem 8.3. Let G denote a generator of
v11ssqm+2(S2p+1), and let
Y 2 v11ssqm+2p1(B1(p))
project to G O ff1. By [50, 2.1],
pY = i*(<@G; ff1; p>) = @(G) O v1 6= 0: 
It is shown in [49] that B1(p) is a direct factor of SU (n)(p)if p < n < 2p, a*
*nd
hence v11ssi(SU (n); p) is given by Theorem 8.1 if p < n < 2p and i 1 or 2 mod
q. This yields the following number theoretic result.
If p is an odd prime, k 1 mod p  1, and p < n < 2p, then the number ep(k; n)
defined in 1.2 equals min(p; p(k  p  pp + pp1)) + 1.
The author has been unable to prove this result without the UNSS. In fact, the
only tractable result for ep(k; n) which follows easily from Definition 1.2 see*
*ms to be
that if n p and k n  1 mod p  1, then ep(k; n) min(n  1; p(k  n + 1) + 1*
*),
which is proved using the Little Fermat Theorem as in [22, p.792].
Using methods similar to those in our proof of Theorem 8.1, Yang ([58]) has
proved the following tractable result for v11ss*(SU (n); p) when p is odd and n
p2  p. Of course this can also be interpreted as a theorem about the numbers
ep(k; n). We emphasize that the proof of Theorem 8.8 does not involve the use of
Theorem 1.3.
Suppose p is odd, k = c + (p  1)d with 1 c < p, and
c + (p  1)b + 1 n c + (p  1)(b + 1)
with 0 b p  1. Define j by 1 j p and d j mod p. Then v11ss2k(SU (n); p)
Section 8 Computing periodic homotopy groups 49
is cyclic of order pe, with
8
>>>min(c + (p  1)j; b + (d  j) + 1)
>>> if b < c and 1 j b
>>>
>>> j c j(p1)
> if c b and 1 j b
>>>
>>>min(c; b + 1 + (d))
>>> if b < c and b < j p
>>>
:
b ifc b and b < j p
One can easily read off from Theorem 8.8 the precise value of the numbers ep(n)
which appeared in Corollary 7.8, yielding the following result for the pexpone*
*nt of
the space SU (n).
If p is odd and n p2 p, then
ae
expp(SU (n)) ep(n) = nn  1 ifoi(pth1)e+r2winseip.+ 1 for some i
When p = 2, UNSS methods of computing v11ss*(; p) become more complicated
because of the jtowers. Then ASS techniques become more useful, as they did
for v11ss*(G2; 2) in [29]. Here we show how to use ASS methods to determine
v11ss*(F4=G2; 2). It is hoped that the result of the calculations of v11ss*(G*
*2; 2) and
v11ss*(F4=G2; 2) might be combined to yield v11ss*(F4; 2), but this involves *
*one
difficulty not yet resolved. Our main reason for including this example is to g*
*ive a
new illustration of this method.
The proof of the following theorem will consume most of the remainder of this
paper. If G denotes an abelian group, then mG denotes the direct sum of m copies
of G.
Let G(n) denote some group of order n.
8
>>>4Z2 i 0 mod 8
>>>Z8 Z8 Z2 i 1 mod 8
> 0 i 3 or 4 mod 8
>>>G(2min(15;(i21)+4)) i 5 mod 8
>>> min(15;(i22)+4)
: Z2 Z2 Z=2 i 6 mod 8
5Z2 i 7 mod 8
There is a fibration
S15 i!F4=G2 p!S23; (8.7)
50 Donald M. Davis Chapter 1
derived in [29, 1.1]. Here and throughout this proof all spaces and spectra are
localized at 2.
In [31], it was shown that for any spherically resolved space Y , there is a f*
*inite
torsion spectrum X satisfying v11ss*(Y ) v11J*(X). In our case, we have
There is a spectrum X such that
(i.) v11J*(X) v11ss*(F4=G2), and
(ii.) there is a cofibration
215P 14! X !223+LP 22; (8.8)
where L equals 0 or a large 2power.
We present in fig. 15 a chart which depicts an initial part of the ASS for v1*
*1J*(X)
if X is as in Proposition 8.11 and L = 0. It depicts the direct sum of the spec*
*tral
sequences for v11J*(215P 14) and v11J*(223P 22), together with one differenti*
*al,
which will be established in Proposition 8.13. The o's are elements from P 14, *
*while
the O's are from P 22. Charts such as these for v11J*(P n) were derived in Sec*
*tion
4.
To see that fig. 15 is also valid when L in Proposition 8.11 is a large 2powe*
*r, we
use the following result.
If L is a large 2power, then the attaching map
222+LP 22! v11215P 14
in Proposition 8.11 has filtration L=2 + 1.
Using results of [40], this implies that a resolution of v11X can be formed fr*
*om
v11215P 14and OEL=2v11223+L P 22, where OEj increases filtrations by j, and t*
*his
yields fig. 15.
Proof of Lemma 8.12. Under Sduality, the generator corresponds to an element
of v11JL+7(P 14^ D(P 22)). This group is isomorphic to
v11JL+6(P 14^ P223): (8.9)
Using methods of [26], one can show that the relevant chart is as in fig. 16, w*
*here the
class indicated with a bigger o is the generator of (8.9), and has filtration L*
*=2 + 1.
Borel ([15]) showed that Sq8(x15) = x23 in H*(F4; Z2). This implies that the
attaching map in F4=G2 and in X is the Hopf map oe, and hence corresponds to
the generator of (8.9). Thus the attaching map has filtration L=2 + 1. 
We can now establish the d2differentials in fig. 15.
There are d2differentials as indicated in fig. 15.
Proof. This follows from the oe attaching map just observed, together with the
observation that the first of the pair of elements that are related by the diff*
*erential
in fig. 15 are v1periodic versions of j23and joe15. 
Section 8 Computing periodic homotopy groups 51
Figure 15. Initial chart for v11ss*(F4=G2)
r

8k + 21 r r
 
 r r
 r r
  A
r r A b
 A
r r A Ab b
 0
rdr Ab r b
  
r br br r
  
r brrb r
 
brbrr

bbrr

bbr
d0= d(k+1)+2
b b
bdb d = d(k)+2
b b b
b b b b

b b b b 
b b b 

b b 
b 8k + 26
Figure 16. The generator of v11JL+6(P14^ P223)
r

r

r r
A 
r ru
AA
rAr r
AA
rArr
A 
rArr

r r

r
52 Donald M. Davis Chapter 1
This d2differential implies there is a nontrivial extension in v11ss8k+26(F4*
*=G2)
as follows.
v11ss8k+26(F4=G2) Z=64.
Proof. This follows from fig. 15 and a standard Toda bracket argument ([50,
2.1]), which in this situation says the following. Let A be the element support*
*ing
the higher of the two d2differentials, and let D be the lowest o in 8k + 26. L*
*et @
denote the boundary morphism in the exact sequence in v11ss*() associated to
the fibration (8.7). Then D lies in the Toda bracket <@(A); j; 2>, and so there*
* exists
an element E 2 v11ss8k+26(F4=G2) such that p*(E) = A O j and i*(D) = 2E. 
As indicated in fig. 15, there are d(k)+2differentials between Otowers in 8k*
* + 22
and 8k + 21, and there are d(k+1)+2differentials between otowers in 8k + 22 a*
*nd
8k+21. This follows just from standard J*()considerations. But there may also*
* be
differentials from the Otower in 8k+22 to the otower in 8k+21. These differen*
*tials
from O to o are determined from the homomorphism
v11ss8k+22(S23) @!v11ss8k+21(S15); (8.10)
which is evaluated in the following result.
The image of the homomorphism (8.10) consists of all multiples of 8 if (k) 7,
and is 0 if (k) > 7.
The following result plays a central role in the proof of Proposition 8.15.
Let (S15)K denote the K*localization as constructed in [42]. There is a commu
tative diagram
ffiS23? @! S15?
?y` ?ye
ffi1 (215P 14^J)h!(S15)K
in which @ is obtained from the fiber sequence (8.7), e is the localization, an*
*d h
induces an isomorphism in ssj() for j = 22 and j 28.
Proof. The map h is constructed as in [29, pp. 669670], using results of [42]*
*. It
induces an isomorphism in ssj() for many other small values of j, but we only *
*care
about values of j which are positive multiples of 22. The obstructions to its b*
*eing
an isomorphism for all small values of j are Z2classes in filtration 1 in Jj(2*
*15P 14)
for j = 19, 23, and 27. The map ` is obtained by obstruction theory, since ffiS*
*23has
cells only in dimensions which are positive multiples of 22. 
Now we prove Proposition 8.15. Let ` be as in Proposition 8.16. The morphism
ss*(`) can be factored as
ss*(ffiS23) ! sss*(ffiS23) ! J*(215P 14):
Section 8 Computing periodic homotopy groups 53
There is a splitting
M
sss*(ffiS23) sss*(S22i):
i>0
We will use the method of [43] to deduce that
sss8k+21(S22) ! J8k+21(215P 14)
sends the v1periodic generator aek to 8 times the generator. Indeed, the stabl*
*e map
S22 !215P 14^ J which induces the morphism factors through 215P 8^ J, from
which it projects nontrivially to 215P78^ J. We then use [43, 2.8] to deduce th*
*at
aek goes to the nonzero element of J8k+21(215 P78). This implies that its image
in J8k+21(215 P 8) is the generator, and this maps to 8 times the generator of
J8k+21(215P 14).
The composite of v1periodic summands of
ss8k+21(ffiS23) ! sss8k+21(ffiS23) ! sss8k+21(S22) (8.11)
is bijective if (k) 7, but is not surjective if (k) > 7. Thus when the composi*
*te
(8.11) is followed into J8k+21(215 P 14), the image of a v1periodic generator *
*is 8
times the generator if (k) 7 and 0 2 Z=16 if (k) > 7. Once we observe that,
in the diagram of Proposition 8.16, h induces an isomorphism in ss8k+21() and e
sends the v1periodic summand isomorphically, we obtain the desired conclusion *
*of
Proposition 8.15. 
The differentials implied by Proposition 8.15 have an interesting and unexpect*
*ed
implication about fig. 15. Since dr from the Otower in 8k + 22 hits the top O *
*in
8k + 21 and the o just above it with the same r, and since dr respects the acti*
*on
of h0, there must be an h0extension between these classes in 8k + 21. If L = 0*
* in
(8.8), then this extension can only be accounted for by a failure of the map (8*
*.8) to
induce a split short exact sequence of A1modules in cohomology. Indeed, we have
If L = 0 in (8.8), then there is a splitting of A1modules
H*X H*(215P 30) H*(223P 6):
This splitting is caused by having Sq2 6= 0 on the class in H*X corresponding to
the top cell of H*(215P 14), i.e.,
Sq2 : H29X ! H31X
is an isomorphism. This is the only way to account for the h0extension in fig.*
* 15.
Proposition 8.17 implies that the h0extensions are present in the chart for va*
*lues
of k ((k) > 7) where they cannot be deduced from differentials. It also implies
54 Donald M. Davis Chapter 1
that there is an h0extension on the top O in 8k + 22. If L > 0 in (8.8), the s*
*ame
conclusion about the charts can be deduced from a more complicated analysis.
It causes fig. 15 to take the form of fig. 17.
Figure 17.Final chart for v11ss*(F4=G2)


k odd r  k even r
  
8k + 21 r r  8k + 21 r r
    
 r r   r r
 r r    r r 
  A     A 
r r A b  r r A b
 A   A
r r A Ab b  r r A Ab b
   J
rrA A b r b  rr A b r b
AA   JJ 
rbDDrAbr r  rbrJ br r
AA   JJ 
rbrArb r  rbrJrb r
AA  J 
bbrrAr  bbrrJr
AA  
bbrAr  bbrr
AA  
bbrAA  bbr
bbAAA  bb
bbAAA  bbd d = d(k)+2

bbAAA b  bb b

bbAAA b b  bb b b
  
bbAbAA b   bb b b 

bbAb   bb b 
  
bb   bb 

b 8k + 26  b 8k + 26
We can read off almost all of Theorem 8.10 from fig. 17. We must show that d6 *
*is
0 on the O's near the bottom in 8k + 23 and 8k + 24. This is done by the argume*
*nt
used to prove Proposition 8.15. The classes involved are present in all spaces *
*in the
diagram in Proposition 8.16, but they are not mapped across by `*, since it fac*
*tors
through sss*(ffiS23).
All that remains is the verification of the abelian group structure. Most of t*
*he
extensions are trivial due to the relation 2j = 0. The extension in 8k + 22 whe*
*n k
is odd was present before the exotic extension was deduced, and remains true. T*
*he
cyclicity of this 27 summand can also be deduced by consideration of the kernel*
* of
the homomorphism in the fibration which defines J, but that seems unnecessary.
Note that no claim is made about the group structure in 8k + 21. 
Section References Computing periodic homotopy groups 55
In 1989, Mimura suggested to the author that he try to calculate v11ss*(G) for
all compact simple Lie groups G. If p is odd, and G = Sp(n) or SO (n), then the
result follows from Theorem 1.3 and [33]. With great effort, v11ss*(Sp(n); 2) *
*was
calculated in [13]. The result involves a surpising pattern of differentials am*
*ong Z2's
from the various spheres which build Sp(n), resulting in [log2(4n=3)] copies of*
* Z2
in certain v11ssi(Sp(n)). Of the classical groups, only v11ss*(SO (n); 2) rem*
*ains. All
torsionfree exceptional Lie groups were handled in [14], using the UNSS. In [2*
*9]
and [12], the torsion cases (G2; 2), (F4; 3), and (E6; 3) were handled. Remaini*
*ng
then are seven cases of (G; p) yet to be calculated. At least a few of these sh*
*ould
lend themselves to the methods of this paper.
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