ON THE 2-PRIMARY v1-PERIODIC HOMOTOPY GROUPS OF SPACES
A.K. BOUSFIELD
Abstract.We develop foundations of a general approach for calculating p-p*
*rimary v1-periodic
homotopy groups of spaces using their p-adic KO-cohomologies and K-cohomo*
*logies with par-
ticular attention to the case p = 2. As a main application, we derive a m*
*ethod for calculating
v1-periodic homotopy groups of simply-connected compact Lie groups using *
*their complex, real,
and quaternionic representation theories. This method has been applied ve*
*ry effectively by D.M.
Davis in recent work. We rely heavily on the v1-stabilization functor 1f*
*rom spaces to spectra.
Roughly speaking, we obtain the p-primary v1-periodic homotopy of a space*
* X from the p-adic
KO-cohomology of 1X, which we obtain from the p-adic KO-cohomology and K*
*-cohomology of
X by a v1-stabilization process under suitable conditions.
Contents
1. Introduction 2
2. The general theory of v1-periodic homotopy groups and the functor 1 *
* 4
3. Pontrjagin duality for united p-adic K-theory *
* 9
4. Even and odd CR-modules 13
5. Even and odd K=p*-local spectra 19
6. Unstable operations in p-adic K-theory *
* 21
7. The v1-stabilization homomorphism 26
8. The v1-stabilizations of odd spheres *
* 29
9. The v1-stabilizations of simply-connected compact Lie groups *
* 33
References 37
__________
Date: March 24, 2004.
Key words and phrases. v1-periodic homotopy groups, v1-stabilizations, p-adic*
* K-theory,
united K-theory, representation theory.
2000 Mathematics Subject Classification. Primary: 55Q51; Secondary: 55N15 55P*
*60, 55S25,
57T20.
Research partially supported by the National Science Foundation.
1
2
1.Introduction
The p-primary v1-periodic homotopy groups v-11ß*X of a space X, as defined by*
* Davis and
Mahowald [26], are a localization of the portion of the homotopy groups of X de*
*tected by p-adic
K-theory. In [19], we showed that the groups v-11ß*X are naturally isomorphic t*
*o stable homotopy
groups ß*øp 1X where øp 1X is the p-torsion part of the spectrum 1X obtained u*
*sing the v1-
stabilization functor 1 constructed in [15], [20], [27], and [31]. Moreover, i*
*n [19], we developed
an approach for calculating v-11ß*X ~=ß*øp 1X from K*(X; ^Zp) via K*( 1X; ^Zp) *
*when p is an
odd prime and X is a suitable space such as a simply-connected finite H-space. *
* This approach
has been applied very successfully by Don Davis to simply-connected compact Lie*
* groups in [24].
After considerable effort, we recently found that most of the results of [19] c*
*an be extended to
the case p = 2 with some modifications and restrictions, and these new results *
*have already been
applied by Don Davis to complete his 13-year project of computing the v1-period*
*ic homotopy groups
of all simply-connected compact Lie groups [25]. In this paper and its sequel, *
*we shall develop the
promised 2-primary extensions of results of [19], giving a general approach for*
* calculating v1-periodic
homotopy groups of suitable spaces. When possible, we work at an arbitrary prim*
*e p, although our
main concern is with p = 2.
We begin in Section 2 by discussing the general theory of v1-periodic homotop*
*y groups and of
the functor 1. This functor carries a pointed space X to a K=p*-local spectrum*
* 1X such that
v-11ß*X ~=ß*øp 1X and, in fact, such that v-11ß*(X; W) ~=[W, 1X]* for each fin*
*ite p-torsion
spectrum W. Thus, the study of v1-periodic homotopy groups may be centered arou*
*nd the spectra
1X. These spectra are especially well-behaved when X is a simply-connected fin*
*ite H-space or
is spherically resolved or, more generally, when 1X has an exponent. In such *
*cases, we show
that 1X is periodic (see Theorem 2.6); we show that X becomes v1-periodically *
*equivalent to the
infinite loop space 1 1X after finite looping (see Theorem 2.10); and we show*
* that the ordinary
homotopy groups of X eventually map splittably onto the v1-periodic homotopy gr*
*oups of X (see
Theorem 2.11).
Since the spectra 1X are K=p*-local, they may be studied by the methods of u*
*nited K-theory
[16]. After establishing a Pontrjagin duality between the united p-adic K-coho*
*mology and K-
homology theories of spectra (see Theorem 3.1), we show that the groups v-11ß*X*
* ~=ß*øp 1X are
determined up to extension by KO*( 1X; ^Zp) together with its Adams operations *
*(see Theorem
3.2).
3
For many interesting spaces X, including simply-connected compact Lie groups,*
* the spectra 1X
have cohomologies K*( 1X; ^Zp) concentrated in even or odd degrees. We show tha*
*t such K=p*-local
spectra 1X are completely classified by their united p-adic K-cohomologies
K*CR( 1X; ^Zp) ~= {K*( 1X; ^Zp), KO*( 1X; ^Zp)}
equipped with the complexification, realification, conjugation, and Adams opera*
*tions (see Theorem
5.3). Moreover, when Kn-1( 1X; ^Zp) = 0, we find that K*CR( 1X; ^Zp) is largely*
* determined by a
small part
Kn( 1X; ^Zp) = {Kn( 1X; ^Zp), KOn( 1X; ^Zp), KOn-4( 1X; ^Zp)}
which we call a -module (see Theorems 4.3 and 4.11). We also show that Kn( 1X;*
* ^Zp) has a
crucial exactness property which facilitates comparisons with other -modules (*
*see Theorem 4.4).
This work allows us to approach the v1-periodic homotopy groups v-11ß*X of suit*
*able spaces X by
seeking to calculate the associated -modules Kn( 1X; ^Zp).
For a space X and integer n, we approach the stable -module Kn( 1X; ^Zp) by *
*starting with
the corresponding unstable -module
eKn(X; ^Zp) = {Ken(X; ^Zp), gKOn(X; ^Zp), gKOn-4(X; ^Zp)}
and dividing out by its " 1-trivial" part. In preparation, we develop an array *
*of unstable operations
in the p-adic KO-cohomologies and K-cohomologies of spaces using Atiyah's Real *
*K-theory (see
Theorems 6.4 and 6.5). We then show that the v1-stabilization homomorphism 1:K*
*en(X; ^Zp) !
Kn( 1X; ^Zp) must annihilate or linearize these unstable operations, allowing u*
*s to calculate Kn( 1X; ^Zp)
from eKn(X; ^Zp) under suitable conditions (see Theorem 7.2). When combined wit*
*h the preceding
work, this gives a v1-stabilization method for calculating v-11ß*X from eK*CR(X*
*; ^Zp) for suitable
spaces X (see 7.6).
To illustrate the v1-stabilization method, we calculate the groups v-11ß*S2n+*
*1at p = 2 and recover
results of Mahowald and Davis [23] (see Theorem 8.9). As a main application of *
*the method, we
obtain an approach for calculating 2-primary v1-periodic homotopy groups of sim*
*ply-connected com-
pact Lie groups using their complex, real, and quaternionic representation theo*
*ries. This approach
grew, in part, from our extensive correspondence with Bendersky and Davis, who *
*were develop-
ing another 2-primary approach using complex representation theory together wit*
*h the Bendersky-
Thompson spectral sequence (see [11]). Our approach eliminates major difficulti*
*es with differentials
and has now been applied very effectively by Davis in [25] as previously noted.*
* For a simply-
connected compact Lie group G, our main result (Therem 9.3) expresses KO*( 1G; *
*^Z2) in terms of
4
the representation theory of G. This result confirms a general conjecture cited*
* in [25, Conjecture
2.2].
This paper provides foundations for a sequel in which we shall continue to de*
*velop 2-primary
extensions of results of [19]. In particular, we shall obtain explicit 2-primar*
*y constructions of the
v1-stabilization 1X and of the localization XK=2for various spaces X, includin*
*g many simply-
connected compact Lie groups.
Throughout this paper, we generally follow the terminology of [21], so that "*
*space" means "sim-
plicial set,ä nd we let Ho* (resp. Hos) denote the homotopy category of point*
*ed spaces (resp.
spectra).
Although we have long been interested in v1-periodic homotopy theory, we were*
* prompted to
develop the present body of work by Martin Bendersky and Don Davis, and we than*
*k them for their
questions and comments.
2.The general theory of v1-periodic homotopy groups and the functor 1
Working at an arbitrary prime p, we first recall the v1-periodic homotopy gro*
*ups v-11ß*X of a
space X and explain how they are captured by the spectrum 1X. We then obtain s*
*tronger results
on v-11ß*X and 1Xwhen X is a simply-connected finite H-space or is spherically*
* resolved or, more
generally, when 1X has an exponent. In such cases, we show that the spectrum *
*1X is periodic;
we show that X becomes v1-periodically equivalent to the infinite loop space 1*
* 1X after finite
looping; and we show that the ordinary homotopy groups of X eventually map spli*
*ttably onto the
v1-periodic homotopy groups of X. These results build on insights of Davis and *
*Mahowald in [23],
[26], and [27].
2.1. The v1-periodic homotopy groups. For a finite p-torsion spectrum W 2 Hos, *
*a v1-map is
a K(1)*-equivalence ( = K=p*-equivalence) !: dW ! W with d > 0 such that K(n)**
*! = 0 for
n > 1, where K(n)*is the n-th Morava K-theory at p. The Hopkins-Smith Periodici*
*ty Theorem (see
[30] or [36]) ensures that each finite p-torsion spectrum W has a v1-map, which*
* becomes unique after
sufficient iteration and in fact becomes natural. Since the sequence W -! dW -*
*! 2dW -! . . .
in Hoseventually desuspends uniquely in Ho*, we may define the v1-periodic homo*
*topy groups of a
space X 2 Ho*relative to W by
v-11ß*(X; W) ~= colimm[ mdW, X]*
5
with naturality in both X 2 Ho*and W 2 Hos. Following Davis and Mahowald [26], *
*we may also
define the (absolute) v1-periodic homotopy groups of X 2 Ho*by
v-11ß*X = colimkv-11ß*+1(X; Z=pk) = colimkv-11ß*(X; SZ=pk)
using the Moore spectra SZ=pk = S0[pke1 with the canonical maps SZ=pk+1! SZ=pk.
2.2. The functor 1. By [20] or by earlier work in [15], [27], or [31], there i*
*s a v1-stabilization
functor or v1-periodic spectrum functor 1: Ho*! Hossuch that:
(i)for a space X and finite p-torsion spectrum W, there is a natural isomorph*
*ism
v-11ß*(X; W) ~= [W, 1X]*;
(ii) 1X is K=p*-local for each space X;
(iii)for a spectrum E, there is a natural equivalence 1( 1 E) ~= EK=p;
(iv) 1 preserves homotopy fiber squares;
(v)for a space X and finite complex A in Ho*, there is a natural equivalence
1(XA) ~= ( 1X)A.
2.3. p-torsion parts and p-completions of spectra. A spectrum E has a natural p*
*-torsion
part øpE ! E given by the homotopy fiber of the localization E ! E[1=p] away fr*
*om p with
øpE ' E ^ øpS ' E ^ S-1Zp1. It also has a natural p-completion E ! ^Epgive*
*n by the
SZ=p*-localization [13] with ^Ep' F(S-1Zp1, E). The functors øp: Hosø Hos: (^-)*
*pare adjoint
and restrict to equivalences between the homotopy categories of p-complete spec*
*tra and of p-torsion
spectra, with the K=p*-local spectra corresponding to the p-torsion K*-local sp*
*ectra. Thus 1X
corresponds to the p-torsion K*-local spectrum øp 1X, and by [19] we have:
Theorem 2.4.For a space X, there is a natural isomorphism v-11ß*X ~= ß*øp 1X.
In some important cases, the spectrum 1X has an exponent and a periodicity, *
*which are auto-
matically inherited by v-11ß*X.
2.5. Spaces with 1-exponents. A space X is said to have a 1-exponent pr if pr*
*' 0: 1X !
1X. Note that whenever two of the spaces in a fiber seqence have 1-exponents*
* (at p), then
so does the third by 2.2(iv). A space is also said to have an eventual H-space*
* exponent pr if
pr ' 0: ( N0X)(p)! ( N0X)(p)for sufficiently large N. This easily implies that*
* X has a 1-
exponent prand holds (for suitable values of r) whenever X is a sphere or a sim*
*ply connected finite
H-space (see [35] and [37]). Thus the following theorem will apply to spherical*
*ly resolved spaces
and to simply-connected finite H-spaces.
6
Theorem 2.6.If a space X has a 1-exponent pr, then the spectrum 1X is periodi*
*c with prqr 1X '
1X where qr= max{8, 2r-1} for p = 2 and qr= 2(p - 1)pr-1for p odd.
Proof.We rely on the KCRT*-Adams spectral sequence of [16] and use the associat*
*ed notation.
Since the united K-homology KCRT* 1X 2 ACRT has exponent pr, it has a canonical*
* periodicity
': qrKCRT* 1X ~= KCRT* 1X, and the element
d2' 2 Ext2,qr+1ACRT(KCRT*X, KCRT*X)
is the obstruction to realizing ' by a map qr 1X ! 1X. Since d2 acts as a de*
*rivation on
compositions and since pr(d2') = 0, we find that d2('pr) = 0, and hence 'prmay *
*be realized by a
map prqr 1X ! 1X. This gives the required periodicity. *
* |__|
Before developing further properties of v1-periodic homotopy groups, we need:
2.7. v1-periodic equivalences. As in [19], we say that a map f :X ! Y in Ho*is *
*a v1-periodic
equivalence when it satisfies the following equivalent conditions:
(i) 1f : 1X ' 1Y ;
(ii)f*:v-11ß*X ~= v-11ß*Y ;
(iii)f*:v-11ß*(X; Z=p) ~= v-11ß*(Y ; Z=p);
(iv))f*:v-11ß*(X; W) ~= v-11ß*(Y ; W) for each finite p-torsion spectrum W.
A v1-periodic equivalence may also be characterized homologically. We let X *
*! X denote the
m-connected cover of a space X, and we call K=p*X the generic K=p-homology o*
*f X when
m 3, since it does not depend on the choice of such m. By [18, Sections 11.5 *
*and 11.11], we have:
Theorem 2.8.A map f in Ho*is a v1-periodic equivalence if and only if f is a g*
*eneric K=p*-
equivalence. In this case, f is also a generic K=p*-equivalence.
2.9. Examples of v1-periodic equivalences. By [18, Corollary 11.2], each K=p*-e*
*quivalence of
H-spaces is a v1-periodic equivalence. Thus, each K=p*-equivalence f of spaces *
*suspends to a v1-
periodic equivalence f because f is a K=p*-equivalence of H-spaces. For exam*
*ple, whenever a
map ff: dA ! A of spaces represents a v1-map 1 ff of spectra, then ff suspend*
*s to a v1-periodic
equivalence ff, although ff itself need not be a v1-periodic equivalence by [3*
*2]. Perhaps the most
striking example of a v1-periodic equivalence is the Snaith map
s: 2n+10S2n+1 -! 1 1 RP2n
7
for n 0 at p = 2, discovered by Mahowald [33] and extended to odd primes by T*
*hompson [39].
This gives
1S2n+1 ' 2n+1( 1 RP2n)K=2
by 2.2 and leads to a computation of the groups v-11ß*S2n+1= ß* 1S2n+1at p = 2 *
*(see [23]
and Theorem 8.9). Proceeding more generally, we now show that each space with a*
* 1-exponent
(including each spherically resolved space and each simply-connected finite H-s*
*pace) becomes v1-
periodically equivalent to an infinite loop space after finite looping.
Theorem 2.10.For a space X with a 1-exponent pr, there is a natural v1-periodi*
*c equivalence
~: mX ! 1 m 1X for sufficiently large m (depending on pr).
This will be proved later in 2.14. We now consider the induced natural homomo*
*rphism
~*:ßiX -! v-11ßiX
defined for sufficiently large i when X has a 1-exponent. To show that ~*is sp*
*littably epic for large
i, we must assume that v-11ß*(X; Z=p), or equivalently v-11ß*X, is of finite ty*
*pe. This condition
holds whenever X is spherically resolved [23], or X is a simply-connected compa*
*ct Lie group (see 9.1
and Theorem 9.3), or X is a rationally associative finite H-space for p odd [19*
*]. Slightly generalizing
a result of [26] or [23], we have:
Theorem 2.11.If X is a space with a 1-exponent pr and with v-11ß*(X; Z=p) of f*
*inite type, then
~*:ßiX ! v-11ß1X is splittably epic for sufficiently large i.
Proof.Whenever an abelian sequence G0! G1! G2. .h.as a finitely generated colim*
*it G1 , the
colimit map Gm ! G1 must be splittably epic for sufficienly large m. Thus, sinc*
*e v-11ß*(X; Z=pr) is
of finite type, the colimit map ßi(X; Z=pr) ! v-11ßi(X; Z=pr) must be splittabl*
*y epic for sufficiently
large i. In the commutative diagram
ßi(X; Z=pr)-@*---!ßi-1X
?? ?
y~* ?y~*
ßi( 1X; Z=pr)@*----!ßi-1( X)
for such i, the left ~* is splittably epic since it is equivalent to the colimi*
*t map, and the bottom @*
is splittably epic since pr ' 0: 1X ! 1X. Hence, the right ~* is also splitta*
*bly epic, and the
lemma follows. |__|
We now prepare to prove Theorem 2.10, relying on work in [20].
8
2.12. The category W. Let d1 be the integer defined in [20] with d1= 3 for p od*
*d and d1= 3, 4,
or 5 (but not known) for p = 2. Let W be the category of which an object is a d*
*1-connected finite
p-torsion complex W 2 Ho*equipped with a v1-map !: mW ! W in Ho*for some m > 0*
*, such
that ! is a v1-periodic equivalence, and of which a map f :(W, !) ! (W0, !0) is*
* a map f :W ! W0
in Ho*with f!i= (!0)j( kf) for some i, j, k > 0. Note that any finite p-torsion*
* complex with a
v1-map in Ho*may be suspended to give an object of W by 2.9, and that the categ*
*ory W is closed
under the suspension functor. Also, for objects (W, !) and (W0, !0) of W, any m*
*ap W ! W0in Ho*
may be finitely suspended to give a map in W.
Lemma 2.13. For an object (W, !) 2 W and space X 2 Ho*, there is a natural v1-p*
*eriodic equiva-
lence h: XW ! 1 ( 1X)W which respects the suspension in W.
Proof.Since the map !: mW ! W is an Lf1-equivalence by [20, Corollary 4.8], th*
*ere is a natural
isomorphism [W, Lf1X] ~=v-11ß0(Lf1X; W). This composes with the natural isomorp*
*hisms
v-11ß0(Lf1X; W) ~= [W, 1Lf1X] ~= [ 1 1 W, Lf1X]
of [20, Corollary 5.9] to induce a natural equivalence 1 1 W ' Lf1W, and hence*
* (Lf1X)W '
1 ( 1X)W by [20, Theorem 5.4]. This combines with the v1-periodic equivalence *
*XW ! (Lf1X)W
to give the desired h. |_*
*_|
2.14. Proof of Theorem 2.10. Choose a sequence of objects (Wi, !i) and maps
ffi: mi+1-mi(Wi, !i) -! (Wi+1, !i+1)
in W for i 1 using Moore spaces
Wi = Mmi(pi) = Smi-1[piemi
with canonical maps ffi:Mmi+1(pi) ! Mmi+1(pi+1) in Ho*. Let q: Wi! Smi be the p*
*inching map.
Using a nullhomotopy pr ' 0: 1X ! 1X, choose a left inverse fl to the map q*:*
* mr 1X !
( 1X)Wr, and let fl :( 1X)Wi ! mi 1X be the induced map for i r. Note that w*
*hen i 2r,
the map fl does not depend on the choice of left inverse. We claim that the com*
*position
* h 1fl
miX q-!XWi -! 1 ( 1X)Wi ---! 1 mi 1X
is a v1-periodic equivalence for i r. For this it suffices to show that the m*
*ap of constant towers
{v-11ß* miX}i -! {v-11ß* 1 mi 1X}i
is a pro-isomorphism and hence an isomorphism. This follows since {v-11ß*(-)}ic*
*arries h to an
isomorphism by Lemma 2.13 and also carries q* and 1 fl to pro-isomorphisms bec*
*ause {ß*(-)}i
9
carries q*: mi 1X ! ( 1X)Wi to a pro-isomorphism. The required map ~ of Theore*
*m 2.10 is
now given by miX ! 1 mi 1X for i = 2r. |_*
*_|
3.Pontrjagin duality for united p-adic K-theory
Working at a prime p, we wish to study the spectra 1X using the methods of u*
*nited K-theory
[16]. In preparation, we now establish a Pontrjagin duality (Theorem 3.1) relat*
*ing the united p-adic
K-cohomology
* *
K*CRT(E; ^Zp) = {K*(E; ^Zp), KO*(E; ^Zp), KT*(E; ^Zp)} def={Kb*E, dKOE, dKTE}
of a spectrum E to the united p-adic K-homology
__ ___ ___
KCRT*(E; Zp1) = {K*(E; Zp1), KO*(E; Zp1), KT*(E; Zp1)} def={K *E, KO*E, KT*E}
~={K*-1(øpE), KO*-1(øpE), KT*-1(øpE)} = KCRT*-1(øpE)
thereby extending previous work of [5], [19, Corollary 2.3], and [40]. We also *
*give a basic application
of this duality showing that the v1-periodic homotopy groups of a space X can b*
*e extracted from the
p-adic KO-cohomology of 1X (Theorem 3.2). For a locally compact Hausdorff abel*
*ian group G, the
Pontrjagin dual G# is given by Homcont(G, R=Z). This restricts to a duality bet*
*ween the categories
of discrete abelian groups and compact Hausdorff abelian groups, with the p-tor*
*sion groups corre-
__ *
* ___ ___
sponding to the p-profinite groups. We consider the p-torsion homologies {K *E,*
* KO*E, KT*E}
* *
and the p-profinite cohomologies {Kb*E, dKOE, dKTE} equipped with stable Adams *
*operations _k
for p-local units k 2 Zx(p)(or more generally for p-adic units k 2 ^Zxp).
__ *
* *-4
Theorem 3.1.For a spectrum E, there are natural dualities eC :bK*E ~=(K *E)#, e*
*R :dKO E ~=
___ *-3 ___
(KO *E)#, and eT: dKT E ~=(KT *E)# such that
__ __ ___ ___
(i)the stable Adams operations _k: K*E ~=K*E, _k: KO*E ~=KO *E, and
___ ___ *-4
_k: KT*E ~=KT*E respectively dualize to _1=k:bK*E ~=bK*E, k2_1=k:dKO E ~=
KdO*-4E, and k_1=k:dKT*-3E ~=dKT*-3E for each k 2 Zx(p);
__ __ ___ ___ ___
(ii)the periodicities B :K*E ~=K*+2E, BR :KO *E ~=KO*+8E, and BT: KT*E ~=
___ *+4 *-4
KT *+4E respectively dualize to B :bK*+2E ~=bK*E, BR :dKO E ~=dKO E,
*+1 *-3
and BT: dKT E ~=dKT E;
___ ___ ___ ___
(iii)the Hopf operations j: KO *E ! KO *+1E and j: KT *E ! KT *+1E re-
*-3 *-4 *-2 *-3
spectively dualize to j: dKO E ! dKO E and j: dKT E ! dKT E;
___ __ __ ___
(iv)the complexification c: KO *E ! K*E and the realification r: K*E ! KO *E
*-4 *-4
respectively dualize to rB2: bK*E ! dKO E and B-2c: dKO E ! bK*E;
10
___ ___ ___ ___ ___
(v)the operations ffl: KO *E ! KT *E, ø :KT *E ! KO *+1E, i :KT *E !
__ __ ___ *-3 *-4
K *E, and fl :K*E ! KT*-1E respectively dualize to ø :dKT E ! dKO E,
*-3 *-3 *-3 *-4
ffl: dKO E ! dKT E, flB2: bK*E ! dKT E, and B-2i :dKT E !
Kb*E.
Before proving this theorem, we apply part (i) to derive the promised result *
*on the v1-periodic
homotopy groups of a space. Let r 2 Zx(p)be a unit which generates (Z=p)x when *
*p is odd and such
that r 3 mod 8 when p = 2.
Theorem 3.2.For a space X 2 Ho*and for a spectrum E 2 Hos, there are natural lo*
*ng exact
sequences
n _r-r2 n -1 # n+1 _r-r2
. .-.! dKO 1X ----!dKO 1X -! (v1 ßn+3X) -! dKO 1X ----!. . .
n _r-r2 n # n+1 _r-r2
. .-.! dKOE ----!dKO E -! (ßn+3øpEK=p) -! dKO E ----!. . .
Proof.For a p-torsion K*-local spectrum F, there is a natural long exact sequen*
*ce
r-1 _r-1
. .-.! KOnF _---!KOnF -! ßn-1F -! KOn-1F ---! . . .
obtained by [13, Corollary 4.4] or by using the KCRT*-Adams spectral sequence o*
*f [16]. After r is
___
replaced by 1=r and KO*F is expressed as KO *+1F, this sequence dualizes to giv*
*e a natural long
exact sequence
n r-2_r-1 n # n+1 r-2_r-1
. .-.! dKOF ------!dKO F -! (ßn+3F) -! dKO F ------!. . .
by Theorem 3.1. The result now follows by taking F = øp 1X or F = øpE and using*
* Theorem
2.4. |__|
Note that the homotopy groups ß*EK=pmay be determined from ß*øpEK=pas in [13,*
* Proposition
2.5]. The proof of Theorem 3.1 will be based on:
3.3. Brown-Comenetz duality. As in [22], for a spectrum E 2 Hos, the Brown-Come*
*netz dual
^cE is the function spectrum F(E, ^cS) where ^cS is determined by the natural e*
*quivalence [Y, ^cS] =
(ß0Y )# for Y 2 Hos. The associated cohomology theory has universal coefficient*
* isomorphisms
(^cE)nY ~=(EnY )#
for all Y 2 Hosand n 2 Z, and thus
ßn(^cE) ~= (ß-nE)#.
11
The Brown-Comenetz functor ^crestricts to a contravariant equivalence from the *
*homotopy category
of spectra whose homotopy groups are finite direct sums of Z=pj's and Zp1's to *
*the homotopy cate-
gory of spectra whose homotopy groups are finite direct sums of Z=pj's and ^Zp'*
*s. For a commutative
ring spectrum R and an R-module spectrum E, note that the Brown-Comenetz dual ^*
*cE inherits an
R-module spectrum structure from E.
We may now view ^c(KZp1), ^c(KOZp1), ^c(KTZp1) as module spectra over the com*
*mutative
ring spectra bKp, dKOp, and dKTp, since they are obtained from the module spect*
*ra KZp1 ' bKpZp1,
KOZp1 ' dKOpZp1, and KTZp1 ' dKTpZp1. Let fflC 2 ß0^c(KZp1), fflR 2 ß-4^c(KOZp1*
*), and
fflT 2 ß-3^c(KTZp1) be the elements corresponding to 1 2 ^Zpunder the isomorphi*
*sms
^Zp~=(ß0KZp1)# ~= (ß4KOZp1)# ~= (ß3KTZp1)#
induced by rB2: ß0KZp1 ~=ß4KOZp1 and ø :ß3KTZp1 ~=ß4KOZp1. Then let eC :bKp!
^c(KZp1) be the bKp-module map with eC(1) = fflC; let eR : -4dKOp! ^c(KOZp1) be*
* the dKOp-
module map with eR(1) = fflR; and let eT: -3dKTp! ^c(KTZp1) be the dKTp-module*
* map with
eT(1) = fflT.
Lemma 3.4. The maps eC :bKp! ^c(KZp1), eR : -4dKOp! ^c(KOZp1), and eT: -3dKTp!
^c(KTZp1) are equivalences.
Proof.Using [16, Section 2.5], we see that: ß*^c(KZp1) is a free ß*bKp-module o*
*n fflC; ß*^c(KOZp1)
is a free ß*dKOp-module on fflR; and ß*^c(KTZp1) is a free ß*dKTp-module on ffl*
*T. Thus the given
maps are ß*-equivalences. |_*
*_|
This lemma combines with 3.3 to give natural duality isomorphisms
eC :K*(E; ^Zp) ~= K*(E; Zp1)#
eR :KO*-4(E; ^Zp) ~= KO*(E; Zp1)#
eT: KT*-3(E; ^Zp) ~= KT*(E; Zp1)#
for a spectrum E.
Lemma 3.5. The parts (ii)-(v) of Theorem 3.1 hold for the above dualities eC, e*
*R, and eT.
Proof.Each of the homology operations in (ii)-(v) is represented by some dKOp-m*
*odule map OE 2
[EZp1, FZp1]* for spectra E, F 2 {K, KO, KT}. The map OE dualizes via Lemma 3.4*
* to a dKOp-
module map ^cOE 2 [Fbp, bEp]* which must be shown equal to some specified dKOp-*
*module map OE02
[Fbp, bEp]*. In each case, we check that ^cOE and OE0induce the same ß*-homomo*
*rphisms, and we
12
conclude that ^cOE = OE0since all dKOp-module maps in [Fbp, bEp]*, except for s*
*ome irrelevant ones in
[dKTp, dKTp]4n+2, are detected by ß* (see [16, Section 1.9]). *
* |__|
We must deal separately with part (i) of Theorem 3.1 since it involves p-adic*
* Adams operations
which are not represented by dKOp-module maps. We first show:
Lemma 3.6. For a spectrum E 2 {K, KO, KT}, the homology E*E is a free E*-module*
* on gener-
ators in degree 0 for E = K, in degree 0 for E = KO, and in degrees 0 and 3 for*
* E = KT.
Proof.This was shown for E = K by Adams-Clarke [4] and is presumably known for *
*E = KO. In
general, we may rely on the formula KCRT*E = ~U(ßCRT*E) of [16, Theorem 8.2], u*
*sing the exact
functor ~U:Inv ! A of [16, Proposition 6.6]. The result of [4] now shows that ~*
*U(Z _-1Z) is
free abelian, and hence ~U(M) is also free abelian for M = Z with _-1 = 1 sinc*
*e M Z _-1Z.
Thus K*E ~=~U(ßC*E) is free abelian for each E 2 {K, KO, KT}, and KO*E ~=~U(ßR**
*E) is the
underlying KO*-module of a free E*-module on 0-dimensional generators. Hence, K*
*CRT*E is a free
CRT-module by [16, Theorem 3.2] on generators in K0E for E = K, in KO0E for E =*
* KO, and in
KT3E for E = KT by the structural results of [16, Section 2.4]. The lemma now f*
*ollows from these
same structural results. |_*
*_|
Lemma 3.7. The maps in [Kbp, bKp]i, [dKOp, dKOp]i, and [dKTp, dKTp]iare all det*
*ected by rational
homotopy groups Q ß* with the exception of the maps in [dKO2, dKO2]ifor i 1*
*, 2 mod 8 and in
[dKT2, dKT2]ifor i 1, 2 mod 4.
Proof.Let E denote K, KO, or KT. Since [Ebp, bEp]i~=[E, -ibEp], it suffices to*
* show that the maps
in [E, N] are detected by Q ß* whenever N is an E-module spectrum with ß0N to*
*rsion-free and
also with ß3N torsion-free when E = KT. There are natural universal coefficient*
* isomorphisms
[E, N] ~= HomCRT(KCRT*E, ßCRT*N) ~= HomE*(E*E, ß*N)
obtained from [16, Section 9.6] (or [3] when E = K or E = KO) using the freenes*
*s results of Lemma
3.6. Thus the rationalization N ! NQ induces a monomorphism [E, N] ! [E, NQ], a*
*nd hence the
maps in [E, N] are detected by ß* Q as required. *
* |__|
3.8. Proof of Theorem 3.1. Using the dualities eC, eR, and eT of Lemma 3.4, we *
*have proved
parts (ii)-(v) of the theorem in Lemma 3.5, and we easily deduce part (i) from *
*Lemma 3.7. |__|
13
4.Even and odd CR-modules
For many interesting spaces X, including simply-connected compact Lie groups,*
* the complex p-
adic K-theory K*( 1X; ^Zp) is concentrated in even or odd degrees. In such case*
*s, the united p-adic
K-theory of 1X is completely captured by
K*CR( 1X; ^Zp) = {K*( 1X; ^Zp), KO*( 1X; ^Zp)}
without KT*( 1X; ^Zp), and in fact is largely captured by a small part of K*CR(*
* 1X; ^Zp) which
we call a -module. In this section, we first recall the underlying theory of *
*CR-modules from
[16, Section 4.7] using cohomological indexing and working over an arbitrary ab*
*elian category. We
then develop some crucial special properties of even and odd CR-modules, and fi*
*nally introduce the
theory of -modules.
4.1. CR-modules. A CR-module over an abelian category M consists of a pair M = *
*{M*C, M*R}
of Z-graded objects in M with operations
B :M*C~=M*-2C t: M*C! M*C BR :M*R~=M*-8R
j: M*R! M*-1R c: M*R! M*C r: M*C! M*R
satisfying the relations
2j = 0 j3= 0 jBR = BRj jr = 0 cj = 0
t2= 1 tB = -Bt rt = r tc = c cBR = B4c
rB4 = BRr cr = 1 + t rc = 2 rBc = j2 rB-1c = 0
We may sometimes write _-1z or z* in place of tz for z 2 M*C. We let CRM deno*
*te the abelian
category of CR-modules over M. For a CR-module M, we call
-1 j
. .-.! M*+1Rj--!M*R-c-!M*CrB---!M*+2R--!. . .
the Bott sequence and call M Bott exact when this chain complex is exact. We al*
*so call M CR-exact
[16, Section 4.7] when it is Bott exact and the chain complex
. .-.! M*+1R=r -j-!M*R=r -j-!M*-1R=r -j-!. . .
is exact. Some examples of Bott exact CR-modules are
K*CR(E; G) = {K*(E; G), KO*(E; G)} KCR*(E; G) = {K*(E; G), KO*(E; G)}
for arbitrary spectra E and coefficients G, where the operations come from stan*
*dard maps for the
spectra K and KO [16, Section 1.9] with Bott exactness shown by [16, Section 1.*
*11]. Whenever
14
K*CR(E; G) or KCR*(E; G) is CR-exact, it prolongs canonically to give the group*
*s KT*(E; G) or
KT*(E; G), which become superfluous [16, Theorem 4.15].
4.2. Even and odd Bott exact CR-modules. Let M be a Bott exact CR-module over an
abelian category M. We call M even or odd when M*Cvanishes in the opposite degr*
*ees, allowing
us to determine M*Cperiodically from a single term MnCwhere Mn-1C= 0. This impl*
*ies that M is
CR-exact since j: MiR=r ~=Mi-1R=r for i n mod 2. Moreover, by the following t*
*heorem, we can
largely determine M*Rfrom the single triad of terms {MnC, MnR, Mn-4R} with oper*
*ations
t: MnC! MnC c: MnR! MnC r: MnC! MnR c0:Mn-4R! MnC q: MnC! Mn-4R
where c0= B-2c and q = rB2.
Theorem 4.3.Suppose M is a Bott exact CR-module over M with Mn-1C= 0 for some n*
*. Then
there are natural isomorphisms
8
>>>MnR for i 0 mod 8
>>> n
>> MRn-\c4
>>>MR for i 4 mod 8
>>>Mn-4=q for i 5 mod 8
>: R
MnR\c for i 7 mod 8
Moreover, the terms Mn+8k-2R~=Mn-2Rand Mn+8k-6R~=Mn-6Rbelong to natural extensi*
*ons
2 B-1c
0 ----! MnR=r--j--!Mn-2R----! MnC\r ----! 0
?? ? ?
yc ?y1 ?yq
2
0 ----!MnC=c0--rB--!Mn-2R-j---!Mn-4R\c0----!0
2 B-3c
0 ----! Mn-4R=q-j---!Mn-6R----!MnC\q----! 0
?? ? ?
yc0 ?y1 ?yr
3 B-1Rj2
0 ----! MnC=c -rB---!Mn-6R----!MnR\c----!0
where the horizontal maps satisfy (rB)(B-1c) = 2, (B-1c)(rB) = 1 - t, (rB3)(B-3*
*c) = 2, and
(B-3c)(rB3) = 1 - t.
Proof.This follows easily from Bott exactness and the CR-module relations. *
* |__|
We shall see later (Theorem 4.11) that the "difficult" terms Mn-2Rand Mn-6Rma*
*y actually be
recovered up to isomorphism (though not functorially) from the given triad {MnC*
*, MnR, Mn-4R}, at
15
least when M is the category of abelian groups or p-profinite abelian groups. W*
*e now obtain a
crucial exactness result for {MnC, MnR, Mn-4R}.
Theorem 4.4.Suppose M is a Bott exact CR-module over M with Mn-1C= 0 for some n*
*. Then
there is an exact sequence
0 1-t (r,q)
. .-.! MnC-(r,q)---!MnR Mn-4R-c-c---!MnC---! MnC----!MnR Mn-4R-! . . .
Proof.This follows from the exact sequences
0 -! MnR=r -c-!MnC=c0-1-t--!MnC\r -q-!Mn-4R\c0-! 0
0 1-t r
0 -! Mn-4R=q -c-!MnC=c ---! MnC\q --!MnR\c -! 0
associated with the ladders of extensions in Theorem 4.3. *
* |__|
In view of the preceding theorems, we introduce:
4.5. -modules. A -module over an abelian category M consists of a triad A = {*
*AC, AR, AH}
of objects in M with operations
t: AC ! AC c: AR ! AC r: AC ! AR c0:AH ! AC q: AC ! AH
satisfying the relations
t2 = 1 cr = 1 + t rc = 2 tc = c rt = r
c0q = 1 + t qc0 = 2 tc0 = c0 qt = q
We may sometimes write _-1z or z* in place of tz for z 2 AC. We let M denote *
*the abelian
category of -modules over M, and we say that a -module A is exact when the ch*
*ain complex
0 1-t (r,q)
. .-.! AC -(r,q)---!AR AH -c-c---!AC ---! AC ----!AR AH -! . . .
is exact. Hence, if two -modules in a short exact sequence are exact, then so *
*is the third. For a
CR-module M 2 CRM and integer n, we obtain a -module
nM = {MnC, MnR, Mn-4R} 2 M
which is exact whenever M is Bott exact with Mn-1C= 0.
16
4.6. Other examples of exact -modules. For a compact Lie group G, there is a *
*-module
R (G) = {R(G), RR(G), RH(G)} consisting of the complex representation ring R(G)*
* with its real
and quaternionic parts RR(G), RH(G) R(G) linked by the standard operations. T*
*his -module
is always exact since it is freely generated by irreducible representations of *
*complex, real,and
quaternionic types (see [2] and [12]). For an abelian group N with involution *
*t: N ~=N, we
let N+ = N\(1 - t) and N+ = N=(1 - t). Then there are exact -modules {N, N+ ,*
* N+} and
{N, N+, N+ } with obvious operations. When t = 1: N ~=N and the map N\2 ! N=2 i*
*s trivial
(e.g. when t = 1: Z=2k ~=Z=2k for k > 1), there are also exact -modules {2N, *
*N, 4N} and
{2N, 4N, N} with obvious operations and there are generally many more.
For an object N 2 M with involution t: N ~=N, we write h+N = ker(1 - t)= im(1*
* + t) and
h-N = ker(1 + t)= im(1 - t) for the associated cohomologies. It is straightforw*
*ard to show:
Lemma 4.7. For an exact -module M 2 M , there are isomorphisms
c + c0:MR=r MH=q ~= h+MC
(r, q): h-MC ~= MR\c MH\c0
We suspect that exact -modules are determined up to isomorphism by their com*
*plex parts
together with their direct sum splittings of h+ and h- terms, but we shall not *
*pursue this here.
Instead, we use these terms to give comparison lemmas for exact -modules and C*
*R-modules.
Lemma 4.8. Suppose f :L ! M is a map of exact -modules over M. Then
(i)f is an isomorphism if and only if fC :LC ! MC is an isomorphism;
(ii)f is epic if and only if fC :LC ! MC and f*:h+LC ! h+MC are both epic;
(iii)f is monic if and only if fC :LC ! MC and f*:h-LC ! h-MC are both
monic.
Proof.In view of Lemma 4.7, part (ii) follows by comparing the sequences LC ! L*
*R ! LR=r ! 0
and MC ! MR ! MR=q ! 0, as well as their quaternionic counterparts. The other p*
*arts follow
similarly. |__|
Lemma 4.9. Suppose f :L ! M is a map of Bott exact CR-modules over M with Ln-1C*
*= 0 and
Mn-1C= 0 for some n. Then
(i)f is an isomorphism if and only if fC :LnC! MnCis an isomorphism;
(ii)f is epic (resp. monic) if and only if f :LnC! MnC, f*:h+LnC! h+MnC,
and f*:h-LnC! h-MnCare all epic (resp. monic).
17
Proof.This follows by combining Lemma 4.8 with Theorems 4.3 and 4.4. *
* |__|
In the remainder of this section, we shall establish a very close corresponde*
*nce between exact
-modules and even or odd Bott exact CR-modules. We first consider
4.10. Adjoints of n. For any n, the functor n: CRM ! M has a left adjoint CR*
*(-, n): M !
CR M where CR(M, n)n-iC~=MC for i even, CR(M, n)n-iC~=0 for i odd, and
8
>>>MR for i 0 mod 8
>>>M =r for i 1 mod 8
>>> R
>0 for i 3, 7 mod 8
>>>
>>>MH for i 4 mod 8
>>>MH=q for i 5 mod 8
:
MC=c for i 6 mod 8
Hence n(CR(M, n)) = M, and if M is an exact -module with h-MC = 0 (i.e. with *
*c and c0
monic), then CR(M, n) is a Bott exact CR-module. The functor n: CRM ! M also *
*has a right
adjoint CR0(-, n): M ! CRM where CR0(M, n)n-iC~=MC for i even, CR0(M, n)n-iC~=*
*0 for i
odd, and 8
>>>MR for i 0 mod 8
>>>0 for i 1, 5 mod 8
>>>
>MH\c0 for i 3 mod 8
>>>
>>>MH for i 4 mod 8
>>>MC\q for i 6 mod 8
:
MR\c for i 7 mod 8
Hence n(CR0(M, n)) = M, and if M is an exact -module with h+MC = 0 (i.e. with*
* r and q
epic), then CR0(M, n) is a Bott exact CR-module.
We now address the general problem of prolonging an exact -module M 2 M to *
*give a Bott
__ __ __n-1
exact CR-module M 2 CRM with nM = M and M C = 0 for some integer n. When h-MC*
* = 0
__
or h+MC = 0, such an M will be given by the above CR(M, n) or CR0(M, n). Howeve*
*r, when
*
* __
M is the exact -module {Z=2, Z=2, Z=2} with c = 1 and q = 1, such a prolongati*
*on M cannot
exist in the category of Z=2-modules. Fortunately, this trouble disappears in o*
*ur preferred abelian
categories.
Theorem 4.11.Suppose M is the category of abelian groups or p-profinite abelian*
* groups. For
__
an exact -module M 2 M and integer n, there exists a Bott exact CR-module M *
*2 CRM with
__ __n-1 __
nM = M and M C = 0. Moreover, M is unique up to (noncanonical) isomorphism.
18
Thus, the isomorphism classes of even or odd Bott exact CR-modules over M cor*
*respond to the
isomorphism classes of exact -modules over M via the functor n for an even or*
* odd n. This
theorem will be proved below in 4.13 using:
4.12. Special resolutions. In the category of abelian groups, we consider the f*
*ollowing elementary
exact -modules: F(C) = {Z tZ, Z, Z} with c = 1 + t and c0= 1 + t; F(R) = {Z,*
* Z, Z} with t = 1,
c = 1, and q = 1; F(H) = {Z, Z, Z} with t = 1, c0= 1, and r = 1; F0(R) = {Z, Z=*
*2, 0} with t = -1
and r onto; and F0(H) = {Z, 0, Z=2} with t = -1 and q onto. A -module is calle*
*d free when it
is a direct sum on copies of F(C), F(R), and F(H); it is called parafree when i*
*t is a direct sum of
copies of F0(R) and F0(H). For an abelian exact -module M, we use Lemma 4.7 to*
* construct a
parafree -module F0 and map F0! M inducing an isomorphism h-F0C~=h-MC. We then*
* find
a free -module F and map F ! M such that F F0 ! M is onto. This determines a*
* short
exact sequence of -modules 0 ! eF! F F0! M ! 0 called a special resolution o*
*f M. The
-module eFis exact with eFCfree abelian and with h-FeC= 0. Hence eFis a free *
*-module by [16,
Proposition 4.8] applied to CR(Fe, 0).
4.13. Proof of Theorem 4.11. It suffices to prove the theorem for abelian group*
*s, since it then
follows for p-profinite abelian groups by Pontrjagin dualization. Let 0 ! eF! F*
* F0! M ! 0 be
a special resolution for the exact -module M. Then the induced map of CR-modul*
*es CR(Fe, n) !
CR(F, n) CR0(F0, n) is monic by Lemma 4.9, and one easily checks that its coker*
*nel is the required
__ __
M . The desired uniqueness follows from a more general property of this M . Nam*
*ely, for any Bott
exact CR-module N with Nn-1C= 0, we claim that each -module map M ! nN prolon*
*gs
__
(nonuniquely) to a CR-module map M ! N. For this, it is fairly straightforward *
*to check that
the induced map F F0 ! nN prolongs (nonuniquely) to a map CR(F, n) CR0(F0,*
* n) ! N,
__
which must be trivial on CR(Fe, n). The desired map M ! N is now obtained by di*
*viding out
CR(F~, n). |__|
We conclude with a technical lemma for later use.
Lemma 4.14. Suppose 0 ! M0! M ! M00! 0 is a short exact sequence of exact -m*
*odules
over an abelian category M with h-M0C= 0 and h-MC = 0. Then there are natural e*
*xact sequences
0 -! M00R\c -! M0C=c -! MC=c -! M00C=c -! 0
0 -! M00R\c -! M0H=q -! MH=q -! M00H=q -! 0
0 -! M00H\c0-! M0C=c0-! MC=c0-! M00C=c0-! 0
19
0 -! M00H\c0-! M0R=r -! MR=r -! M00R=r -! 0
Proof.Since M0R\c = 0 and MR\c = 0, we obtain the first exact sequence by the s*
*erpent lemma.
We then obtain the second exact sequence from the first by using the isomorphis*
*m of the left and
middle kernels in the ladder of exact sequences
0
0 ----! M0H=q--c--!M0C=c----!M0C=M0+C----!0
?? ? ?
y ?y ?y
0
0 ----! MH=q --c--!MC=c ----!MC=M+C ----!0
where the right kernel is trivial. We obtain the third and fourth exact sequenc*
*es similarly. |__|
Roughly speaking, the exact sequences in this lemma combine to give a long ex*
*act sequence for
the real parts of the associated Bott exact CR-modules.
5. Even and odd K=p*-local spectra
Using results of [16], we now obtain an algebraic classification of the K=p*-*
*local spectra E whose
complex K-cohomologies K*(E; ^Zp) are concentrated in even or odd degrees. This*
* classification will
depend only on the CR-modules K*CR(E; ^Zp) together with their stable Adams ope*
*rations and will
apply to most of the spectra E = 1X of interest on this paper.
5.1. Stable p-adic Adams modules. By a finite stable p-adic Adams module we mea*
*n a finite
abelian p-group G with automorphisms _k: G ~=G for k 2 Zx(p)such that:
(i)_1= 1 and _j_k = _jkfor all j, k 2 Zx(p);
(ii)for a sufficiently large integer n, the condition j k mod pn implies _j=*
* _k
on G.
By a stable p-adic Adams module we mean the topological inverse limit of an inv*
*erse system of
finite stable p-adic Adams modules. Such a module G has an underlying p-profini*
*te abelian group
structure with continuous automorphisms _k: G ~=G for k 2 Zx(p). In fact, a sta*
*ble p-adic Adams
module is just the Pontrjagin dual of a stable p-torsion Adams module in the se*
*nse of [14, Section
1] or [16, Section 5.1]. We let ^Abe the abelian category of stable p-adic Adam*
*s modules, and we
let ~Si:^A! ^A, for i 2 Z, be the functor with ~SiG equal to G as a group but w*
*ith _k: ~SiG ~=~SiG
equal to ki_k: G ~=G for k 2 Zx(p).
20
5.2. A^CR-modules. By an ^ACR-module we mean a CR-module M of stable p-adic Ada*
*ms modules
with _-1 = t in M*Cand with _-1 = 1 in M*R, where the homomorphisms
B :~SM*C~=M*-2C BR :~S4M*R~=M*-8R j: M*R! M*-1R
c: M*R! M*C r: M*C! M*R
are all maps in ^A. Equivalently, an ^ACR-module is just the Pontrjagin dual of*
* a p-torsion ACR-
module in the sense of [16, Section 5.5], where the duality is taken with respe*
*ct to the componentwise
rules of Theorem 3.1. We let ^ACRbe the abelian category of ^ACR-modules. The m*
*ain examples
of ^ACR-modules are the cohomologies
K*CR(E; ^Zp) = {K*(E; ^Zp), KO*(E; ^Zp)}
for arbitrary spectra E, which are Pontrjagin dual to the p-torsion ACR-modules
KCR*(E; Zp1) = {K*(E; Zp1), KO*(E; Zp1)} ~={K*-1(øpE), KO*-1(øpE)} = KCR*-1(øpE)
by Theorem 3.1. We can now give our main classification theorem for even or od*
*d K=p*-local
spectra.
Theorem 5.3.Suppose M is an even or odd Bott exact ^ACR-module. Then there exis*
*ts a K=p*-
local spectrum E with K*CR(E; ^Zp) ~=M, and E is unique up to (noncanonical) eq*
*uivalence.
Proof.The Pontrjagin dual M# is a p-torsion Bott exact ACR-module which prolong*
*s canonically
to a p-torsion CRT-exact ACRT-module by [16, Lemma 4.14]. Hence, there exists a*
* p-torsion K*-
local spectrum X with KCR*(X; Zp1) ~=KCR*-1X ~=M# by [16, Theorem 10.1], and th*
*e spectrum
E = ^Xphas the desired properties. It is unique by Theorem 5.4 below. *
* |__|
Theorem 5.3 shows that the homotopy types of even or odd K=p*-local spectra c*
*orrespond to the
isomorphism classes of even or odd Bott exact ^ACR-modules. We have used:
Theorem 5.4.Suppose E and F are K=p*-local spectra with Kn-1(E; ^Zp) = 0 and Kn*
*-1(F; ^Zp) =
0 for some n. Then, for each ^ACR-module homomorphism OE: K*CR(F; ^Zp) ! K*CR(E*
*; ^Zp), there
exists a map f :E ! F with OE = f*.
Proof.It suffices to prove the corresponding result for the p-torsion K-local s*
*pectra øpE and øpF,
and that result follows from [16, Section 9.8] by dualization as in the proof o*
*f Theorem 5.3. |__|
We remark that the map f in this theorem is generally not unique. For instanc*
*e, there is a map
f 6= 0: SK=2! SK=2of order 2 with f* = 0 on K*CR(SK=2; ^Z2).
21
6.Unstable operations in p-adic K-theory
For a space X and integer n, we may approach the stable -module
Kn( 1X; ^Zp) = {Kn( 1X; ^Zp), KOn( 1X; ^Zp), KOn-4( 1X; ^Zp)}
in favorable cases by starting with the corresponding unstable -module
eKn(X; ^Zp) = {Ken(X; ^Zp), gKOn(X; ^Zp), gKOn-4(X; ^Zp)}
and dividing out by its " 1-trivial" part. In preparation, we now discuss vario*
*us unstable operations
in p-adic K-theory. Although some of these operations are well-known, others ma*
*y not be, and we
shall explain how they may be constructed using Atiyah's Real K-theory [7]. We *
*start by recalling:
6.1. ~-rings without identity. A ~-ring without identity consists of a commutat*
*ive ring A without
identity together with functions ~m :A ! A for m = 1, 2, 3, . .w.here ~1(a) = a*
* and where the
usual expressions for ~m(a + b), ~m(ab), and ~m~n(a) hold when a, b 2 A and m, *
*n 1 (see Atiyah-
Tall [10]). We note that such an A may be viewed as the augmentation ideal of a*
* ~-ring Z A
with identity 1 2 Z. We call such an A semigraded when it has the form A = A0 *
*A1 with the
Z=2-gradation properties that for all elements x, y 2 A0, u, v 2 A1, and m, k *
* 1, the following
conditions hold: xy 2 A0, xu 2 A1, uv 2 A0, ~mx 2 A0, ~2ku 2 A0, and ~2k-*
*1u 2 A1. Of
course, uv = vu instead of uv = -vu for u, v 2 A1. For a compact Lie group G, w*
*e note that
the augmentation ideals eR(G) and eRR(G) eRH(G) are ~-rings without identity,*
* where the latter is
semigraded. We may view Re (G) = {Re(G), eRR(G), eRH(G)} as a prototype for:
6.2. ~-rings. A ~-ring A consists of a ~-ring AC without identity and a semig*
*raded ~-ring
AR AH without identity together with a -module structure on {AC, AR, AH} suc*
*h that the
following conditions hold for all elements z, w 2 AC, x, y 2 AR, u, v 2 AH, and*
* n, k 1:
(i)c(xy) = (cx)(cy), c(uv) = (c0u)(c0v), c0(xu) = (cx)(c0u), and (zw)* = z*w*
where -* denotes t-;
(ii)(rz)x = r(z(cx)), (rz)u = q(z(c0u)), (qz)x = q(z(cx)), and (qz)u = r(z(c0u*
*));
(iii)c(~mx) = ~m(cx), c(~2ku) = ~2k(c0u), c0(~2k-1u) = ~2k-1(c0u), and
(~mz)*= ~m(z*);
(iv)the operation ~OE:AC ! AR given by ~OE(z) = ~2(rz) - r(~2z) has the proper-
ties c~OE(z) = z*z, O~E(zw) = (~OEz)(~OEw), and O~E(z + w) = ~OEz + ~OEw + *
*r(z*w);
22
(v)using the operation ~OE:AC ! AR, we have
k-1X
~2k(qz) = ~2k(rz) = r(~2kz) + ~OE(~kz) + r (~iz)(~2k-iz*)
i=1
k-1X
~2k-1(rz) = r(~2k-1z) + r (~iz)(~2k-1-iz*)
i=1
k-1X
~2k-1(qz) = q(~2k-1z) + q (~iz)(~2k-1-iz*).
i=1
6.3. p-adic ~-rings. We say that a ~-ring A = {AC, AR, AH} is of finite type *
*when AC, AR,
AH are finitely generated as abelian groups, and we say that A is fl-nilpotent *
*when AC and AR AH
are fl-nilpotent, i.e., when they are nilpotent with vanishing operations flm f*
*or sufficiently large m
(see [17, Section 4]). For a ~-ring A of finite type and a fixed prime p, the *
*tensor product with the
~-ring ^Zpgives a ~-ring A ^Zpwhose underlying -module is p-profinite and w*
*hose operations
are all continuous. By a weak (resp. strong) p-adic ~-ring, we mean the topolo*
*gical inverse limit
limff(Aff ^Zp) of an inverse system of ~-rings Aff ^Zpwhere each Affis of fi*
*nite type (resp. of
finite type and fl-nilpotent). For a strong p-adic ~-ring B, we note that ^Zp *
*BC and ^Zp BR BH
are p-adic ~-rings in the sense of [17, Section 5].
Our main topological examples of p-adic ~-rings will be given by
eKn(X; ^Zp) = {Ken(X; ^Zp), gKOn(X; ^Zp), gKOn-4(X; ^Zp)}
for a space X and integer n. We define an internal multiplication * on eKn(X; ^*
*Zp) by z * w = zw
when n = 0 and z * w = 0 when n 6= 0 for elements z, w 2 Ken(X; ^Zp). We also *
*define an
n n-4
internal multiplication * on gKO (X; ^Zp) gKO (X; ^Zp) by the following form*
*ulae for elements
n n-4
x, y 2 gKO (X; ^Zp) and u, v 2 gKO (X; ^Zp): (i) x * y = jnxy when n 0 and x*
* * y = 0 when
n < 0; (ii) x * u = jnxu when n 0 and x * u = 0 when n < 0; and (iii) u * v =*
* B-1Rjnuv when
*
n 0 and u * v = 0 when n < 0. Note that jn = 0 in gKO(X; ^Zp) unless p = 2 an*
*d n 2.
Theorem 6.4.For a space X 2 Ho* and integer n, eKn(X; ^Zp) has a natural weak p*
*-adic ~-
ring structure with the above internal multiplication, where the structure is s*
*trong whenever X is
connected or n 6= 0. Moreover, the ~-ring eKn(X; ^Zp) is isomorphic to eK0( |n*
*|X; ^Zp) for n
0, while the ~-ring eKn(X; ^Zp) is isomorphic to eK0( |n|X; ^Zp) for all n. Fi*
*nally, the operation
~OE:eKn(X; ^Zp) ! gKOn(X; ^Zp) is trivial for n > 0.
23
This will be proved in 6.10, and it provides a wide array of exterior power o*
*perations in eK*(X; ^Zp)
* *
* n
and gKO(X; ^Zp). The following theorem expresses the operation ~OE:eKn(X; ^Zp) *
*! gKO(X; ^Zp) for
0
n 0 in terms of a more basic operation OE: eKn(X; ^Zp) ! gKO(X; ^Zp).
Theorem 6.5.For a space X 2 Ho* and n 0, there is a natural operation OE: eKn*
*(X; ^Zp) !
gKO0(X; ^Zp) with the following properties for elements z, w 2 eKn(X; ^Zp):
(i)cOE(z) = Bn(z*z);
(ii)OE(zw) = (OEz)(OEw);
(iii)OE(z + w) = OEz + OEw + rBn(z*w);
(iv)OE(Bz) = OE(z);
(v)O~E(z) = j|n|OE(z).
This extends a result of Seymour [38] and will be proved in 6.11 using:
6.6. Atiyah's Real K-theory. In [7], Atiyah introduced a common generalization *
*KR(-) of real
and complex K-theory. It applies to a compact Real space Y , which consists of *
*a compact Hausdorff
space Y equipped with a map ø :Y ! Y such that ø2 = 1. A Real vector bundle ove*
*r Y consists
of a complex vector bundle q: E ! Y equipped with a map ø :E ! E such that ø2 =*
* 1 and
øq = qø with ø :Ey! Eøyantilinear for each y 2 Y . The ring KR(Y ) is then obta*
*ined by applying
the Grothendieck construction to the semiring of Real vector bundles on Y . Ati*
*yah-Segal [9] and
Dupont [28] extended the ring KR(Y ) to a semigraded ring KM(Y ) = KR(Y ) KH(*
*Y ) using the
same definitions, but with the condition ø4 = 1 on vector bundles in place of ø*
*2 = 1, where the
summand KH(Y ) is generated by the vector bundles with ø2 = -1, which are calle*
*d Symplectic.
For a compact Real space Y , we now obtain a -module K (Y ) = {K(Y ), KR(Y )*
*, KH(Y )}
with operations as follows:
(i)t: K(Y ) ! K(Y ) is defined on vector bundles by t(E) = ø*E~;
(ii)c: KR(Y ) ! K(Y ) and c0:KH(Y ) ! K(Y ) are defined on vector bundles
by forgetting the ø-actions;
(iii)r: K(Y ) ! KR(Y ) and q: K(Y ) ! KH(Y ) are defined on vector bundles
by sending E to E ø*E~with the natural Real or Symplectic ø-action.
When Y is connected or has a specified base component (closed under ø), we defi*
*ne augmentations
ffl: K(Y ) ! Z, ffl: KR(Y ) ! Z, and ffl: KH(Y ) ! Z sending vector bundles to *
*their complex
dimensions over the base component, and we let eK (Y ) = {Ke(Y ), gKR(Y ), gKH(*
*Y )} be the -
module of augmentation kernels. We now obtain a precursor to Theorem 6.4.
24
Theorem 6.7.For a compact Real space Y with a specified base component, the -m*
*odules K (Y )
and eK(Y ) have natural ~-ring structures. Moreover, if Y is a finite complex,*
* then K (Y ) and
Ke (Y ) are of finite type, and if Y is a connected finite complex, then eK(Y )*
* is fl-nilpotent.
Proof.In [28, Theorem 2], Dupont showed that the classical Splitting Principle *
*[8, Corollary 2.7.11]
generalizes to Real and Symplectic vector bundles over compact Real spaces, usi*
*ng Real and Sym-
plectic "line bundlesö f complex dimension 1. Thus, we obtain ~-ring structur*
*es on K(Y ) and
KR(Y ) KH(Y ) by the usual constructions for vector bundles, and these give *
*~-ring structures
on K (Y ) and eK(Y ) by straightforward arguments. When Y is a finite complex, *
*K*(Y ) is of finite
type, and hence KR(Y ) is finitely generated abelian by Segal's spectral sequen*
*ce (see [38, Theorem
3.1]). More generally, KR(Y ) KH(Y ) is finitely generated abelian since it is *
*additively isomorphic
to KR(Y x S3,0) by [28, Theorem 1]. When Y is a connected finite complex, the i*
*deal eK(Y ) and
the kernel of c: gKR(Y ) ! eK(Y ) are nilpotent by [38, Theorem 3.1]. Hence, gK*
*R(Y ) gKH(Y ) is
also nilpotent. Moreover, the elements of gKR(Y ) and gKH(Y ) must have finite *
*fl-dimension by the
Splitting Principle, since a Real or Symplectic line bundle ! has has flm (! - *
*1) = 0 for m > 1 and
has flm (1 - !) = (1 - !)m = 0 for sufficiently large m. Hence, eK(Y ) is fl-ni*
*lpotent. |__|
We also obtain a precursor to Theorem 6.5. This applies to a compact Real sp*
*ace Y with
basepoint (fixed under ø), and it involves the compact Real spaces Y "xY and Y *
*"^Y given by Y x Y
with ø(y1, y2) = (øy2, øy1) and by Y ^ Y with ø(y1^ y2) = øy2^ øy1.
Theorem 6.8.For a compact Real space Y with basepoint, there are natural operat*
*ions ÖE:K(Y ) !
KR(Y "xY ) and ÖE:eK(Y ) ! gKR(Y "^Y ) with the following properties on element*
*s of K(Y ) or eK(Y ):
(i)cÖE(z) = z*x z;
(ii)ÖE(zw) = (ÖEz)(ÖEw);
(iii)ÖE(z + w) = ÖEz + ÖEw + r(z*x w);
(iv) *ÖE(z) = ~OE(z) where is the diagonal Y ! Y "xY or Y ! Y "^Y .
Proof.The operation ÖE:K(Y ) ! KR(Y "xY ) is defined on vector bundles by sendi*
*ng E to ø*E~ E
with the twisting ø-action, and this induces an operation ÖE:eK(Y ) ! gKR(Y "^Y*
* ). The properties
(i)-(iv) are easily verified on vector bundles. *
* |__|
As a final preparation for our main proofs, we use Real K-theory to approach:
6.9. The p-adic K-cohomology of spaces. For m, n 0, let m,nbe the pointed Re*
*al (m + n)-
sphere obtained as the 1-point compactification of Rm+n = Rm x Rn with ø(x, y) *
*= (-x, y). If X
25
is a pointed finite complex then the results of Atiyah [7], Atiyah-Segal [9], D*
*upont [28], or Seymour
[38] give a natural isomorphism of -modules
( 0,|n|
eKn(X; Z) ~= eK( ^ X) for n 0
eK( n,0^ X) for n 0.
Moreover, the internal multiplication in eKn(X; Z) (defined by the formulae pre*
*ceding Theorem 6.4)
agrees with the Real K-theoretic multiplication in eK( 0,|n|^ X) or eK( n,0^ X)*
*. This follows
easily for n 0 and follows for n > 0 since the diagonal n,0! n,0^ n,0= 2n,*
*0is equivariantly
homotopic to the standard inclusion and hence induces the operator
2n 2n-4 n n-4
jn: gKO (X) gKO (X) -! gKO(X) gKO (X)
by [7, Proposition 3.2]. More generally, if X is a pointed complex (possibly in*
*finite), then there is a
natural isomorphism of p-adic -modules
( 0,|n|
^Zfor n 0
eKn(X; ^Zp) ~= limffeK( ^ Xff) p
limffeK( n,0^ Xff) ^Zpfor n 0.
where Xffranges over the finite pointed subcomplexes of X. Moreover, the intern*
*al multiplication
in eKn(X; ^Zp) agrees with the limit of Real K-theoretic multiplications.
6.10. Proof of Theorem 6.4. By Theorem 6.7 and 6.9, eKn(X; ^Zp) has a natural w*
*eak p-adic ~-
ring structure obtained as an inverse limit of the structures of eK( 0,|n|^ Xff*
*) ^Zpor eK( n,0^
Xff) Z^pfor the finite pointed Xff X. This structure is strong when X is conne*
*cted (with a single
vertex for simplicity) or n 6= 0, since the complexes 0,|n|^ Xffand n,0^ Xffa*
*re then connected.
n
Moreover, the operation ~OE:eKn(X; ^Zp) ! gKO (X; ^Zp) is trivial for n > 0 by *
*Theorem 6.8, since
the diagonals
: n,0-! n,0"^ n,0~= n,n
: n,0^ Xff-! ( n,0^ Xff) "^( n,0^ Xff)
are equivariantly nullhomotopic. *
*|__|
6.11. Proof of Theorem 6.5. For a pointed finite complex W and n 0, there is *
*a natural
0
operation OE: eKn(W; Z) ! gKO(W; Z) given by the composition
* ~
Ke( 0,|n|^W) -ÖE-!gKR(( 0,|n|"^ 0,|n|) ^ (W^"W)) (h^-)----!gKR( |n|,|n|^W) ~=gK*
*R(W)
where h: |n|,|n|~= 0,|n|"^ 0,|n|is the standard equivariant homeomorphism and *
*where ~ comes
*
* 0
from Atiyah's (1, 1)-Periodicity Theorem [7]. Moreover, the operations OE: eKn(*
*W; Z) ! gKO(W; Z)
satisfy the conditions of Theorem 6.5(i)-(v) by Theorem 6.8 and by [7, Proposit*
*ion 3.2], since the
diagonal : 0,|n|! 0,|n|"^ 0,|n|is equivalent to the standard inclusion 0,|n*
*| |n|,|n|. The
26
desired operations OE: eK(X; ^Zp) ! gKO0(X; ^Zp) are now obtained by tensoring *
*with ^Zpand passing
to inverse limits. |__|
7.The v1-stabilization homomorphism
For a space X and prime p, we now introduce the v1-stabilization homomorphism*
* 1:Ke*CR(X; ^Zp) !
K*CR( 1X; ^Zp) and explain how it may be used to determine K*CR( 1X; ^Zp) under*
* suitable condi-
tions (see 7.6). This will be applied to odd spheres and to simply-connected co*
*mpact Lie groups in
Sections 8 and 9.
7.1. The v1-stabilization homomorphism. For a K=p*-local spectrum E (such as bK*
*por dKOp)
and an integer n, the cohomology eEn(X) of a pointed space X is represented by *
*the space E_n=
1 ( nE), which has 1E_n' nE by 2.2. Thus there is a natural v1-stabilization*
* homomorphism
1:Een(X) -! En( 1X)
sending each f :X ! E_nto 1f : 1X ! nE. Equivalently, this is obtained by app*
*lying En to
the map 1X ! 1 1 1 X ' ( 1 X)K=pinduced by the adjunction unit X ! 1 1 X. *
*The
homomorphism 1 respects the cohomology suspension oe, so that the diagram
eEn(X) ---1-! En( 1X)
?? ?
yoe ~=?yoe
eEn-1( X)--1--!En-1( 1 X)
commutes, and hence an element x 2 eEn(X) has 1x = 0 whenever oeix = 0 for som*
*e i > 0. Any
element x 2 eEn(X) with 1x = 0 is called 1-trivial. For K=p*-local spectra *
*D, E 2 Hosand
integers m, n 2 Z, let !: eDm(X) ! eEn(X) be a natural cohomology operation wit*
*h representing
map !: D_m! E_n. Then ! induces a commutative diagram
eDm(X)---1-!Dm ( 1X)
?? ?
y! ?y 1!
Een(X)---1-!En( 1X)
and we note that 1 preserves stable cohomology operations, that is, it gives *
*1! = whenever
! = 1 .
For a pointed space X, the homomorphisms
1:Ke*(X; ^Zp) -! K*( 1X; ^Zp)
27
1:KgO*(X; ^Zp) -! KO*( 1X; ^Zp)
now combine to give the v1-stabilization homomorphism
1:Ke*CR(X; ^Zp) -! K*CR( 1X; ^Zp)
of p-profinite CR-modules. For each integer n, this restricts to a homomorphism
1:Ken(X; ^Zp) -! Kn( 1X; ^Zp)
of p-profinite -modules, which we may use to determine Kn( 1X; ^Zp) and eventu*
*ally K*CR( 1X; ^Zp)
under favorable conditions.
Theorem 7.2.For a pointed space X and integer n, suppose that Kn-1( 1X; ^Zp) = *
*0 and suppose
that M eKn(X; ^Zp) is a 1-trivial p-profinite -submodule such that eKn(X; ^*
*Zp)=M is an exact
-module with 1:Ken(X; ^Zp)=MC ~=Kn( 1X; ^Zp). Then 1:Ken(X; ^Zp)=M ~=Kn( 1X;*
* ^Zp).
Proof.This follows by Lemma 4.8 since the -module Kn( 1X; ^Zp) is exact by The*
*orem 4.4. |__|
To construct the needed 1-trivial elements in eKn(X; ^Zp), we use operations*
* obtained from
Theorems 6.4 and 6.5.
Lemma 7.3. For a pointed space X and integers n and k with k > 0, the internal *
*~-ring operations
n n-4
~pk, _pk, and `p in eKn(X; ^Zp) and in gKO (X; ^Zp) gKO (X; ^Zp) are all ann*
*ihilated by 1.
0
Moreover, when n 0, the operation OE: eKn(X; ^Zp) ! gKO(X; ^Zp) is annihilate*
*d by 1.
Here, `p is the natural ~-ring operation with _px = xp+ p`px as in [17]. This*
* lemma will be
proved in 7.8.
7.4. Examples of 1-trivial elements. For a pointed space X, we obtain from 7.1*
* and Lemma
*
7.3 the following useful examples of 1-trivial elements in eK*(X; ^Zp) and gKO*
*(X; ^Zp): all graded
*
decomposables in eK*(X; ^Zp) and their realifications in gKO(X; ^Zp); all grade*
*d decomposables in
gKO*(X; ^Zp); all image elements of ~p or `p in eKn(X; ^Zp) and in gKOn(X; ^Zp)*
* gKOn-4(X; ^Zp) for
0
n = -1, 0, 1; all image elements of OE: eKn(X; ^Zp) ! gKO(X; ^Zp) for n = -1, 0*
*; and all elements
generated by the preceding ones using the CR-module operations. We may obtain o*
*ther examples
by relaxing the conditions on n, but these are generally superfluous.
28
7.5. Kb 1-good spaces. Dividing eK*(X; ^Zp) by its submodule of known 1-trivia*
*l elements, we
obtain a v1-stabilization homomorphism
1:Q^K*(X; ^Zp)=`p -! K*( 1X; ^Zp)
as in [19, Lemma 7.10], where ^QK*(X; ^Zp) denotes the p-profinite quotient of *
*eK*(X; ^Zp) by its
graded decomposables, and where ^QK*(X; ^Zp)=`pdenotes the Bott periodic quotie*
*nt of ^QK*(X; ^Zp)
by `p^QK0(X; ^Zp) and `p^QK-1(X; ^Zp). A pointed space X will be called bK 1-go*
*od if
1:Q^K*(X; ^Zp)=`p ~=K*( 1X; ^Zp).
Here, we could equivalently replace `p by ~p since each `px is congruent to (-1*
*)p+1~px modulo
decomposables. We also note that the operation `p in eK-1(X; ^Zp) corresponds t*
*o the operation _p
of [19] in eK1(X; ^Zp). Working at an odd prime p in [19, Theorem 9.2], we prov*
*ed the bK 1-goodness
of an arbitrary 1-connected H-space X with H*(X; Q) associative and with H*(X; *
*Z(p)) finitely
generated over Z(p). As explained below in 8.1 and 9.1, we can also prove the b*
*K 1-goodness of an
odd sphere and of a simply-connected compact Lie group at the prime p = 2. In t*
*hese cases and
others, we may apply:
7.6. The general v1-stabilization method. Suppose that X is a bK 1-good pointed*
* space with
Kn-1( 1X; ^Zp) = 0 for a suitable n. Then, under favorable conditions, we may *
*apply a ver-
sion of Theorem 7.2 to determine Kn( 1X; ^Zp); then apply Theorems 4.3 and 4.11*
* to determine
K*CR( 1X; ^Zp); and finally apply Theorem 3.2 or methods of united K-theory to *
*determine the
v1-periodic homotopy groups v-11ß*X ~=ß*øp 1X. This general method should provi*
*de additional
information on the spectrum 1X since the ^ACR-module K*CR( 1X; ^Zp) determines*
* the homotopy
type of 1X by Theorem 5.3.
We conclude this section by proving Lemma 7.3 using the following 1-triviali*
*ty criterion, which
applies to a natural cohomology operation !: eDm(X) ! eEn(X) with representing *
*map !: D_m!
E_nfor K=p*-local spectra D, E 2 Hosand integers m, n 2 Z. Using the p-torsion *
*subgroup func-
tor tp, we say that tpß*E_nhas exponent ps if pstpßiE_n= 0 for all i > 0. We a*
*lso say that
!*:ß*D_m=tp! ß*E_n=tp becomes boundlessly p-divisible if, for each power pj, it*
* becomes pjdivisi-
ble in sufficiently high dimensions.
Lemma 7.7. Suppose that tpß*E_nhas exponent ps for some s > 0; suppose that !*:*
*ß*D_m=tp !
ß*E_n=tp becomes boundlessly p-divisible; and suppose that the maps in [D, E]m-*
*n are all detected
29
by Q ß*. Then 1! = 0 and the image elements of !: eDm(X) ! eEn(X) are 1-tri*
*vial for each
space X.
Proof.For each k > 0, the image of !*:ß*(D_m; Z=pk) ! ß*(E_n; Z=pk) has exponen*
*t p2sin suffi-
ciently high dimensions, and thus the image of !*:v-11ß*D_m! v-11ß*E_nhas expon*
*ent p2s. Hence,
by Theorem 2.4, the image of øp 1!*:ß*øp 1D_m! ß*øp 1E_nhas exponent p2s, and t*
*he image of
1!*:ß* 1D_m! ß* 1E_nhas exponent p4s. Since the maps in [D, E]m-n are detected*
* by Q ß*,
we conclude that 1! = 0, and the lemma follows. *
* |__|
7.8. Proof of Lemma 7.3. When X is a sphere, the operations ~pk, _pk, and `p ar*
*e easily de-
termined modulo torsion, since they are preserved by complexification and are k*
*nown in the ~-ring
Ken(X; ^Zp) ~=eK0( |n|X; ^Zp). The results that 1~pk= 0, 1_pk = 0, and 1`p *
*= 0 now fol-
low by Lemma 7.7, using Lemma 3.7 to verify the Q ß* condition. When n 0, t*
*he operation
cOE: eKn(X; ^Zp) ! eK0(X; ^Zp) is annihilated by 1since the elements cOE(z) = *
*Bn(z*z) are 1-trivial
by 7.1. Thus, c*( 1OE) = 0 in the Bott exact sequence
[KZ^p, KO^Zp]n-1-j*-![KZ^p, KO^Zp]n -c*-![KZ^p, KZ^p]n
and 1OE = 0 since j*= 0. |__|
8. The v1-stabilizations of odd spheres
We now illustrate the v1-stabilization method of 7.6 by applying it to an odd*
* sphere S2n+1at
the prime p = 2. In particular, we show that S2n+1is bK 1-good; we determine th*
*e 2-adic united
K-theory and other homotopical properties of 1S2n+1; and we recover the v1-per*
*iodic homotopy
groups v-11ß*S2n+1~=ß*ø2 1S2n+1which were originally determined by Mahowald usi*
*ng other
methods (see [23]). For simplicity, we rely on certain results of Mahowald and *
*Thompson to show
the bK 1-goodness of S2n+1at p = 2, although we hope to give a more general sel*
*f-contained account
in a subsequent paper.
Theorem 8.1.For n 1, the sphere S2n+1is bK 1-good at an arbitrary prime p.
Proof.At p odd, this follows by [19, Theorem 9.2]. At p = 2, we have 1S2n+1' *
*2n+1( 1 RP2n)K=2
by 2.9, and hence
(
Ki( 1S2n+1; ^Z2) ~= 0 for i = 0
Z=2n for i = -1
30
by [1]. Since these groups agree with ^QKi(S2n+1; ^Z2)=`2, it suffices to show *
*that 1:Ke-1(S2n+1; ^Z2) !
K-1( 1S2n+1; ^Z2) is onto, and this follows since the map from 1S2n+1to ( 1 S2*
*n+1)K=2has ho-
motopy cofiber ( 1 D2S2n+1)K=2with K0( 1 D2S2n+1; ^Z2) = 0 by [34] and 2.2. *
* |__|
Focusing on the case p = 2, we now apply our v1-stabilization method to deter*
*mine KO*( 1S2n+1; ^Z2)
from the -module eK-1(S2n+1; ^Z2) or eK1(S2n+1; ^Z2) with its internal operati*
*on `2 = -~2. We
deal separately with the four possible cases of n modulo 4.
8.2. Determining KO*( 1S2n+1; ^Z2) for n 1 mod 4 . We consider the -module e*
*K-1(S2n+1; ^Z2) ~=
{^Z2, ^Z2, ^Z2} with its operations t = 1, c = 2, r = 1, c0= 1, q = 2, `2C= 2n,*
* `2R= 2n, and `2HR= 2n-1,
using the notation `2C, `2R, and `2HRfor the complex, real, and quaternionic-to*
*-real components of
`2. This -module has a 1-trivial submodule
M = {2n^Z2, 2n-1^Z2, 2n+1^Z2} = {im`2C, im`2HR, imq`2C}
giving an exact quotient -module {Z=2n, Z=2n-1, Z=2n+1} whose complex componen*
*t goes isomor-
phically to K-1( 1S2n+1; ^Z2) by Theorem 8.1. Thus, we obtain v1-stabilization *
*isomorphisms
K-1( 1S2n+1; ^Z2) ~= eK-1(S2n+1; ^Z2)=M ~= {Z=2n, Z=2n-1, Z=2n+1},
and we can now apply Theorems 4.3 and 4.11 to determine K*CR( 1S2n+1; ^Z2). In*
* fact, using
our knowledge of the stable Adams operations _k in eK-1(S2n+1; ^Z2) for k 2 Zx(*
*2), we find that
KOi( 1S2n+1; ^Z2) is: Z=2n-1for i = -1 with _k = kn+1; Z=2n+1for i = -5 with *
*_k = kn+3;
Z=2 for i = -3, -4, -6, -7 with _k = 1; and 0 for i = -2, -8.
8.3. Determining KO*( 1S2n+1; ^Z2) for n 2 mod 4. We consider the -module eK*
*1(S2n+1; ^Z2) ~=
{^Z2, ^Z2, ^Z2} with its operations t = 1, c = 2, r = 1, c0= 1, q = 2, `2C= 2n,*
* `2R= 2n, and `2HR= 2n-1.
Then, as in 8.2, we determine K*CR( 1S2n+1; ^Z2) and find that KOi( 1S2n+1; ^Z2*
*) is: Z=2n-1for
i = 1 with _k = kn; Z=2n+1for i = -3 with _k = kn+2; Z=2 for i = -1, -2, -4, -5*
* with _k = 1;
and 0 for i = 0, -6.
8.4. Determining KO*( 1S2n+1; ^Z2) for n 3 mod 4. We consider the -module eK*
*-1(S2n+1; ^Z2) ~=
{^Z2, ^Z2, ^Z2} with its operations t = 1, c = 1, r = 2, c0= 2, q = 1, `2C= 2n,*
* `2R= 2n, and `2HR= 2n+1.
This -module has a 1-trivial submodule
M = {2n^Z2, 2n^Z2, 2n^Z2} = {im`2C, im`2R, imq`2C}
31
giving an exact quotient -module {Z=2n, Z=2n, Z=2n} whose complex component go*
*es isomorphi-
cally to K-1( 1S2n+1; ^Z2) by Theorem 8.1. Thus, we obtain v1-stabilization iso*
*morphisms
K-1( 1S2n+1; ^Z2) ~= eK-1(S2n+1; ^Z2)=M ~= {Z=2n, Z=2n, Z=2n},
and we can now apply Theorems 4.3 and 4.11 to determine K*CR( 1S2n+1; ^Z2). In *
*fact, using our
knowledge of _k in eK-1(S2n+1; ^Z2) for k 2 Zx(2), we find that KOi( 1S2n+1; ^Z*
*2) is: Z=2n for i = -1
with _k = kn+1; Z=2n for i = -5 with _k = kn+3; Z=2 for i = -2, -4 with _k = 1;*
* Z=2 Z=2
for i = -3 with _k = 1 by [41, Lemma 4.5]; and 0 for i = -6, -7, -8.
8.5. Determining KO*( 1S2n+1; ^Z2) for n 0 mod 4. We consider the -module eK*
*1(S2n+1; ^Z2) ~=
{^Z2, ^Z2, ^Z2} with its operations t = 1, c = 1, r = 2, c0= 2, q = 1, `2C= 2n,*
* `2R= 2n, and `2HR= 2n+1.
Then, as in 8.4, we determine K*CR( 1S2n+1; ^Z2) and find that KOi( 1S2n+1; ^Z2*
*) is: Z=2n for i = 1
with _k = kn; Z=2n for i = -3 with _k = kn+2; Z=2 for i = 0 or -2 with _k = 1; *
*Z=2 Z=2 for
i = -1 with _k = 1 by [41, Lemma 4.5]; and 0 for i = -4, -5, -6.
We could now apply Theorem 3.2 to determine the v1-periodic homotopy groups v*
*-11ß*S2n+1
up to extension. However, to circumvent extension problems, we prefer to treat*
* 1S2n+1as a
K-theoretic two-cell spectrum built from:
8.6. The K-theoretic sphere and pseudosphere. Suppose that X is a K*-local spec*
*trum with
K0X ~=Z, K1X ~=0, and XQ ' SQ. Then, by [16, Proposition 10.6], either X ' SK o*
*r X ' TK for
the spectrum T = S0[je2[2e3. We call TK the K-theoretic pseudosphere and find t*
*hat KO*TK
is a free KO*-module on a generator of degree 4, instead of the usual degree 0.*
* We note that the
above classification implies that TK ^TK ' SK , and hence the functor TK ^- act*
*s as an equivalence
on the homotopy category of K*-local spectra. We let g: SK ! TK be the bottom c*
*ell map and
find that it gives the doubling homomorphism on K0 using the standard isomorphi*
*sms K0SK ~=Z
and K0TK ~=Z. Moreover, g generates the group [SK , TK ] ~=ß0TK ~=Z by [13, Cor*
*ollary 4.4]. We
can now give the promised K-theoretic two-cell model for the homotopy type of *
*1S2n+1.
Theorem 8.7.The spectrum 2n+1 1S2n+1is equivalent to the homotopy fiber of 2n-*
*1g: SK=2!
TK=2 for n 1, 2 mod 4 and of 2n: SK=2! SK=2 for n 0, 3 mod 4 .
Proof.Using the results of 8.2-8.5, it is straightforward to show that the ^ACR*
*-module K*( 2n+1 1S2n+1; ^Z2)
is isomorphic to K*(Fn; ^Z2) for the required homotopy fiber Fn, and hence 2n+*
*1 1S2n+1' Fn by
Theorem 5.3. |__|
32
The above model for 1S2n+1might also be obtained from Yosimura's analysis of*
* ( 1 RP2n)K
in [41] or from Mahowald and Thompson's work in [34]. To describe the resulting*
* homotopy groups,
we let 2(j) denote the greatest power of 2 dividing j (where 2(0) = +1).
Theorem 8.8.For each i, we have:
8
>>>^Z2 if i = -1
>>>^Z Z=2 if i = 0
>>> 2
> Z=2 Z=2if i 1 mod 8
>>>
>>>Z=8 if i 3 mod 8
>>>0 if i 4, 5, 6 mod 8
: (j)+4
Z=2 2 if i = 8j - 1 with j 6= 0
8
>>>^Z2 if i = 0, -1
>>>
>><0 if i 0, 1, 2 mod 8 with i 6= 0
ßiTK=2 ' > Z=8 if i 3 mod 8
>>>Z=2 if i 4, 6 mod 8
>>>Z=2 Z=2if i 5 mod 8
>:
Z=2 2(j)+4if i = 8j - 1 with j 6= 0
Proof.The results for ßiSK=2follow from [13, Corollary 4.5], while those for ßi*
*TK=2follow from
Theorem 3.2 except in the case i 5 mod 8 where there is an extension problem.*
* To solve this
problem, and for later use, we note that TK=2^ S=2 ' TK ^ (S=2)K ' 4(S=2)K by *
*Theorem 5.3 or
[16, Proposition 10.5]. We then deduce the required splitting of ßiTK=2for i *
*5 mod 8 from the
orders of the groups ß*(TK=2^ S=2) ~=ß*-4(S=2)K calculated using Theorem 3.2. *
* |__|
We can now recover the result of Mahowald and Davis [23, p.1041] on the v1-pe*
*riodic homotopy
groups of S2n+1at p = 2.
Theorem 8.9 (Mahowald and Davis).If n 1, 2 mod 4, then
8
>>>Z=2 if i 0, 5 mod 8
min(3,n+1)
>>:Z=2 Z=2 if i 2, 3 mod 8
Z=2min(n-1, 2(j)+4)if i = 8j - 2 or 8j - 1
33
If n 0, 3 mod 4, then
8
>>>Z=2 Z=2 Z=2 if i 0, 1 mod 8
>>>
>>Z=8 if i 3 mod 8
>>>0 if i 4, 5 mod 8
>>>Z=2min(n, 2(j)+4)if i = 8j - 2
>:
Z=2 Z=2min(n, 2(j)+4)if i = 8j - 1
Proof.The map g*: KOi(TK=2; ^Z2) ! KOi(SK=2; ^Z2) is given by 1 for i 0 mod 8*
*, given by
4 for i 4 mod 8, and given by 0 otherwise. Hence, using Theorems 3.2 and 8.8,*
* we find that
g*:ßiSK=2! ßiTK=2is given by (1, 0) for i = 0, given by 4 for i 3 mod 8, give*
*n by 1 for i 7
mod 8, and given by 0 otherwise. Using the homotopy fiber sequence ( S=2)K=2! S*
*K=2-g!TK=2
and the fact that S=2 has exponent 4, we can now easily calculate the homotopy *
*groups
v-11ß3+iS3 ~=ßi( 3 1S3) ~= ßi( S=2)K=2
to confirm the theorem for n = 1. We can next calculate the homotopy groups v-1*
*1ß2n+iS2n+1~=
ßi( 2n+1 1S2n+1) for n > 1 up to extension by using the homotopy exact sequence*
*s from Theorem
8.7. Finally we conclude that all of these extensions split since the homomorph*
*isms
(1 ^ 2n-1g)*:ß*(S=2 ^ SK=2) -! ß*(S=2 ^ TK=2)
(1 ^ 2n)*:ß*(S=2 ^ SK=2) -! ß*(S=2 ^ SK=2)
are trivial for n > 1, as seen from the exponents of the homotopy groups ß*(S=2*
*^SK=2) ~=ß*(S=2)K=2
and ß*(S=2 ^ TK=2) ~=ß*( 4S=2)K=2in each dimension. *
* |__|
9.The v1-stabilizations of simply-connected compact Lie groups
Finally, we apply the v1-stabilization method to simply-connected compact Lie*
* groups at the prime
p = 2. For such a group G, our main result (Theorem 9.3) will express KO*( 1G; *
*^Z2) in terms of the
representation theory of G, assuming that G is bK 1-good. Since we can prove th*
*e bK 1-goodness
of all simply-connected compact Lie groups, this will confirm our general conje*
*cture, which Davis
presented in [25, Conjecture 2.2] and used so effectively to calculate v1-perio*
*dic homotopy groups.
We start by discussing:
9.1. The bK 1-goodness of simply-connected compact Lie groups. In [19, Theorem *
*9.2], we
proved that each simply-connected compact Lie group G is bK 1-good at an odd pr*
*ime p; in fact, we
gave explicit constructions of GK=pand 1G. After considerable effort, we recen*
*tly showed that this
work extends to the prime p = 2 provided that G satisfies a certain Technical C*
*ondition involving
34
its representation ring. This result is made useful by recent work of Don Davis*
* [25, Theorem 1.3]
showing that a simply-connected compact simple Lie group satisfies our Technica*
*l Condition if and
only if it is not E6or Spin(4k+2) with k not a 2-power. Hence, all simply-conne*
*cted compact simple
Lie groups are bK 1-good, except possibly for E6and Spin(4k+2) with k not a 2-p*
*ower. Fortunately,
we can prove that these remaining groups are bK 1-good by a careful analysis of*
* (E6=F4)K=2and
by fibration arguments. We plan to include a detailed account of this work in a*
* subsequent paper
showing the bK 1-goodness of all simply-connected compact Lie groups. To state *
*our main theorem,
we use:
9.2. Indecomposables of representation rings. For a simply-connected compact Li*
*e group G,
we first let Q(G) = eR(G)=Re(G)2 denote the indecomposables of the complex repr*
*esentation ring
R(G). We then let QR(G) Q(G) and QH(G) Q(G) denote the real and symplectic *
*indecompos-
ables given by the images of eRR(G) and eRH(G) in Q(G). These indecomposables c*
*ombine to give
a ~-ring
Q (G) = {Q(G), QR(G), QH(G)}
whose structure is inherited from the ~-ring {Re(G), eRR(G), eRH(G)} of 6.1 an*
*d 6.2. Since Q (G)
has trivial multiplication, its operations ~k are additive for k 1. By standa*
*rd results presented in
[12, Sections II.6 and VI.4] or [25, Theorem 2.3], Q(G) is a finitely generated*
* free abelian group on
generators ~æ= æ - dimæ where æ ranges over the basic representations of G; mor*
*eover, Q (G) is a
free -module on the generators ~æ-dimæ where æ ranges over the complex, real, *
*and quaternionic
basic representations in Q(G), QR(G), and QH(G) respectively. In particular, Q *
*(G) is an exact
-module such that c: QR(G) Q(G), c0:QH(G) Q(G), QR(G) \ QH(G) = im(1 + t),*
* and
QR(G) + QH(G) = ker(1 - t)for the conjugation t: Q(G) ! Q(G).
We can now state our main theorem in a form derived directly from Davis [25, *
*Conjecture 2.2].
* *
For brevity, we write Q = Q(G), QR = QR(G), QH = QH(G), and dKO(-) = KO (-; ^Z2*
*).
Theorem 9.3.If G is a simply connected compact Lie group which is bK 1-good at *
*p = 2, then
there is a long exact sequence of abelian groups
0 ~2 1
. .-.! 0 -! dKO( 1G) -! Q=(QR + QH) ---!Q=QR -! dKO( 1G)
2 ~2
-! 0 -! QH=(QR \ QH) -! dKO( 1G) -! QR \ QH ---!QH
3 4 ~2
-! dKO( 1G) -! 0 -! 0 -! dKO( 1G) -! Q=(QR \ QH) ---!Q=QH
5 ~2
-! dKO( 1G) -! (QR + QH)=(QR \ QH) ---!QR=(QR \ QH)
35
-! dKO6( 1G) -! QR + QH -~2--!QR -! dKO7( 1G) -! 0
8 ~2
-! 0 -! dKO( 1G) -! Q=(QR + QH) ---!Q=QR -! . . .
which continues by Bott periodicity. Moreover, for any integer i and odd intege*
*r k 1, the Adams
2i-1 2i-2
operation _k in dKO ( 1G) and dKO ( 1G) corresponds to k-i_k (or equivalent*
*ly k-i+1~k)
in the Q-terms under the morphisms of the exact sequence.
As explained in 7.6 and [25], this leads to calculations of the v1-periodic h*
*omotopy groups v-11ß*G.
In cases where G satisfies the Technical Condition (see 9.1), we originally obt*
*ained the above exact
*
sequence (tensored with ^Z2) as the dKO-cohomology exact sequence of a stable (*
*co)fiber sequence
coming from our explicit construction of 1G at p = 2. We now proceed to prove *
*Theorem 9.3 in
general using:
9.4. The v1-stabilization of Q (G). For a simply-connected compact Lie group G,*
* we first obtain
a -module homomorphism
ff: eR(G) -! K-1( 1G; ^Z2)
by composing the canonical ~-ring homomorphism eR(G) ! eK0(BG; ^Z2) with the *
*-module
homomorphisms
eK0(BG; ^Z2)-1---!K0( 1BG; ^Z2)
?? ?
yoe ~=?yoe
eK-1(G; ^Z2)-1---!K-1( 1G; ^Z2)
This ff factors through the quotient homomorphism eR(G) i Q (G) since it vanish*
*es on the terms
Re(G)2, eRR(G)2, eRH(G)2, eRR(G)ReH(G), rRe(G)2, qRe(G)2, and OERe(G) by 7.5. W*
*e let
ff: Q (G) -! K-1( 1G; ^Z2)
denote the induced v1-stabilization homomorphism for the exact -module Q (G).
9.5. The exact -modules Q0(G) and Q00(G). We now modify the exact -module Q (*
*G) =
{Q(G), QR(G), QH(G)} to give an exact -module
Q0(G) = {Q(G), QR(G) + QH(G), QR(G) \ QH(G)}
with the obvious operations c: QR(G)+QH(G) Q(G), c0:QR(G)\QH(G) Q(G), t: Q(*
*G) !
Q(G), r = 1 + t: Q(G) ! QR(G) + QH(G), q = 1 + t: Q(G) ! QR(G) \ QH(G). We note*
* that
Q0(G) has epic q since QR(G) \ QH(G) equals the image of 1 + t: Q(G) ! Q(G). Th*
*e monic
operation ~2:Q(G) ! Q(G) now induces a -module monomorphism ~2:Q0(G) ! Q (G) s*
*ince
36
~2t = t~2, ~2QR(G) QR(G), ~2QH(G) QR(G), and ~2(QR(G) \ QH(G)) QR(G) \ QH*
*(G).
This determines an exact -module Q00(G) = coker~2 belonging to a short exact s*
*equence
2 00
0 -! Q0(G) -~-!Q (G) -! Q (G) -! 0
where Q00(G) has monic c by Lemma 4.14 since Q0(G) has epic q. The v1-stabiliza*
*tion homomor-
phism ff: Q (G) ! K-1( 1G; ^Z2) now factors through the quotient homomorphism Q*
* (G) i
Q00(G) since it vanishes on image elements of ~2 by 7.4, and we let
00 -1
ff: Q (G) -! K ( 1G; ^Z2)
denote the induced v1-stabilization homomorphism.
Lemma 9.6. If G is a simply-connected compact Lie group which is bK 1-good at p*
* = 2, then
ff: Q00(G) ~=K-1( 1G; ^Z2) and K0( 1G; ^Z2) ~=0. Moreover, for any odd integer *
*k 1, the Adams
operations {_k, _k, _k} in K-1( 1G; ^Z2) correspond to {_k, _k, k2_k} in Q00(G).
Proof.Since G is bK 1-good, there are isomorphisms
1:QbK*(G; ^Z2)=~2 ~=K*( 1G; ^Z2)
which reduce to isomorphisms ff: Q00(G) ~=K-1( 1G; ^Z2) and 0 ~=K0( 1G; ^Z2) by*
* [6] or [29].
Thus ff: Q00(G) ~=K-1( 1G; ^Z2) as in the proof of Theorem 7.2, and the lemma f*
*ollows easily. |__|
We can now essentially determine K*CR( 1G; ^Z2) from the exact -module Q00(G*
*) using the
notation Q00= Q00(G), Q00R= Q00R(G), and Q00H= Q00H(G). We write A#B for an ab*
*elian group
belonging to a short exact sequence 0 ! A ! A#B ! B ! 0.
Lemma 9.7. If G is a simply-connected compact Lie group which is bK 1-good at p*
* = 2, then there
are natural isomorphisms
8 00
>>>QR for m 7 mod 8
>>> 00
>>>QR=r for m 6 mod 8
>>>(Q00=c0)#(Q00\c0)for m 5 mod 8
><00 0 H
KOm ( 1G; ^Z2) ~= >QH\c00 for m 4 mod 8
>>>QH for m 3 mod 8
>>>Q00=q for m 2 mod 8
>>> H00
>>>Q =c for m 1 mod 8
:0 for m 0 mod 8
Moreover, for any integer i and odd integer k 1, the Adams operation _k in KO*
*2i-1( 1G; ^Z2)
and KO2i-2( 1G; ^Z2) corresponds to the operation k-i_k in the Q-terms.
37
Proof.By 4.1 and Lemma 9.6, K*CR( 1G; ^Z2) is a Bott exact CR-module with K0( 1*
*G; ^Z2) ~=0
and K-1( 1G; ^Z2) ~=Q00(G) where c is monic in Q00(G). Hence, the result follow*
*s immediately
from Theorem 4.3. |__|
9.8. Proof of Theorem 9.3. We first obtain exact sequences
2 00
0 -! Q=(QR + QH) -~-!Q=QR -! Q =c -! 0
00
0 -! QH=(QR \ QH) -! QH=q -! 0
2 00
0 -! QR \ QH -~-!QH -! QH -! 0
00 0 ~2 00 0
0 -! QH\c -! Q=(QR \ QH) --! Q=QH -! Q =c -! 0
00 0 ~2 00
0 -! QH\c -! (QR + QH)=(QR \ QH) --! QR=(QR \ QH) -! QR=r -! 0
2 00
0 -! QR + QH -~-!QR -! QR -! 0
by applying Lemma 4.14 to the short exact sequence 0 ! Q0(G) ! Q (G) ! Q00(G) !*
* 0 of
-modules. We then obtain the desired long exact sequence by patching the above*
* exact sequences
together and applying Lemma 9.7. The theorem now follows easily. *
* |__|
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Department of Mathematics, University of Illinois at Chicago, Chicago, Illino*
*is 60607
E-mail address: bous@uic.edu