EQUIVARIANT FORMS OF CONNECTIVE K THEORY
J.P.C.GREENLEES
Abstract. We construct a Cp-equivariant form of connective K-theory with *
*several good
properties.
1. Introduction
In the non-equivariant world, connective K-theory is an extremely useful coho*
*mology
theory. It is not much harder to calculate than periodic K-theory, but can be *
*a lot more
powerful. However, there does not appear to be any abstract explanation for the*
* importance
of the theory. To qualify, such an explanation would tell me how to construct (*
*or at least
recognize) an equivariant version of the theory.
I still do not understand the abstract nature of the theory, but I do have a *
*list of desirable
properties for an equivariant form of connective K-theory. This paper arises s*
*ince none
of the constructions I previously knew had all of these properties. Initially,*
* I suspected
the properties were inconsistent, but in fact they almost determine the constru*
*ction. The
resulting cohomology theory needs further investigation: it may be expected to*
* improve
certain existing results proved with periodic equivariant K-theory and connecti*
*ve K-theory
of the Borel construction. At present the construction is restricted to groups *
*of prime order.
Any treatment for arbitrary finite groups will have to use a more geometric con*
*struction.
Let G be a group of order p.
Properties 1.1. Consider the following properties of a G-spectrum E.
1. E is a split ring spectrum, and nonequivariantly ku.
2. E[v-1] ' K (equivariantly) where v is the degree 2 Bott element arising fr*
*om the split
structure.
3. E is complex orientable (equivariantly).
4. EG*is concentrated in even degrees.
5. E*Gis a Noetherian ring.
Property 1 simply states that E is an equivariant form of connective K-theory*
*, and is
therefore not negotiable. The other properties are in decreasing order of impor*
*tance. Note
that complex orientability is enough to ensure that a completion theorem holds;*
* provided
E*Gis Noetherian this states in particular that
E*(BG+) = (E*G)^I
where I = ker(E*G-! E* = ku*) is the augmentation ideal.
The purpose of this note is to prove the following theorem.
Theorem 1.2. There is a G-spectrum ku with all five of the properties listed *
*in 1.1. It has
coefficient ring
ku*G= R(G)[v; y]=(vy = O(ff); yae)
1
2 J.P.C.GREENLEES
where ff is the natural representation of G, O(ff) = 1 - ff is its K-theory Eul*
*er class and
ae = 1 + ff + . .+.ffp-1 is the regular representation. The Bott element v is i*
*n degree 2, and
the element y is in degree -2.
Remark 1.3. Note that the coefficient ring is R(G) in each positive even degr*
*ee, and Jn in
degree -2n. This is the Rees ring of R(G) with respect to the augmentation idea*
*l J [10],
and the associated projective scheme is that of the blowup of Spec(R(G)) at the*
* subscheme
defined by J. It is tempting to hope that for any finite group there is a -eq*
*uivariant
form of ku whose coefficient ring is the Rees ring of R() with respect to J. Ho*
*wever such
an equivariant form cannot have all of the properties listed in 1.1, since thes*
*e imply the
completion theorem holds. This is inconsistent with known values of ku*(B+), si*
*nce
Rees(R(); J)^J= Rees(R()^J; J^J)
is in even degrees, whilst ku*(B+) is often nonzero in odd degrees (for example*
* if is_
elementary abelian of rank 3) [12, 8]. *
* |__|
The rest of the paper is layed out as follows. We begin by discussing some kn*
*own equivari-
ant forms of ku, and showing they do not have the required properties. Next, we*
* summarize
facts about periodic K-theory. This ensures we have a good example in mind, al*
*lows us
to introduce notation, and is an input to the verification of Property 2. We t*
*hen spend
two sections showing how much of the construction is forced by complex stabilit*
*y and the
comparison with periodic K-theory; this is logically unnecessary, but I believe*
* it makes the
construction more compelling. Finally we turn to the construction and propertie*
*s of a good
equivariant form of ku.
I am grateful to R.R.Bruner for his instruction on connective K-theory and hi*
*s explanation
of various calculations, and to J.P.May, N.P.Strickland, P.R.Turner and J.Wolbe*
*rt for useful
conversations.
Contents
1. Introduction *
* 1
2. Examples 3
3. Periodic K-theory and associated notation. *
* 4
4. Complex stability *
* 5
5. The complete part *
* 7
6. Relation to periodic K-theory *
* 8
7. The solenoidal ku-module. *
* 9
8. Construction of equivariant connective K theory *
* 11
9. The first properties of equivariant connective K theory *
* 12
10. Multiplicative properties of equivariant connective K-theory. *
* 13
11. Complex orientability of equivariant connective K-theory. *
* 15
12. Highly structured products *
* 17
References *
*18
EQUIVARIANT FORMS OF CONNECTIVE K THEORY 3
2. Examples
We summarize three known ways to construct equivariant versions of ku. We sho*
*w that
none of them have all five properties listed in 1.1, thereby establishing the n*
*eed for the
subsequent sections in which an equivariant form with these properties is const*
*ructed.
Example 2.1. The connective cover K<0> of equivariant K-theory.
This spectrum is constructed in the usual way as the fibre of the map K -! K(*
*-1; -1]
killing ssn and ssGnfor all n 0. The resulting spectrum K<0> certainly has Pro*
*perties 1, 2
and 4. We shall see in the Section 4 that no connective theory can be complex s*
*table. Also,_
since K<0> is bounded below, the geometric fixed point spectrum GK<0> is contra*
*ctible. |__|
Example 2.2. May's spectrum MU-inf G1(ku) [11], obtained as an inflation of MU*
*-modules.
Formal manipulations show that since ku is an MU-algebra up to homotopy, the *
*MU-
induced spectrum is a split MU-algebra up to homotopy and hence a split ring sp*
*ectrum
up to homotopy. In particular it is a module over equviariant MU and hence com*
*plex
orientable. It thus has Properties 1 and 3. We shall show that it also has Prop*
*erty 4 but
not Property 2. Its geometric fixed points are enormous:
G(MU-inf G1(ku)) = (GMU) ^MU ku = ku ^ (BU+[z; z-1])^(p-1);
*
* __
and hence this is different from the previous example. *
* |__|
Example 2.3. The inflation infG1(ku) obtained by change of universe from ku.
This spectrum is commonly written i*ku if i denotes the inclusion of the fixe*
*d points in a
chosen complete G-universe.
It is no surprise that such a crude construction is very unsuccessful. The r*
*esult is con-
nective and therefore not complex stable (see Section 4), so does not have Prop*
*erty 3. It
does not have Property 2 either, since infG1(ku)[1=v] = infG1(K); the fact that*
* infG1(K) 6= K
is well known, and one argument can be obtained by adapting the following calcu*
*lation of
ssG*(infG1(ku)).
Consider homotopy groups and see that Property 4 also fails. To calculate ssG*
**(infG1(ku))
we use the long exact sequence arising from
EG+ ^ infG1(ku) -! infG1(ku) -! "EP ^ infG1(ku):
The first term has homotopy groups ku*(BG+), and the last term has homotopy gro*
*ups
Z[v]. By considering the map in degree zero we see that last map is surjective*
*, so that
ssG*(infG1(ku)) = Z[v] ku*(BG+). It is well known (and discussed in Section 5 *
*below) that
ku*(BG+) is non-zero and torsion in odd degrees.
It may be worth comparing this to the first example. Since ku = S0[e2[: :,:th*
*e inflation
infG1(ku) has a similar cell decomposition, and its zeroth homotopy is A(G). In*
*deed, from
the fact that periodic K-theory is split, we have a map infG1(ku) -! K<0>. In d*
*egree 0 it
is the permutation representation homomorphism A(G) -! R(G). When p = 2 this is*
* an
isomorphism, and since it is a map of A(G)[v] modules, we find that the homotop*
*y groups__
of the fibre are the odd degree (torsion) part of ku*(BG+). *
* |__|
4 J.P.C.GREENLEES
3.Periodic K-theory and associated notation.
Our basic tool is the Tate pullback square [7]. In the case of periodic K-the*
*ory it reads
K -! K ^ "EP
# o2 #
F (EG+; K) -! t(K)
This method is useful because we have simpler descriptions of the two spectra o*
*n the right,
and F (EG+; K) only depends on non-equivariant K-theory.
We need some more notation. Recall that KG0= R(G) and KG*= R(G)[v; v-1], wher*
*e v
is the Bott element of degree 2. Associated to the representation ring there ar*
*e a number of
other algebraic objects. Most familiar perhaps is the augmentation ideal J = ke*
*r(R(G) -!
Z). The regular representation ae generates the_ideal of J-power torsion elemen*
*ts, which is
additively isomorphic to Z; the quotient ring R = R(G)=(ae)_will be important *
*to us. A
source of some confusion is that J is isomorphic_to R as an R(G)-module, howeve*
*r neither
the exact sequence 0 -! (ae) -! R(G) -! R - ! 0 nor the exact sequence 0 -! J -!
R(G) -! Z -! 0 are split over_R(G). The effect of J-completion is trivial on Z *
*and (ae),
and p-completion on J and R; the sequences still do not split after completion.
For the group of order p we let ff denote a faithful one dimensional_represen*
*tation, so
that R(G) = Z[ff]=(ffp - 1), ae = 1 + ff + ff2 + . .+.ffp-1, and R may be ident*
*ified with the
ring generated by a primitive p-th root of unity in the complex numbers. The Eu*
*ler class
O = O(ff) = 1 - ff also plays a role. Note that, for any i, O(ffi) is a multipl*
*e of O(ff), and
vice versa if i is prime to p. Thus (O) is the ideal generated by any non-zero *
*Euler class.
Lemma 3.1. (i) The geometric fixed points of periodic K-theory are given by
__
GK ' KR [1=p]
(ii) The Tate spectrum of periodic K-theory is given by
Y __^
t(K) ' "EP ^ 2nHR p[1=p]:
2n
Proof: (i) To calculate the coefficients of GX for any complex stable X we have
ss*(GX) = ssG*(E"P ^ X) = ssG*(S1ff^ X) = ssG*(X)[1=O(ff)]:
*
* __
Since O(ff) = 1 - ff, a short calculation shows that the coefficient ring of *
*GK is R[1=p] in
each even degree. Furthermore GK is a non-equivariant K-algebra up to homotopy;*
* since
the coefficients are free over Z[1=p] the result follows. *
* __
(ii) This is a special case of [6, A.5] or [7, 13.1]. *
* |__|
For periodic K-theory, both v and O act isomorphically, so it is important to*
* be clear that
we use the following convention.
Convention 3.2. Multiplication by v is used to relate notation for homotopy gr*
*oups in
different dimensions.
EQUIVARIANT FORMS OF CONNECTIVE K THEORY 5
When we come to the analogous situation for ku, multiplication by v will not *
*be an
isomorphism, whereas multiplication by O will be. The correct action of v will *
*then need to
be chosen by reference to the action of O. This is recorded in the following le*
*mma.
__
Lemma 3.3. Multiplication by O = O(ff) = 1 - ff on R has the property that it*
*s (p - 1)st
power is p times an isomorphism.
__
Proof: The module R has basis 1; ff; : :;:ffp-2. If we write out the matrix of *
*O(ff) = 1 - ff
with respect to this basis we see that it has determinant p, and it is easy to *
*see that_(1-ff)p-1
is divisible by p. *
* |__|
4. Complex stability
In this section we assume that E is complex stable and non-equivariantly boun*
*ded below
(for example if Properties 1 and 3 of 1.1 hold). This tells us rather a lot abo*
*ut the coefficients
of E and its geometric fixed point spectrum.
The complex stability condition states that for complex representations V the*
*re is a Thom
isomorphism
EG*(SV ^ X) ~=EG*(S|V |^ X)
(where |V | is the trivial representation of the same dimension as V ). A great*
* deal of infor-
mation comes from complex stability in conjunction with the fact that E is none*
*quivariantly
bounded below.
Thus we may choose a faithful one dimensional complex representation V and co*
*nsider its
cell structure in the following two cofibre sequences
1-Rg
G+ -! G+ -! C
and
C -! S0 -! SV ;
where g is a generator of G and Rg denotes right multiplication by g.
The first cofibre sequence lets us calculate the homology of C.
Lemma 4.1. There is an isomorphism
EG*(C) ~=ku* ku*
of ku*-modules.
Proof: Since G acts trivially on the nonequivariant homotopy ku* we see (1 - Rg*
*)*_is_the
zero map of EG*. *
* |__|
Now the inclusion S0 -! SV together with complex stability induces multiplica*
*tion by
the Euler class
O(V ) : EGk-! EGk-2:
Thus the second cofibre sequence together with our calculation for C gives the *
*following
powerful restriction.
6 J.P.C.GREENLEES
Corollary 4.2. For k -1 the Euler class is an isomorphism
~= G __
O(V ) : EGk-! Ek-2: |__|
It is thus convenient temporarily to ignore Convention 3.2, and let P = EG-2a*
*nd Q = EG-1
denote the typical even and odd groups in the negative homotopy of EG*. This g*
*ives the
homotopy groups of the geometric fixed point spectrum.
Corollary 4.3. The homotopy groups of the geometric fixed point spectrum of E a*
*re GE2k=
P and GE2k+1= Q. Furthermore the action of v (as an endomorphism of P or Q) is *
*in-__
dependent of degree. *
* |__|
The sequence in positive degrees takes the form
. . .
O G
Z -! EG3-! E1 -!
O G
Z -! EG2-! E0 -!
O G
Z -! EG1-! E-1 -!
O G
Z -! EG0-! E-2 -! 0
:
If EG*is all in even degrees then we obtain
O 2 G 2
0 -! ku* -! EG*-! E* -! ku* -! 0
and EG*= ku* O(V )EG*. Now let
Ok+1 G
I2k= im(EG2k-! E-2 = P );
and
Ok+1 G
K2k= ker(EG2k-! E-2 = P );
so that I0 = P and K0 = Z. Now compare two exact sequences:
0 -! K2k+2 -! EG2k+2 -! I2k+2 -! 0
# # #
0 -! K2k -! EG2k -! I2k -! 0:
The right hand vertical is injective, and the central vertical has kernel and c*
*okernel Z. We
conclude
(i) I2k=I2k+2is a cyclic group, and
(ii) K2k+2= Z K02k, where K02k= K2k or K02k Z = K2k according to whether the c*
*yclic
group in (i) is infinite or finite. In any case we see by induction that K2k+2i*
*s a free abelian
group with rank either that of K2kor one more. Furthermore, if P Q is finite d*
*imensional,
the cyclic group in (i) is eventually finite and so the rank of the kernel K2k *
*is eventually
constant.
EQUIVARIANT FORMS OF CONNECTIVE K THEORY 7
5. The complete part
If E is split with underlying non-equivariant spectrum ku, we have the pullba*
*ck square
E -! E ^ "EP
# o2 #
F (EG+; E) -! t(E)
k k
F (EG+; ku) t(ku)
The point here is the two equalities at the bottom: we have written ku because *
*the spectra
are independent of which equivariant form of ku has been used. For definitenes*
*s, and to
emphasize the logic of the construction, we should use infG1ku. In this sectio*
*n we make
the groups concerned explicit. In the present context it seems most appropriate*
* to give the
descriptions as abelian groups, with v action. If lu is the principal Adams su*
*mmand, we
have
ku = lu _ 2lu _ : :_:2(p-2)lu
and
lu* = Z[vp-1]:
Furthermore
8
< Z Z^p ifn 0 is a multiple of2(p - 1)
ssGn(F (EG+; lu)) = lu-n(BG+) = Z^p for other even n
: 0 otherwise
Here vp-1 acts isomorphically on the Z factor, and also on the Z^pfactor when t*
*he codomain
is in degree 0 or more. In lower even degrees vp-1 acts as multiplication by p.
ae
Z^p ifn is even
ssGn(t(lu)) = t(lu)Gn=
0 otherwise
In all even degrees vp-1 acts as multiplication by p.
The fibre of F (EG+; lu) -! t(lu) is lu ^ EG+. Its homotopy groups can be ded*
*uced from
the long exact sequence and the fact that it is connective:
8
< Z ifn 0 is a multiple of2(p - 1)
ssGn(lu ^ EG+) = lun(BG+) = Z=pj+1 ifn = 2j(p - 1) + 2s + 1 with0 s < p *
*- 1
: 0 otherwise
By taking suitable wedges of suspensions of lu we obtain the corresponding calc*
*ulations
for ku. For comparison with periodic K-theory we note that as abelian groups R*
*(G)^J=
Z (Z^p)p-1. However the formal group descriptions K*(BG+) = Z[v; v-1][[y]]=([p*
*](y)) and
ku*(BG+) = Z[v][[y]]=([p](y)) (where [p](y) = (1-(1-vy)p)=v ), show that in deg*
*rees n 0,
the map ku-n(BG+) -! K-n(BG+) is isomorphic.
8 ^
< R(G)J_ ifn 0 is even
ssGn(F (EG+; ku)) = ku-n(BG+) = R ^p ifn < 0 is even
:
0 otherwise
__^
Here v acts isomorphically on the Z factor, and also on the Rp factor when the *
*codomain is
in degree 0 or more. In lower even degrees the action is more complicated: we a*
*re therefore
8 J.P.C.GREENLEES
__^
violating Convention 3.2 in writing Rp for the negative dimensional homotopy gr*
*oups.
ae__^
ssGn(t(ku)) = t(ku)Gn= R p ifn is even
0 otherwise
In all even degrees vp-1 acts as p times an isomorphism.
Lemma 5.1. We may choose a basis so that v acts on t(ku)G*as multiplication b*
*y O. This
may be represented in accordance with Convention 3.2 by
__^ __
t(ku)G2n= O-nR p: |__|
The fibre of F (EG+; ku) -! t(ku) is ku ^ EG+. Its homotopy group ssGn(ku ^ E*
*G+) is as
follows:
8
< Z ifn 0 is even
kun(BG+) = (Z=pj+1)s+1 (Z=pj)p-s-2 ifn = 2j(p - 1) + 2s + 1 with0 s < p -*
* 1
: 0 otherwise
In other words, in odd positive degrees kun(BG+) is the sum of p - 1 cyclic gro*
*ups, and as
n increases, they take turns to increase in order by a factor of p.
The Mayer-Vietoris sequence associated to the pullback square becomes a short*
* exact
sequence if we assume (Property 4) that homotopy is concentrated in even degree*
*s:
0 -! EG2k-! P ku-2k(BG+) -! t(ku)G2k-! 0:
In negative degrees it tells us (as we learnt before in Section 4) that EG2k~=P*
* . If k 0 it
becomes
__^
0 -! EG2k-! P R(G)^J-! Rp -! 0:
6.Relation to periodic K-theory
Assuming Property 2, we have a map i : E -! K which becomes an equivalence wh*
*en v
is inverted. Assuming complex stability, in negative even degrees it is i-2k : *
*P = EG-2k-!
KG-2k= R(G). This map is not zero since if we pass to the limit as k -! 1 (usin*
*g O(ff))
we reach
__
i-1 : P = ss-2k(GE) -! ss-2k(GK) = R(G)[1=O(ff)] = R[1=p]:
If we now invert v, we obtain an equivalence (since this is the same as first i*
*nverting v
and then O(ff)). Hence i-2k maps P onto a non-trivial_subgroup of R(G) with_ran*
*k p - 1.
This is necessarily Z-projective, and hence P = P 0 R additively, where R = R(G*
*)=(ae) as
above. This shows that there is essentially a unique minimal equivariant form o*
*f connective
K-theory with Properties 1.1. __
Furthermore, we may deduce the action of v on R by using Lemma 3.3, bearing i*
*n mind_
that in Section 4 multiplication by O was used to identify the various copies o*
*f R in different
dimensions._In_other words if we normalize_so_that in degree 0 the image of P i*
*n R(G)[1=O]
is R R[1=p] the image in degree -2k is OkR . This then follows Convention 3.2
EQUIVARIANT FORMS OF CONNECTIVE K THEORY 9
7. The solenoidal ku-module.
The purpose of this section is to construct and study a candidate for the geo*
*metric fixed
point spectrum of equivariant connective K-theory.
After our algebraic_calculations, and especially Section 6, we see that its c*
*oefficients should
be the_Z[v]-algebra R [y; y-1; v]=(v = Oy): we have also met this as the R(G)[v*
*]-submodule
of R[1=p][v; v-1] generated by the elements_O-nvn._Since v acts in the solenoid*
*al fashion_by
multiplication with Oy we call this module R_Sol, and emphasize that it is addi*
*tively R in
each even degree. Since v - Oy is regular, RSol admits a presentation
__ -1 v-Oy__ -1 __
0 -! R[y; y ; v] -! R[y; y ; v] -! RSol -! 0
by free Z[v]-modules. Now ku is an E1 -ring spectrum by infinite loop space th*
*eory, and
hence an algebra over the sphere spectrum in the sense of [4]. Working_in the *
*homotopy
category of ku-modules [4] we may therefore define a ku-module kuR Sol by copyi*
*ng this:
__ -1 v-Oy __ -1 __
kuR [y; y ] -! kuR [y; y ] -! kuR Sol:
We begin with the more elementary properties.
__
Lemma 7.1. (i) The homotopy groups of kuR Sol are
__ __
ssG*(kuR Sol) ~=R Sol
__
as ku* = Z[v] modules, and thus R in each even degree, and 0 in each odd degree.
(ii) Inverting p we have an equivalence
__ __
kuR Sol[1=p] ' KR [1=p]
Proof: Part (i) is immediate from the definition and the fact_that v-Oy is a re*
*gular element.
*
* __
Part (ii) comes from the fact that O is an isomorphism of R[1=p]. *
* |__|
__
The defining cofibre sequence for kuR Sol gives the following calculation.
Lemma 7.2. Given a ku-module M we have a short exact sequence
__ -1 __ __ *
* __
0 -! Ext1ku*(kuR Sol*; M*) -! [kuR Sol; M]ku -! Hom ku*(kuR Sol*; M*) -! 0: *
* |__|
__
Corollary 7.3. The ku-module kuR Sol is determined up to equivalence by its hom*
*otopy_as
a ku*-module. *
* |__|
__
The critical property for us is that kuR Sol^psplits as a product of Eilenber*
*g-MacLane
spectra. From coefficients alone_one sees this is_not a splitting_of ku-modules*
*. This contrasts
with the fact that, neither kuR Sol, nor even kuR Sol[1=p] ' KR [1=p] splits as*
* a product of
Eilenberg-MacLane spectra.
To obtain a splitting we need maps to Eilenberg-MacLane spectra. Fortunately,*
* the Z[v]-
module (HZ^p)*(ku) is known and determined by its effect in homotopy.
10 J.P.C.GREENLEES
Q
Lemma 7.4. [1, III.16.5] For r 0 define a numerical function by m(r) = pp[*
*r=p-1].
(i) The homology of ku is additively Z in each even degree 0, and as a Z[v]-mo*
*dule
HZ*(ku) HQ*(ku) = Q[v] it is generated by vr=m(r).
(ii) The cohomology HZ*(ku) is additively Z in each even degree. Furthermore, t*
*he degree
map
HZ*(ku) = [ku; HZ]* -! Hom (ku*; HZ*)
is injective, and the image of a generator in cohomological degree 2r is multip*
*lication by
m(r) as a map Z = ku2r -! (HZ)0 = Z. As a submodule of HQ*(ku) = Q[v]* =
Q{v0; v1; v2; : :}:(with vvr+1= vr; v-1 = 0) it is additively generated by the *
*elements_m(r)vr
for r 0. *
*|__|
The splitting arises even after localizing at p.
__
Proposition 7.5. The p-local spectrum kuR Sol(p)splits as a wedge of Eilenberg-*
*MacLane
spectra: __ Y __
kuR Sol(p)' 2nHR (p):
n
Proof: It suffices to construct maps
__ 2n __
2n : kuR Sol(p)-! HR (p)
which induce an isomorphism_in ss2n. We_use multiplication by powers of y to id*
*entify each
even homotopy group of kuR Sol(p)with R(p). If the map 2n exists, the composite*
* 2m2n
__ y-m __ -1 __ 2n __
2mkuR (p)-! kuR (p)[y; y ] -! kuR Sol(p)-! HR (p)
must be multiplication by Om-n provided m n.
We reverse this deduction to give a construction of 2n. By 7.4 the map
__ 2n __ __ __ __
ss2n : [2mkuR (p); HR (p)] -! Hom (R (p); R(p)) = R(p)
is injective and has image consisting of multiples of p[(m-n)=p-1]. Since Om-n *
*is divisible by
p[(m-n)=p-1], the map 2m2nexists as required. Now assemble the 2m2ninto a map
__ -1 2n __
02n: kuR (p)[y; y ] -! HR (p):
This has the property that the composite
__ -1 v-Oy __ -1 2n __
kuR (p)[y; y ] -! kuR (p)[y; y ] -! HR (p)
is trivial in homotopy, and hence trivial by 7.4. Hence 02nextends to a map 2n*
*. The __
extension is unique since H*(ku) is in even degrees. *
* |__|
Remark 7.6. (i) Because of the fact that t(ku) = F (EG+; ku) ^ "EP is p-local*
*, the calcula-
tion 7.4 is also behind the splitting of t(ku) into Eilenberg-MacLane spectra.
(iii) It is therefore an arithmetic coincidence that has led to t(ku)G splittin*
*g for the group
of order p. For_a general group it is natural to expect the proper Tate spectr*
*um tP(ku)
to be like kuR Sol in character. Similarly if is the cyclic group of order p2 *
*it should not be
hard to identify the -spectrum t(ku) exactly, using the equivariant form of ku *
*constructed
here.
EQUIVARIANT FORMS OF CONNECTIVE K THEORY 11
8. Construction of equivariant connective K theory
In this section we construct a G-spectrum E, which is the best available equi*
*variant form
of connective K-theory. Ultimately this should be called simply ku, but, until *
*its properties
are justified, we use the more neutral name E. We shall see in Sections 9 to 11*
* that it has
Properties 1.1 as described in Section 1.
For the construction, we use the pullback square
E -! "EP ^ E ' "EP ^ GE
# o2 #
F (EG+; E) -! t(E)
k k
F (EG+; ku) t(ku)
We already know the spectra at the bottom: indeed, we have written ku because t*
*he spectra
are independent of which equivariant form of ku has been used. For definitenes*
*s, and to
emphasize the logic of the construction we should use infG1ku. The homotopy gro*
*ups con-
cerned were described in the_Section 5; geometrically, by [3] we know that t(ku*
*) is a wedge
^
of suspensions of "EP ^ HR p, one in each even degree. In view of our Conventio*
*n 3.2, the
action by v should give comparison between different degrees, so we should disp*
*lay this more
visibly. Note that we have a comparison map t(ku) -! t(K), and we may build_thi*
*s into
^
the notation by writing HM for an Eilenberg-MacLane spectrum where M Rp[1=p]. *
*Now,
__^ __^ __^ __^
as R(G)-modules OkR p~=R p, so abstractly H(OkR p) ' HR p. Thus we write
Y __^
t(ku) ' "EP ^ 2nH(O-nR p);
n
thereby implicitly giving a map to
Y __^
t(K) ' "EP ^ 2nH(O-nR p[1=p]):
n
To complete the construction, we need only_specify GE and give a map to t(ku). *
* If we
are to define E so as to ensure GE = kuR Sol, the right hand vertical in the Ta*
*te pullback
square will need to be a map
_ __^
"EP ^ ku__RSol ' "EP ^ E -! t(E) ' "EP ^ 2nH(O-nR p):
n
It is equivalent to get a non-equivariant map
__ _ 2n -n__^
: kuR Sol -! H(O R p):
n
This must be done so as to be compatible with the known map for periodic K-theo*
*ry. In
other words we must obtain a square
__ W __^
kuR Sol -! n2nH(O-nR p)
__ # #
W __^
KR [1=p] -! n2nH(R p[1=p])
12 J.P.C.GREENLEES
which in ss2n is the pullback square of inclusions
__ __^
O-nR -! O-nR p
__ # #_
^
R [1=p] -! Rp[1=p]:
We may take to be the completion map followed by the splitting of 7.5.
Definition 8.1. We define the principal equivariant form of ku to be the G spec*
*trum E
given by the pullback square
__
E -! "EP ^ kuR Sol
# o2 # 1 ^
F (EG+; ku) -! t(ku)
9.The first properties of equivariant connective K theory
In this section we begin to investigate the properties of the spectrum E defi*
*ned in 8.1. We
calculate its coefficients, show that it is split and that it is well related t*
*o periodic K-theory.
Lemma 9.1. The spectrum E is split with underlying non-equivariant spectrum k*
*u.
Proof: The spectrum E is certainly non-equivariantly ku since the two right han*
*d entries in
the Tate pullback square are non-equivariantly contractible.
To show that E is split, we must construct a map s : infG1(ku) -! E. Indeed, *
*we have
a natural map s0: infG1(ku)_-! F (EG+; infG1(ku)) = F (EG+; ku), and we_choose *
*the map
s00: infG1(ku) -! E"P ^ kuR Sol corresponding to the unit map ku -! kuR Sol of *
*non-
equivariant spectra.
By construction, the composites of s0and s00into t(ku) are both induce the in*
*clusion of
Z[v] in homotopy. Since homotopy detects such maps by 7.4, s0and s00are compati*
*ble and
hence combine to give a map s as required. Furthermore this specifies s uniquel*
*y: there are
no maps of odd degree infG1(ku) -! t(ku) because (HZ^p)*(ku) is zero in odd deg*
*rees by __
7.4. The map s is a non-equivariant equivalence because s0is by construction. *
* |__|
Proposition 9.2. The spectrum E has Property 2:
E[1=v] = K:
Proof: To obtain a comparison map we need to compare the defining pullback for *
*E with
the pullback for periodic K-theory.
We now need only form
Q __^ __
F (EG+; ku) -! t(ku) ' "EP ^ 2n2nH(O-nR p) - "EP ^ kuR Sol
# # #
Q __^ __
F (EG+; K) -! t(K) ' "EP ^ 2n2nHR p[1=p] - E"P ^ KR [1=p]
Here the left hand square exists and commutes by the definition of Tate cohomol*
*ogy._We
have_chosen E so_that_the right hand vertical comes from the natural map kuR So*
*l - !
kuR Sol[1=p] ' KR [1=p], where the last equivalence was proved in 7.1. The resu*
*lting right
hand square induces a commutative square of homotopy groups by construction; si*
*nce the
codomain is rational, this ensures the square itself is homotopy commutative.
EQUIVARIANT FORMS OF CONNECTIVE K THEORY 13
Finally, we need to know the resulting map E -! K becomes an equivalence when*
* v is
inverted. Of course v is already invertible in the second row, so it is suffici*
*ent (though not
necessary) to show that if we invert v in the first row we obtain the second. *
*Since vp-1
is p times an isomorphism, this is holds in the middle and on the right. Finall*
*y we claim
F (EG+; ku)[1=v] = F (EG+; ku[1=v]). This is a calculation not a formality, bu*
*t we have_
already reported the relevant facts in Section 5 *
* |__|
We calculate the coefficient group of E, giving it a ring structure as a subr*
*ing of K*G. In
the next section we show this agrees with the product arising from the product *
*we construct
on E.
Proposition 9.3. The coefficient ring of the principal equivariant form of conn*
*ective K-
theory is
E*G= R(G)[v; y]=(vy = O; yae);
where v is of degree 2, y is of degree -2, ae is the regular representation and*
* O = 1 - ff.
Proof: We first observe that the map E -! K induces an injective map on coeffic*
*ients.
Indeed, we consider the induced map of Mayer-Vietoris sequences
__ __
0 -! E*G -! ku*(BG+) RSol -! R Sol^p -! 0
# #_ __ #
0 -! K*G -! K*(BG+) RSol[1=p] -! R Sol^p[1=p]-! 0:
__ __
Note that it suffices to observe that ku*(BG+) -! K*(BG+) and kuR Sol* -! KR [1*
*=p]*
are injective: this was shown in Section 5 and Lemma 7.1.
This lets us describe E*Gas a subring of K*G= R(G)[v; v-1]. In degrees 0 and*
* above
ku*(BG+)_-! K*(BG+) is an isomorphism, and any element of the_kernel of R(G)^J
R [1=p] -! R(G)^J[1=p] necessarily has second component in R. Since t(ku) -! t(*
*K) is also
injective in homotopy we see E*G-! K*Gis an isomorphism in_positive degrees.
In negative degrees the maps ku*(BG+) -! t(ku)G*and kuR Sol* -! t(ku)G*are bo*
*th
injective, so the coefficient group EG-2nis their intersection
__^ n__ n__ __
EG-2n= OnR p\ O R = O R : |__|
Remark 9.4. A useful consequence of this calculation is that if L is a free m*
*odule over E*G
on a single generator of degree d then x 2 L is a generator if and only if (i) *
*x is a generator
of L[1=v] and (ii) x is in degree d.
10. Multiplicative properties of equivariant connective K-theory.
In this section we show that E is a commutative ring spectrum and that the ma*
*p E -! K
is a map of ring spectra.
Proposition 10.1. The spectrum E is a split ring spectrum with underlying non-e*
*quivariant
spectrum ku.
Proof: We need to check that E is a ring spectrum, and that the splitting is a *
*map of ring
spectra. The trick here is to recognize that the difficulties at and away from *
*p are different.
14 J.P.C.GREENLEES
We can treat the problems separately and then assemble them using the arithme*
*tic Hasse
pullback square
E -! E[1=p]
# o2 # 1 ^
E^p -! E^p[1=p]
The main point is that this gives an exact sequence for calculating [E^i; E]G.
Lemma 10.2. There is an exact sequence
0 -! [E^i; E]G -! [E^i; E[1=p]]G [E^i; E^p]G -! [E^i; E^p[1=p]]G
Proof: It remains to verify that [E^i; E^p[1=p]]G*is concentrated in even degre*
*es.
Note first that by 9.3, the group E*Gconsists of finitely generated abelian g*
*roups in even
degrees, so the coefficients of E^p[1=p] are finite dimensional vector spaces o*
*ver Qp in even
degrees. Since rational G-spectra split as products of Eilenberg-MacLane spectr*
*a by [7, Ap-
pendix A], both E^i Q and E^p[1=p] are wedges of rational Eilenberg-MacLane spe*
*ctra in
even degrees. The result follows since the category of rational G-Mackey functo*
*rs_has_global
dimension 0. *
* |__|
It thus suffices to construct products ^pon E^pand [1=p] on E[1=p] that are s*
*uitably
compatible. Indeed this gives us
^p ^
E ^ E -! E^p^ E^p-! Ep
and similarly for [1=p], which we must show agree when composed with the natura*
*l map
into the rational spectrum E^p[1=p]. Thus compatibility may be verified in homo*
*topy.
__
First consider the p-completion. By 7.5, the map : E"P ^ kuR Sol -! t(ku) b*
*ecomes
an equivalence on p-completion, so that E^p' F (EG+; ku^p). Since non-equivaria*
*nt ku is a
commutative and associative ring spectrum the same is true for E^p. Indeed, if *
*we use the
highly structured inflation of Elmendorf-May [5], we can construct E^pas a ku-a*
*lgebra.
For E[1=p] we may adopt a more naive approach. The analogue of the following *
*lemma is
false without inverting p.
Lemma 10.3. For s 0 the group [E^s[1=p]; E[1=p]]G is the kernel of
__ ^s __ Y __ ^s 2n*
* __^
[ku^s[1=p]; ku[1=p]] x [kuR Sol [1=p]; kuR Sol[1=p]] -! [kuR Sol [1=p]; *
*HR p[1=p]]
n
Proof: This follows from the Mayer-Vietoris sequence of the defining pullback s*
*quare. We
need to calculate maps into the three corners. First note that for any G-spectr*
*a X and Y we
have [X[1=p]; Y [1=p]]G = [X[1=p]; Y ]G. This avoids difficulties in using the *
*smash function
adjunction. We may now calculate
[E^s[1=p]; F (EG+; ku)]*G=[E^s^ EG+[1=p]; ku]*G
= [F (EG+; ku)^s^ EG+[1=p]; ku]*G
= [ku^s^ BG+[1=p]; ku]
__
[E^s[1=p]; E ^ "EP]*G=[GE^s[1=p];_kuR Sol]*_
= [kuR Sol^s[1=p]; kuR Sol]*
EQUIVARIANT FORMS OF CONNECTIVE K THEORY 15
and Q __^
[E^s[1=p]; t(E)]*G=[GE^s[1=p];_ nH2nRp ]*
Q __^
= [kuR Sol^s[1=p]; nH2nRp ]*
Finally_we need to_know_the relevant odd dimensional part is zero. This follow*
*s since
^ __^
[kuR Sol^s[1=p]; HR p[1=p]] is zero in odd degrees:_indeed, Rp [1=p] is rationa*
*l, and so it suf-
fices to observe the rational homotopy of kuR Sol^s is in even degrees, as foll*
*ows_from the
case s = 1. *
* |__|
To define a product_[1=p] we must find a compatible pair of products 0[1=p] (*
*on ku[1=p])
and 00[1=p] (on kuR Sol[1=p]). To obtain 0[1=p] we invert p on the usual produ*
*ct 0 :
ku^ku -! ku. For 00[1=p] we take geometric_fixed points of the usual product on*
* equivariant
K-theory and use the fact that GK ' kuR Sol[1=p]. To check these are compatible*
* we need
only verify they agree in rational homotopy.
This shows that E is a commutative ring spectrum. We need to check the splitt*
*ing map
of 9.1 is a ring_map. The map s0is a ring map integrally. At p we have only def*
*ined the ring
structure on kuR Sol via its equivalence with t(ku), and away from p, the map s*
*00is a ring
map because of the square
__
infG1(ku)[1=p]-! kuR Sol[1=p]
# __ #'
__
infG1(K)[1=p]-! KR [1=p]: |__|
Lemma 10.4. The map E -! E[1=v] ' K constructed above is a map of ring spectr*
*a.
Proof: This follows by comparing pullback squares, since K may be shown to be a*
* ring_
spectrum by the same method as we used for E. *
* |__|
11.Complex orientability of equivariant connective K-theory.
Finally we establish that the spectrum E has a canonical complex orientation.
Proposition 11.1. The spectrum E is complex stable.
Proof: We need to find a Thom class for Sfifor all one dimensional representati*
*ons fi. This
is trivial if fi is G-fixed, and the other spheres are homeomorphic as G-spaces*
*, so it suffices
to deal with fi = ff. One way to do this is to construct equivalences E ^ Sff' *
*E ^ S2. We do
this again by using the defining pullback square, so it suffices to find vertic*
*al equivalences in
the diagram
W __^ *
* __
S2 ^ F (EG+; ku) -! S2 ^ t(ku) ' S2 ^ "EP ^ n 2nH(O-nR p) - S2 ^ "EP ^ k*
*uR Sol
a # A b # B c #
W __^ *
* __
Sff^ F (EG+; ku) -! Sff^ t(ku) ' Sff^ "EP ^ n 2nH(O-nR p) - Sff^ "EP ^ k*
*uR Sol
so that the squares A and B commute.
For the square B, we note that Sff^ "EP ' E"P, so the idea is to take b and c*
* to be
multiplication by O. For b this amounts to a shift of terms in the wedge of_Eil*
*enberg-MacLane_
spectra, and for c we use 7.2 to deduce there is a unique ku-map Gc : 2kuR Sol *
*-! kuR Sol
16 J.P.C.GREENLEES
inducing multiplication by O in homotopy._ To see the resulting square B is com*
*mutative
note_that the codomain_is E"P ^ kuR Sol^pas a ku-module. Now, by 7.2 again, ku*
*-maps
2kuR Sol -! kuR Sol^pare detected by their effect in homotopy.
It remains to check we may find an equivalence a so as to make the square A c*
*ommute.
Noting that EG+ ' holim S(kff)+, we view the map F (EG+; ku) -! t(ku) = F (E"P;*
* ku ^
! k
EG+) as
holim D(S(kff)+) ^ EG+ ^ ku -! holim D(Skff) ^ EG+ ^ ku;
k k
so to obtain a map it suffices to use a fixed Thom equivalence EG+ ^ku^S2 ' EG+*
* ^ku^Sff,
smash it with the maps D(S(kff)+) -! D(Skff) and pass to inverse limits. This g*
*ives a map
on inverse limits, unique up to maps which are zero on homotopy; we make an arb*
*itrary choice
of one. Finally, we need to know that our two descriptions of the map b are con*
*sistent. This
is the statement that v = Oy, since O is also the ku-Euler class of ff: in othe*
*r words we need
the diagram
W __^
holim 3BG-kff^ ku - ' n2n+2H(O-nR p)
k
O # # O
W __^
holim BG-kff^ ku - ' n2nH(O-nR p)
k
to commute. Up to this point we have not needed to know anything more about the*
* horizontal
equivalence than their effect in homotopy. We now claim it may be chosen so tha*
*t the above
diagram commutes. Indeed a map
__^ _ 2n+2 -n__^
2i: 2iH(O-iRp ) -! H(O R p)
n
will be part of an equivalence provided its component 2i2iin the ith factor is *
*an isomorphism.
Now the composite of 2i2iand the Thom isomorphism may have a non-zero component*
* in
degree 2(i+j) for j 0; we may choose 2(i+j)2ito cancel this, because the Thom *
*isomorphism_
is an equivalence on the 2(i + j)th factor. *
* |__|
To treat complex orientations, recall that there is a cofibre sequence CP (V *
*) -! CP (V
-1 *
ff) -! SV ff . A complex orientation is an element x 2 EG(CP (U)+) so that (i) *
*x restricts
to zero on CP (ffl)+ = *+ and (ii) x restricts to a generator on (CP (ffl ff);*
* CP (ff)).
Proposition 11.2. The spectrum E is complex orientable.
Proof: Periodic K-theory has a natural orientation, xK 2 K2G(CP (U)+); the rest*
*riction of
this to K2(CP (ffl ff); CP (ffl)) ~=K2G(Sff) is the Bott class. It suffices to*
* show that xK is the
image of an element x 2 E*G(CP (U)+); indeed, since E*G-! K*Gis injective it fo*
*llows that
its restriction to CP (ffl)+ is zero, and by Remark 9.4 it follows that the res*
*triction of x(ffl) to
(CP (ffl ff); CP (ff)) is a generator.
To see that xK lifts to E theory we choosena complete flag ffl = V 1 V 2nV 3 *
*. .i.n U.
We argue by induction that the class xVK on CP (V n) lifts to a class xV in E *
*theory. This
is immediate by degree for n = 1. For the inductive step observe that the cofib*
*re sequence
-1
CP (V ) -! CP (V ff) -! SV ff , together with complex stability and concentrat*
*ion in
EQUIVARIANT FORMS OF CONNECTIVE K THEORY 17
even degrees allows us to deduce that we have a split short exact sequence in b*
*oth E theory
and K theory:
-1 * *
0 - ! E*G(SV ff ) -! EG(CP (V ff)) -! EG(CP (V )) - ! 0
# # #
-1 * *
0 - ! K*G(SV ff ) -! KG(CP (V ff)) -! KG(CP (V )) - ! 0:
From the top exact sequence xV lifts to a class __xV,ffand because the left han*
*d vertical is
surjective in the relevant degree we can add an adjustment term to obtain xV ff*
*mapping_to
xVKff. *
* |__|
12. Highly structured products
The author does not know if E admits the structure of a commutative S0-algebr*
*a, or that
of an algebra over infG1ku. However, we have a partial result in this direction*
*, in the spirit
of those of McClure [9]. This result is independent of 10.1.
McClure works in the language of E1 -ring spectra, so we pause to translate h*
*is results
into the language of [4]. I am grateful to J.P.May for guidance. Up until now*
*, we have
assumed that G-spectra are indexed on a complete G-universe U. We now need to c*
*onsider
naive G-spectra indexed on a G-fixed universe, such as UG. We have the forgetfu*
*l functor
N : G-spectra-! Naive G-spectra:
The results of [4] are independent of universe, and apply equally well to naive*
* G-spectra.
We refer to the 0-sphere spectrum in the relevant category as S0; as is usually*
* the way
with suspension spectra, this means something different in the categories of no*
*n-equivariant
spectra, of G-spectra and of naive G-spectra. We shall say that a (genuine) G-s*
*pectrum E is
a naive S0-algebra if NE is an S0-algebra. Since passage to Lewis-May fixed poi*
*nts factors
through N, this means that EG is an S0-algebra.
By the work of Elmendorf-May [5], since ku is a non-equivariant S0-algebra, t*
*he inflation
infG1ku may be taken to be an equivariant S0-algebra. Hence F (EG+; ku) may be *
*taken to
be an S0-algebra. McClure's results state that if E is is an equivariant S0-alg*
*ebra then the
Tate spectrum t(E) is never an equivariant S0-algebra (unless it is contractibl*
*e), but it is a
naive S0-algebra.
Theorem 12.1. The spectrum E may be constructed so that NE is homotopy equiva*
*lent to
an S0-algebra, and the map NE -! NK is homotopic to a map of S0-algebras.
Proof: We still use the defining pullback square, but this time in the category*
* of naive
S0-algebras.
__
Lemma 12.2. The non-equivariant spectrum kuR Sol may be constructed as an S0-*
*algebra.
__ __ __
Proof: The spectrum may be obtained from kuR Sol[1=p], kuR Sol^pand kuR Sol^p[1*
*=p] by
an arithmetic_Hasse square._
Now kuR Sol[1=p] ' KR [1=p] has the structure of_a ku-algebra_since K may be *
*constructed
from ku by Bousfield localization. The spectra kuR Sol^pand kuR Sol^p[1=p] are *
*generalized
Eilenberg-MacLane spectra and hence S0-algebras.
18 J.P.C.GREENLEES
Replacing the_two maps in the fork by fibrations to ensure the right homotopy*
* type, we_
construct kuR Sol as a pullback of S0-algebras. *
* |__|
__ __^
The map : kuR Sol -! nHR p may be constructed as a map of S0-algebras since
__
it is p-completion._ It follows from [9] that E"G ^ kuR Sol is a naive S0-algeb*
*ra, and the
map "EG ^ kuR Sol -! t(ku) is a map of naive S0-algebras. Furthermore McClure s*
*hows
F (EG+; ku) -! t(ku) is also a map of naive S0-algebras. Because t(ku) is a pr*
*oduct of
Eilenberg-MacLane spectra, the two implicit S0-algebra structures on it my be t*
*aken to
agree. Replacing the two maps in the fork of the defining pullback by fibration*
*s to ensure_
the right homotopy type, we construct NE as a pullback of S0-algebras. *
* |__|
References
[1]J.F.Adams "Stable homotopy and generalized homology." Chicago Univ. Press (*
*1974)
[2]D. Bayen and R.R.Bruner "Real connective K-theory and the quaternion group.*
*" Trans. American
Math. Soc. 348 (1996) 2201-2216
[3]D.Davis and M.E.Mahowald "The spectrum (P ^ bo)-1 " Math. Proc. Cambridge P*
*hil. Soc. 96 (1984)
85-93
[4]A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May. Rings, Modules and *
*Algebras in Stable Homotopy
Theory, Volume 47 of Amer. Math. Soc. Surveys and Monographs. American Math*
*ematical Society, 1996.
[5]A. D. Elmendorf and J. P. May "Algebras over the equivariant sphere spectru*
*m." J. Pure and Applied
Algebra (1997)
[6]J.P.C.Greenlees "Tate cohomology in commutative algebra" J. Pure and Applie*
*d Algebra 94 (1994)
59-83
[7]J.P.C.Greenlees and J.P.May "Generalized Tate cohomology." Mem. American Ma*
*th. Soc. 543 (1995)
viii+178pp
[8]D.C.Johnson and W.S.Wilson "On a theorem of Ossa." Proc. American Math. Soc*
*. (to appear)
[9]J.E.McClure "E1 -ring structures for Tate spectra." Proc. American Math. So*
*c. 124 (1996) 1917-1922
[10]H.Matsumura "Commutative ring theory" Cambridge Univ. Press (1986)
[11]J.P.May "Equivariant and non-equivariant module spectra." Preprint (1996) 1*
*5pp
[12]E.Ossa "Connective K-theory of elementary abelian groups" Lecture Notes in *
*Maths 1375 Springer-
Verlag (1989) 269-275
School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK.
E-mail address: j.greenlees@sheffield.ac.uk