PRODUCTS ON MU-MODULES
N. P. STRICKLAND
Abstract.Elmendorf, Kriz, Mandell and May have used their technology of m*
*odules over
highly structured ring spectra to give new constructions of MU-modules su*
*ch as BP, K(n) and
so on, which makes it much easier to analyse product structures on these *
*spectra. Unfortunately,
their construction only works in its simplest form for modules over MU[1_*
*2]*that are concentrated
in degrees divisible by 4; this guarantees that various obstruction group*
*s are trivial. We extend
these results to the cases where 2 = 0 or the homotopy groups are allowed*
* to be nonzero in all
even degrees; in this context the obstruction groups are nontrivial. We s*
*hall show that there are
never any obstructions to associativity, and that the obstructions to com*
*mutativity are given
by a certain power operation; this was inspired by parallel results of Mi*
*ronov in Baas-Sullivan
theory. We use formal group theory to derive various formulae for this po*
*wer operation, and
deduce a number of results about realising 2-local MU*-modules as MU-modu*
*les.
1.Introduction
A great deal of work in algebraic topology has exploited the generalised coho*
*mology theory
MU*(X) (for spaces X), which is known as complex cobordism; good entry points t*
*o the literature
include [2, 26, 25, 29]. This theory is interesting because of its connection w*
*ith the theory of formal
group laws (FGL's), starting with Quillen's fundamental theorem [23, 24] that M*
*U* is actually
the universal example of a ring equipped with an FGL.
Suppose that we have a graded ring A* equipped with an FGL. In the cases disc*
*ussed below, the
FGL involved will generally be the universal example of an FGL with some intere*
*sting property.
Examples include the rings known to topologists as BP *, P (n)*, K(n)* and E(n)*
**; see Section 2
for the definitions. It is natural to ask whether there is a generalised cohomo*
*logy theory A*(X)
whose value on a point is the ring A*, and a natural transformation MU*(X) -! A*
**(X), such
that the resulting map MU* -!A* carries the universal FGL over MU* to the given*
* FGL over
A*. This question has a long history, and has been addressed by a number of dif*
*ferent methods
for different rings A*. The simplest case is when A* is obtained from MU* by in*
*verting some
set S of nonzero homogeneous elements, in other words A* = S-1MU*. In that case*
* the functor
A*(X) = A* MU* MU*(X) is a generalised cohomology theory on finite complexes, *
*which can
be extended to infinite complexes or spectra by standard methods. For example, *
*given a prime
p one can invert all other primes to get a cohomology theory MU*(p)(X). Cartier*
* had previously
introduced the notion of a p-typical FGL and constructed the universal example *
*of such a thing
over BP *, which is a polynomial algebra over Z(p)on generators vk in degree -2*
*(pk-1) for k > 0.
It was thus natural to ask our "realisation question" for A* = BP *. Quillen [*
*23] constructed
an idempotent self map ffl: MU*(p)(X) -! MU*(p)(X), whose image is a subring, w*
*hich we call
BP *(X). He showed that this is a cohomology theory whose value on a point is t*
*he ring BP *,
and that the FGL's are compatible in the required manner. This cohomology theor*
*y was actually
defined earlier by Brown and Peterson [6] (hence the name), but in a less struc*
*tured and precise
way. It is not hard to check that we again have BP *(X) = BP * MU* MU*(X) when *
*X is finite.
This might tempt us to just define A*(X) = A* MU* MU*(X) for any A*, but unfor*
*tunately
this does not usually have the exactness properties required of a generalised c*
*ohomology theory.
Another major advance was Landweber's determination [13] of the precise conditi*
*ons under which
A* MU* MU*(X) does have the required exactness properties, which turned out to*
* be natural
ones from the point of view of formal groups. However, there are many cases of *
*interest in which
Landweber's exactness conditions are not satisfied, and for these different met*
*hods are required.
Many of them are of the form A* = (S-1MU*)=I for set S of homogeneous elements *
*and some
1
2 N. P. STRICKLAND
homogeneous ideal I S-1MU*. For technical reasons things are easier if we ass*
*ume that I is
generated by a regular sequence, in other words I = (x1, x2, . .).and xk is not*
* a zero-divisor in
(S-1MU*)=(xj | j < k). If A* arises in this way, we say that it is a localised *
*regular quotient
(LRQ) of MU*. If S = ; we say that A* is a regular quotient of MU*. The first*
* advance in
this context was the Baas-Sullivan theory of cobordism of manifolds with singul*
*arities [4]. Given
a regular quotient A* of MU*, this theory constructed a cohomology theory A*(X)*
*, landing in
the category of MU*-modules, and a map MU*(X) -! A*(X). Unfortunately, the deta*
*ils were
technically unwieldy, and it was not clear whether A*(X) was unique or whether *
*it had a natural
product structure, and if so whether it was commutative or associative. Some of*
* these questions
were addressed by Shimada and Yagita [27], Mironov [18] and Morava [20], largel*
*y using the
geometry of cobordisms. Another idea was (in special cases, modulo some technic*
*al details) to
calculate the group of all natural transformations A*(X) A*(X) -!A*(X) and th*
*en see which
of them are commutative, associative and unital. This was the approach of Würgl*
*er [33, 31, 32];
much more recently, Nassau has corrected some inaccuracies and extended these r*
*esults [22, 21].
Baas-Sullivan theory eventually yielded satisfactory answers for rings of the*
* form MU*=x,
but the work involved in handling ideals with more than one generator remained *
*rather hard.
The picture changed dramatically with the publication of [9] by Elmendorf, Kriz*
*, Mandell and
May (hereafter referred to as EKMM), which we now explain. Firstly, the natural*
* home for our
investigation is not really the category of generalised cohomology theories, bu*
*t rather Boardman's
homotopy category of spectra [2, 16], which we call B. There is a functor 1 fr*
*om finite complexes
to B, and any cohomology theory A*(X) on finite complexes is represented by a s*
*pectrum A 2 B
in the sense that An(X) = [ 1 X, nA] for all n and X. The representing spectru*
*m A is unique
up to isomorphism [5, 1], and the isomorphism is often unique. There have been *
*many different
constructions of categories equivalent to B. The starting point of [9] was EKMM*
*'s construction
of a topological model category M with a symmetric monoidal smash product, whos*
*e homotopy
category is equivalent to B. This was previously feared to be impossible, for *
*subtle technical
reasons [14]. EKMM were also able to construct a version of MU which was a stri*
*ctly commutative
monoid in M, which allowed them to define the category MMU of MU-modules. They *
*showed how
to make this into topological model category, and thus defined an associated ho*
*motopy category
DMU . This again has a symmetric monoidal smash product, which should be though*
*t of as a sort
of tensor product over MU. They showed that the problem of realising LRQ's of M*
*U* becomes
very much easier if we work in DMU (and then apply a forgetful functor to B if*
* required). In fact
their methods work when MU is replaced by any strictly commutative monoid R in *
*M such that
R* is concentrated in even degrees. They show that if A* is an LRQ of R* and 2 *
*is invertible in
A* and A* is concentrated in degrees divisible by 4, then A can be realised as *
*a commutative and
associative ring object in DR.
In the present work, we will start by sharpening this slightly. The main poi*
*nt here is that
EKMM notice an obstruction to associativity in A4k+2, so they assume that these*
* groups are
zero. Motivated by a parallel result in Baas-Sullivan theory [19], we show that*
* the associativity
obstructions are zero even if the groups are not (see Remark 3.10). We deduce t*
*hat if A* is an
LRQ of R* and 2 is invertible in A* then A can be realised as a commutative and*
* associative
ring in DR, in a way which is unique up to unique isomorphism (Theorem 2.6). We*
* also prove a
number of subsidiary results about the resulting ring objects.
The more substantial part of our work is the attempt to remove the condition *
*that 2 be invertible
in A*, without which the results become somewhat more technical. We show that t*
*he obstruction
to defining a commutative product on R=x is given by eP(x) for a certain power *
*operation eP:Rd -!
R2d+2=2. This was again inspired by a parallel result of Mironov [19]. We deduc*
*e that if A* =
S-1R*=I is an LRQ of R* without 2-torsion and eP(I) I (mod 2) then A* is agai*
*n uniquely
realisable (Theorem 2.7). When A* has 2-torsion we have no such general result *
*and must proceed
case by case. Again following Mironov, we show that when R = MU, the operation *
*ePcan be
computed using formal group theory. We considerably extend and sharpen Mironov'*
*s calculations,
using techniques which I hope will be useful in more general work on power oper*
*ations. Using these
results, we show that many popular LRQ's of MU*(2)have almost unique realisatio*
*ns as associative,
PRODUCTS ON MU-MODULES 3
almost commutative rings in DMU . See Theorems 2.12 and 2.13 for precise state*
*ments. The
major exceptions are the rings BP * and E(n)*, but we show that even these b*
*ecome uniquely
realisable as commutative rings in DMU if we allow ourselves to modify the usua*
*l definition slightly.
We call the resulting spectra BP 0and E(n)0; they are acceptable substitutes*
* for BP and
E(n) in almost all situations.
2.Statement of Results
We use the category M of S-modules as constructed in [9]; we recall some deta*
*ils in Section 8.
The main point is that M is a symmetric monoidal category with a closed model s*
*tructure whose
homotopy category is Boardman's homotopy category of spectra. We shall refer to*
* the objects of
M simply as spectra.
Because M is a symmetric monoidal category, it makes sense to talk about stri*
*ctly commutative
ring spectra; these are essentially equivalent to E1 ring spectra in earlier fo*
*undational settings.
Let R be such an object, such that R* = ß*R is even (by which we mean, concentr*
*ated in even
degrees). We also assume that R is q-cofibrant in the sense of [9, Chapter VII]*
* (if not, we replace R
by a weakly equivalent cofibrant model). The main example of interest to us is *
*R = MU . There
are well-known constructions of MU as a spectrum in the earlier sense of Lewis *
*and May [15],
with an action of the E1 operad of complex linear isometries. Thus, the results*
* of [9, Chapter II]
allow us to construct MU as a strictly commutative ring spectrum.
One can define a category MR of R-modules in the evident way, with all diagra*
*ms commuting at
the geometric level. After inverting weak equivalences, we obtain a homotopy ca*
*tegory D = DR,
referred to as the derived category of MR. We shall mainly work in this derived*
* category, and
the category R = RR of ring objects in D (referred to in [9] as R-ring spectra)*
*. All our ring
objects are assumed to be associative and to have a two-sided unit. Thus, an ob*
*ject A 2 R has an
action R ^S A -!A which makes various diagrams commute at the geometric level, *
*and a product
A ^S A -!A that is geometrically compatible with the R-module structures, and i*
*s homotopically
associative and unital. We also write R* for the category of algebras over the *
*discrete ring R*.
We write Re*for the category of even R*-algebras, and Rc*for the commutative on*
*es, and similarly
Rec*, Re, Rc and Rec.
Definition 2.1.Let A* be an even commutative R*-algebra without 2-torsion. A st*
*rong realisa-
tion of A* is a commutative ring object A 2 Recwith a given isomorphism ß*(A) '*
* A*, such that
the resulting map
R(A, B) -!R(A*, ß*(B))
is an isomorphism whenever B 2 Recand B* has no 2-torsion. We say that A* is st*
*rongly realisable
if such a realisation exists.
Remark 2.2. It is easy to see that the category of strongly realisable R*-algeb*
*ras is equivalent
to the category of those A 2 Recfor which ß*(A) is strongly realisable. In part*
*icular, any two
strong realisations of A* are canonically isomorphic.
Our main aim is to prove that certain R*-algebras are strongly realisable, an*
*d to prove some
more ad hoc results for certain algebras over MU*=2.
Definition 2.3.A localised regular quotient (LRQ) of R* is an algebra A* over R*
** that can be
written in the form A* = (S-1R*)=I, where S is any set of (homogeneous) element*
*s in R* and I
is an ideal which can be generated by a regular sequence. We say that A* is a p*
*ositive localised
regular quotient (PLRQ) if it can be written in the form (S-1R*)=I as above, wh*
*ere I can be
generated by a regular sequence of elements of nonnegative degree.
Remark 2.4. If A* is an LRQ of R* and B* is an arbitrary R*-algebra then R*(A*,*
* B*) has
at most one element. Suppose that A is a commutative ring object in A 2 Recwit*
*h a given
isomorphism ß*(A) ' A*. It follows that A is a strong realisation of A* if and *
*only if: whenever
there is a map A* -!ß*(B) of R*-algebras, there is a unique map A -!B in Rec.
4 N. P. STRICKLAND
Remark 2.5. Let S be a set of homogeneous elements in R*. Using the results of *
*[9, Section
VIII.2] one can construct a strictly commutative ring spectrum S-1R and a map R*
* -! S-1R
inducing an isomorphism S-1ß*(R) -!ß*(S-1R). Results of Wolbert show that DS-1R*
* is equiva-
lent to the subcategory of DR consisting of objects M such that each element of*
* S acts invertibly
on ß*(M). Using this it is easy to check that any algebra over S-1R* is strongl*
*y realisable over
R if and only if it is strongly realisable over S-1R. For more discussion of th*
*is, see Section 4.
We start by stating a result for odd primes, which is relatively easy.
Theorem 2.6.If A* is an LRQ of R* and 2 is a unit in A* then A* is strongly rea*
*lisable.
This will be proved as Theorem 4.11.
Our main contribution is the extension to the case where 2 is not inverted. O*
*ur results involve
a certain öc mmutativity obstruction" _c(x) 2 ß2|x|+2(R)=(2, x), which is defin*
*ed in Section 3. In
Section 10, we show that when d 0 this arises from a power operation eP:ßd(R)*
* -!ß2d+2(R)=2.
This result was inspired by a parallel result of Mironov in Baas-Sullivan theor*
*y [19]. The restriction
d 0 is actually unneccessary but the argument for the case d < 0 is intricate*
* and we have no
applications so we have omitted it. In Section 5 we show how to compute this po*
*wer operation
using formal group theory, at least in the case R = MU. The first steps in this*
* direction were also
taken by Mironov [19], but our results are much more precise.
By Remark 2.5 we also have a power operation eP:ßd(S-1R) -! ß2d+2(S-1R)=2. Th*
*is is in
fact determined algebraically by the power operation on ß*R, as we will see in *
*Section 5.
Our result for the case where A* has no 2-torsion is quite simple and similar*
* to the case where
2 is inverted.
Theorem 2.7.Let A* = (S-1R*)=I be a PLRQ of R* which has no 2-torsion. Suppose *
*also that
eP(I) S-1R* maps to 0 in A*=2. Then A* is strongly realisable.
This will be proved as Theorem 4.12.
We next recall the definitions of some algebras over MU* which one might hope*
* to realise as
spectra using the above results. First, we have the rings
kU* := Z[u] |u| = 2
KU* := Z[u 1]
H* := Z (in degree zero)
HF* := Fp (in degree zero).
These are PLRQ's of MU* in well-known ways. Next, we consider the Brown-Peterso*
*n ring
BP* := Z(p)[vk | k > 0] |vk| = 2(pk - 1).
We take v0 = p as usual. There is a unique p-typical formal group law F over th*
*is ring such that
XF k
[p]F(x) = expF(px) +F vkxp .
k>0
(Thus, our vk's are Hazewinkel's generators rather than Araki's.) We use this F*
*GL to make BP*
into an algebra over MU* in the usual way. We define
P (n)*:= BP*=(vi| i < n) = Fp[vj | j n]
B(n)*:= v-1nBP*=(vi| i < n) = v-1nFp[vj | j n]
k(n)*:= BP*=(vi| i 6= n) = Fp[vn]
K(n)*:= v-1nBP*=(vi| i 6= n) = Fp[vn1]
BP *:= BP*=(vi| i > n) = Z(p)[v1, . .,.vn]
E(n)*:= v-1nBP*=(vi| i > n) = Z(p)[v1, . .,.vn-1, vn1]
These are all PLRQ's of BP*, and it is not hard to check that BP* is a PLRQ of *
*MU(p)*, and
thus that all the above rings are PLRQ's of MU(p)*.
PRODUCTS ON MU-MODULES 5
We also let wk 2 ß2(pk-1)MU denote the bordism class of a smooth hypersurface*
* Wpkof degree
k
p in CPp . It is well-known that In = (wi| i < n) is the smallest ideal modulo *
*which the universal
formal group law over MU* has height n, and that the image of In in BP* is the *
*ideal (vi| i < n).
In fact, we have
X X
[Wm ]xm dx = [p]F(x)d logF(x) = [p]F(x) [CP m]xm dx.
m>0 m 0
Moreover, the sequence of wi's is regular, so that MU*=In is a PLRQ of MU*.
One can also define PLRQ's of MU[1_6]* giving rise to various versions of ell*
*iptic homology, but
we refrain from giving details here. If we do not invert 6 then the relevant ri*
*ngs seem not to be
LRQ's of MU*. If we take R = MU^pthen we can make Zp[vn] into an LRQ of R* in s*
*uch a way
that the resulting formal group law is of the (non-p-typical) type considered b*
*y Lubin and Tate
in algebraic number theory. We can also take R = LK(n)MU and consider [E(n)*as *
*an LRQ of
R* via the Ando orientation [3] rather than the more usual p-typical one. We le*
*ave the details of
these applications to the reader.
The following proposition is immediate from Theorem 2.6.
Proposition 2.8.If p > 2 and R = MU or R = MU(p)then kU(p)*, KU(p)*, H(p)*, HFp*
**, BP*,
P (n)*, B(n)*, k(n)*, BP *, E(n)* and MU*=In are all strongly realisable.
After doing some computations with the power operation eP, we will also prove*
* the following.
Proposition 2.9.If R = MU then kU*, KU*, H* and HF* are strongly realisable. If*
* R = MU(2)
then kU(2)*, KU(2)*, H(2)*and BP* are strongly realisable.
The situation is less satisfactory for the rings BP * and E(n)* at p = 2. *
*For n > 1, they
cannot be realised as the homotopy rings of commutative ring objects in D. Howe*
*ver, if we kill off
a slightly different sequence of elements instead of the sequence (vn+1, vn+2, *
*. .)., we get a quotient
ring that is realisable. The resulting spectrum serves as a good substitute for*
* BP in almost all
arguments.
Proposition 2.10.If R = MU(2)and n > 0, there is a quotient ring BP 0*of BP**
* such that
1.The evident map
Z(2)[v1, . .,.vn] -!BP* -!BP 0*
is an isomorphism.
2.BP 0*is strongly realisable.
3.We have BP 0*=In = k(n)* = BP*=(vi| i 6= n) as MU*-algebras.
Moreover, the ring E(n)0*= v-1nBP 0*is also strongly realisable. If n = 1 t*
*hen we can take
BP <1>0*= BP <1>*.
This is proved in Section 7.
The situation for MU*=2 and algebras over it is also more complicated than fo*
*r odd primes.
Definition 2.11.Throughout this paper, we write ø for the twist map X ^ X -!X ^*
* X, for any
object X for which this makes sense. We say that a ring map f :A -!B in R is ce*
*ntral if
OE O ø O (f ^ 1) = OE O (f ^ 1): A ^ B -!B,
where OE: B ^ B -! B is the product. We say that B is a central A-algebra if th*
*ere is a given
central map A -!B.
Theorem 2.12. When R = MU(2), there is a ring MU=In 2 R with ß*(MU=In) = MU*=In,
and derivations Qi:MU=In -! 2i+1-1MU=In for 0 i < n. If OE is the product on *
*MU=In we
have
OE O ø - OE = wnOE O (Qn-1^ Qn-1).
6 N. P. STRICKLAND
This is proved in Section 7. There are actually many non-isomorphic rings wit*
*h these properties.
We will outline an argument that specifies one of them unambiguously.
We get a sharper statement for algebras over P (n)*.
Theorem 2.13. When R = MU(2), there is a central BP -algebra P (n) = BP ^ MU=In*
* 2 R and
an isomorphism ß*P (n) = P (n)*. This has derivations Qi:P (n) -! 2i+1-1P (n) f*
*or 0 i < n.
If OE is the product on P (n) we have
OE O ø - OE = vnOE O (Qn-1^ Qn-1).
If B is another central BP -algebra such that
8
><{0, 1} ifk = 0
ßkB = >0 if0 < k < |vn|
:{0, vn} ifk = |vn|
then either there is a unique map P (n) -!B of BP -algebras, or there is a uniq*
*ue map P (n) -!
Bop. Analogous statements hold for B(n), k(n) and K(n) with BP replaced by v-1n*
*BP , BP 0
and E(n)0respectively.
This is also proved in Section 7. Related results were announced by Würgler i*
*n [33], but there
appear to be some problems with the line of argument used there. A correct proo*
*f on similar lines
has recently been given by Nassau [21, 22].
3.Products on R=x
Suppose that x 2 Rd is not a zero-divisor (so d is even). We then have a cofi*
*bre sequence in
the triangulated category D:
dR x-!R æ-!R=x fi-! d+1R.
Because x is not a zero divisor, we have ß*(R=x) = R*=x. In particular, ßd+1R=x*
* = 0 (because
d + 1 is odd), and thus æ*:[R=x, R=x] ' [R, R=x]. It follows that R=x is unique*
* up to unique
isomorphism as an object under R.
We next set up a theory of products on objects of the form R=x. Apart from th*
*e fact that
all such products are associative, our results are at most minor sharpenings of*
* the those in [9,
Chapter V].
Observe that (R=x)(2)is a cell R-module with one 0-cell, two (d + 1)-cells an*
*d one (2d + 2)-cell.
We say that a map OE: (R=x)(2)-!R=x is a product if it agrees with æ on the bot*
*tom cell, in other
words OE O (æ ^ æ) = æ: R -!R=x.
The main result is as follows.
Proposition 3.1. 1.All products are associative, and have æ as a two-sided unit.
2.The set of products on R=x has a free transitive action of the group R2d+2=*
*x (in particular,
it is nonempty).
3.There is a naturally defined element _c(x) 2 ß2d+2(R)=(2, x) such that R=x *
*admits a commu-
tative product if and only if _c(x) = 0.
4.If so, the set of commutative products has a free transitive action of ann(*
*2, R2d+2=x) = {y 2
R2d+2=x | 2y = 0}.
5.If d 0 there is a power operation eP:Rd -!R2d+2=2 such that _c(x) = eP(x)*
* (mod 2, x) for
all x.
Proof.Part (1) is proved as Lemma 3.4 and Proposition 3.8. In part (2), the fac*
*t that products
exist is [9, Theorem V.2.6]; we also give a proof in Corollary 3.3, which is sl*
*ightly closer in spirit
with our other proofs. Parts (3) and (4) form Corollary 3.12. Part (5) is expla*
*ined_in more detail
and proved in Section 10. *
* |__|
From now on we will generally state our results in terms of eP(x) instead of _c*
*(x), as that is the
form in which the results are actually applied.
PRODUCTS ON MU-MODULES 7
Lemma 3.2. The map x: dR=x -!R=x is zero.
Proof.Using the cofibration
dR æ-! dR=x fi-! d+1R
and the fact that ßd+1(R=x) = ßd+1(R)=x = 0, we find that æ*:[R=x, R=x]d -! [R,*
* R=x]d =
ßd(R=x) is injective. It is clear that x gives zero on the right hand side, so *
*it_is_zero on the left
hand side as claimed. *
* |__|
Corollary 3.3.There exist products on R=x.
Proof.There is a cofibration dR=x x-!R=x 1^æ--!(R=x)(2). The lemma tells us th*
*at the first map__
is zero, so 1 ^ æ is a split monomorphism, and any splitting is clearly a produ*
*ct. |__|
Lemma 3.4. If OE: (R=x)(2)-! R=x is a product then æ is a two-sided unit for OE*
*, in the sense
that
OE O (æ ^ 1) = OE O (1 ^ æ) = 1: R=x -!R=x.
Proof.By hypothesis, OE O (æ ^ 1): R=x -! R=x is the identity on the bottom cel*
*l of R=x. We
observed earlier that [R=x, R=x] ' [R, R=x], and it follows that OEO(æ^1) = 1._*
*Similarly OEO(1^æ) =
1. |__|
Remark 3.5. EKMM study products for which æ is a one-sided unit, and our defini*
*tion of prod-
ucts is a priori even weaker. It follows from the lemma that EKMM's products ar*
*e the same as
ours and have æ as a two-sided unit.
Lemma 3.6. Let A 2 D be such that x: dA -!A is zero. Then the diagram
R=x _ R=x (æ^1,1^æ)------!(R=x)(2)fi^fi--! 2d+2R
induces a left-exact sequence
[ 2d+2R, A] -![(R=x)(2), A] -![R=x _ R=x, A].
Similarly, the diagram
*
* (3)
(R=x)(2)_ (R=x)(2)_ (R=x)(2)(æ^1^1,1^æ^1,1^1^æ)--------------!(R=x)(3)*
*fi--! 3d+3R
gives a left-exact sequence
[ 3d+3R, A] -![(R=x)(3), A] -![(R=x)(2)_ (R=x)(2)_ (R=x)(2), A].
Proof.Consider the following diagram:
R=x
|æ^1|
1^æ |fflffl
R=x ______//(R=x)(2)
KKK
fi|| |fi^1|fi^fiKKKKK
fflffl| |fflffl KK%%
d+1R __æ_//_ d+1R=xfi_//_ 2d+2R
We now apply the functor [-, A] and make repeated use of the cofibration
dR x-!R æ-!R=x fi-! d+1R.
The conclusion is that all maps involving fi become monomorphisms, all maps inv*
*olving æ become
epimorphisms, and the bottom row and the middle column become short exact. The *
*first claim
8 N. P. STRICKLAND
follows by diagram chasing. For the second claim, consider the diagram
(R=x)(2)_ (R=x)(2)
|(æ^1^1,1^æ^1)|
1^1^æ fflffl|
(R=x)(2)________//_(R=x)(3)P
PPP
fi^fi|| |fi^fi^1|fi^fi^fiPPPPPPP
fflffl| fflffl| P((P
2d+2R_____æ___// 2d+2R=x__fi___// 3d+3R
We apply the same logic as before, using the first claim (with A replaced by F *
*(R=x,_A))_to see
that the middle column becomes left exact. *
* |__|
We next determine how many different products there are on R=x.
Lemma 3.7. If OE is a product on R=x and u 2 ß2d+2(R)=x = [ 2d+2R, R=x] then OE*
*0= OE + u O
(fi ^fi) is another product. Moreover, this construction gives a free transitiv*
*e action of ß2d+2(R)=x
on the set of all products.
Proof.Let P be the set of products. As (fi ^fi)O(æ^æ) = 0, it is clear that the*
* above construction
gives an action of ß2d+2(R)=x on P . Now suppose that OE, OE02 P . We need to s*
*how that there is
a unique u: 2d+2R -!R=x such that OE0= OE + u O (fi ^ fi). Using the unital pr*
*operties of OE and
OE0given by Lemma 3.4, we see that
(OE0- OE) O (æ ^ 1) = (OE0- OE) O (1 ^ æ) = 0.
Because of Lemma 3.2, we can apply Lemma 3.6 to see that OE0- OE = u O (fi ^ fi*
*) for_a unique
element u, as claimed. *
* |__|
Proposition 3.8.Any product on R=x is associative.
Proof.Let OE be a product, and write
ffi := OE O (OE ^ 1 - 1 ^ OE): (R=x)(3)-!R=x,
so the claim is that ffi is nullhomotopic. Using the unital properties of OE we*
* see that
ffi O (æ ^ 1 ^ 1) = ffi O (1 ^ æ ^ 1) = ffi O (1 ^ 1 ^ æ) = 0.
Using Lemma 3.6, we conclude that ffi = u O (fi ^ fi ^ fi) for a unique element*
* u 2 [ 3d+3R,_R=x] =
ß3d+3(R)=x = 0 (because 3d + 3 is odd). Thus ffi = 0 as claimed. *
* |__|
Remark 3.9. The corresponding result in Baas-Sullivan theory was already known *
*(this is proved
in [18] in a form which is valid when R* need not be concentrated in even degre*
*es, for example
for R = MSp).
Remark 3.10. The EKMM approach to associativity is essentially as follows. They*
* note that
R=x has cells of dimension 0 and d+1, so (R=x)(3)has cells in dimensions 0, d+1*
*, 2d+2 and 3d+3.
The map ffi vanishes on the zero-cell and ßd+1(R=x) = ß3d+3(R=x) = 0 so the onl*
*y obstruction to
concluding that ffi = 0 lies in ß2d+2(R=x). EKMM work only with LRQ's that are *
*concentrated
in degrees divisible by 4, so the obstruction goes away. We instead use Lemma 3*
*.6 to analyse the
attaching maps in (R=x)(3); implicitly, we show that the obstruction is divisib*
*le by x and thus is
zero.
We now discuss commutativity.
Lemma 3.11. There is a natural map c from the set of products to ß2d+2R=x such *
*that c(OE) = 0
if and only if OE is commutative. Moreover,
c(OE + u O (fi ^ fi)) = c(OE) - 2u.
PRODUCTS ON MU-MODULES 9
Proof.Let ø :(R=x)(2)-!(R=x)(2)be the twist map. Clearly, if OE is a product th*
*en so is OE O ø.
Thus, there is a unique element v 2 ß2d+2R=x such that
OE O ø = OE + v O (fi ^ fi).
We define c(OE) := v. Next, recall that the twist map on 2d+2R = d+1R ^ d+1R*
* is homotopic
to (-1), because d + 1 is odd. It follows by naturality that (fi ^ fi) O ø = ø *
*O (fi ^ fi) = -fi ^ fi.
Consider a second product OE0= OE + u O (fi ^ fi). We now see that
OE0O ø = OE + v O (fi ^ fi) - u O (fi ^ fi) = OE0+ (v - 2u) O (fi *
*^ fi).
Thus c(OE0) = c(OE) - 2u as claimed. *
* |___|
Corollary 3.12.There is a naturally defined element _c(x) 2 ß2d+2(R)=(2, x) suc*
*h that R=x ad-
mits a commutative product if and only if _c(x) = 0. If so, the set of commutat*
*ive products has a
free transitive action of the group ann(2, ß2d+2(R)=x) := {y 2 ß2d+2(R)=x | 2y *
*= 0}. In particular,
if ß*(R)=x has no 2-torsion then there is a unique commutative product.
Proof.We choose a product OE on R=x and define _c(x) := c(OE) (mod 2). This is *
*well-defined, by
the lemma. If _c(x) 6= 0 then c(OE0) 6= 0 for all OE0, so there is no commutati*
*ve product. If _c(x) = 0
then c(OE) = 2w, say, so that OE0= OE + w O (fi ^ fi) is a commutative product.*
* In this case, the
commutative products are precisely the products of the form OE0+ z O (fi ^ fi) *
*where 2z =_0, so they
have a free transitive action of ann(2, ß2d+2(R)=x). *
* |__|
Next, we consider the Bockstein operation:
__ d+1
fi:= æfi :R=x -! R=x.
Definition 3.13.Let A 2 R be a ring, with product OE: A ^ A -! A. We say that *
*a map
Q: A -! kA is a derivation if we have
Q O OE = OE O (Q ^ 1 + 1 ^ Q): A(2)-!A.
__
Proposition 3.14.The map fiis a derivation with respect to any product OE on R=*
*x.
__ __ __
Proof.Write ffi := fiO OE - OE O (fi^ 1 + 1 ^ fi), so the claim is that ffi = 0*
*. It is easy to see that
ffi O (æ ^ 1) = ffi O (1 ^ æ) = 0, so by Lemma 3.6 we see that ffi factors thro*
*ugh a unique_map_
2d+2R -! d+1R=x. This is an element of ßd+1(R)=x, which is zero because d + 1 *
*is odd. |__|
We end this section by analysing maps out of the rings R=x.
Proposition 3.15.Let A 2 Re be an even ring. If x maps to zero in ß*A then ther*
*e is precisely
one unital map f :R=x -!A, and otherwise there are no such maps. If f exists an*
*d OE is a product
on R=x, then there is a naturally defined element dA(OE) 2 ß2d+2(A) such that
(a)dA(OE) = 0 if and only if f is a ring map with respect to OE.
(b)dA(OE + u O (fi ^ fi)) = dA(OE) + u.
(c)If A is commutative then 2dA(OE) = c(OE) 2 ß2d+2A.
Proof.The statement about the existence and uniqueness of f follows immediately*
* from the
cofibration dR x-!R æ-!R=x fi-! d+1R, and the fact that ßd+1A = 0. Suppose tha*
*t f exists; it
follows easily using the product structure on A that x: dA -!A is zero. Now le*
*t _ be the given
product on A, and let OE be a product on R=x. Consider the map
ffi := _ O (f ^ f) - f O OE: (R=x)(2)-!A.
By the usual argument, we have ffi = v O (fi ^ fi) for a unique map v : 2d+2R -*
*! A. We define
dA(OE) := v 2 ß2d+2A. It is obvious that this vanishes if and only if f is a ri*
*ng map, and that
dA(OE + u O (fi ^ fi)) = dA(OE) + u.
Now suppose that A is commutative, so _ = _ O ø. On the one hand, using the *
*fact that
(fi ^fi)Oø = -fi ^fi we see that ffi -ffi Oø = 2dA(OE)O(fi ^fi). On the other h*
*and, from the definition
of ffi and the fact that _ O ø = _, we see that
ffi - ffi O ø = f O (OE - OE O ø) = c(OE) O (fi ^ fi).
10 N. P. STRICKLAND
Because (fi ^ fi)*:ß2d+2A -![(R=x)(2), A] is a split monomorphism, we conclude *
*that_2dA(OE) =
c(OE) in ß2d+2A. *
*|__|
4. Strong realisations
In this section we assemble the products which we have constructed on the R-m*
*odules R=x to
get products on more general R*-algebras. We will work entirely in the derived *
*category D, rather
than the underlying geometric category. All the main ideas in this section come*
* from [9, Chapter
V].
We start with some generally nonsensical preliminaries.
Definition 4.1.Given a diagram A f-!C -g B in R, we say that f commutes with g *
*if and only
if we have
OEC O (f ^ g) = OEC O ø O (f ^ g): A ^ B -!C.
Note that this can be false when f = g; in particular A is commutative if and o*
*nly if 1A commutes
with itself.
The next three lemmas become trivial if we replace D by the category of modul*
*es over a
commutative ring, and the smash product by the tensor product. The proofs in th*
*at context can
easily be made diagrammatic and thus carried over to D.
Lemma 4.2. If A and B are rings in R, then there is a unique ring structure on *
*A ^ B such that
the evident maps A i-!A ^ B -j B are commuting ring maps. Moreover, with this p*
*roduct, (i,_j)
is the universal example of a commuting pair of maps out of A and B. *
* |__|
Lemma 4.3. A map f :A ^ B -!C commutes with itself if and only if f O i commute*
*s with itself
and f Oj commutes with itself. In particular, A^B is commutative if and only if*
* i_and j commute
with themselves. *
*|__|
Lemma 4.4. If A and B are commutative, then so is A ^ B, and it is the coproduc*
*t_of A and B
in Rc. |_*
*_|
Corollary 4.5.Suppose that
o A and B are strong realisations of A* and B*.
o The ring A* R* B* has no 2-torsion.
o The natural map A* R* B* -!ß*(A ^ B) is an isomorphism.
Then A ^ B is a strong realisation of A* R* B*. *
* |___|
We next consider the problem of realising S-1R*, where S is a set of homogene*
*ous elements
of R*. If S is countable then we can construct an object S-1R 2 D by the method*
* of [9, Section
V.2]; this has ß*(S-1R) = S-1ß*(R). If we want to allow S to be uncountable th*
*en it seems
easiest to construct S-1R as the finite localisation of R away from the R-modul*
*es {R=x | s 2 S};
see [17] or [11, Theorem 3.3.7]. In either case, we note that S-1R is the Bousf*
*ield localisation
of R in D with respect to S-1R. We may thus use [9, Section VIII.2] to construc*
*t a model of
S-1R which is a strictly commutative algebra over R in the underlying topologic*
*al category of
spectra. The localisation functor involved here is smashing, so results of Wolb*
*ert [30] [9, Section
VIII.3] imply that DS-1R is equivalent to the full subcategory of DR consisting*
* of R-modules M
for which ß*(M) is a module over S-1R*. This makes the following result immedia*
*te.
Proposition 4.6.Let S be a set of homogeneous elements of R*, and let A* be an *
*algebra over __
S-1R*. Then A* is strongly realisable over R if and only if it is strongly real*
*isable over S-1R. |__|
This allows us to reduce everything to the case S = ;.
Now consider a sequence (xi) in R*, with products OEion R=xi. Write Ai= R=x1^*
* . .^.R=xi,
and make this into a ring as in Lemma 4.2. There are evident maps Ai-! Ai+1, so*
* we can form
the telescope A = holim-!Ai.
i
PRODUCTS ON MU-MODULES 11
Lemma 4.7. If M 2 D and I = (x1, x2, . .). R* acts trivially on M and r 0 th*
*en [A(r), M] =
lim[-A(r)i, M].
i
Proof.This will follow immediately from the Milnor sequence if we can show that*
* lim1-[A(r)i, M]* =
*
* i
0. For this, it suffices to show that the map æ*:[B ^ R=xi, M] -![B, M] is surj*
*ective for all B.
This follows from the cofibration |xi|B xi-!B -!B ^ R=xiand the fact that xiac*
*ts_trivially on
M. |__|
Proposition 4.8.Let (xi) be a sequence in R*, and OEi a product on R=xi for eac*
*h i. Let A be
the homotopy colimit of the rings Ai= R=x1 ^ . .^.R=xi, and let fi:R=xi-! A be *
*the evident
map. Then there is a unique associative and unital product on A such that maps *
*fiare ring maps,
and ficommutes with fj when i 6= j. This product is commutative if and only if *
*each ficommutes
with itself. Ring maps from A to any ring B biject with systems of ring maps gi*
*:R=xi-! B such
that gicommutes with gj for all i 6= j.
Proof.Because R=xi admits a product, we know that xi acts trivially on R=xi. Be*
*cause A has
the form R=xi^ B, we see that xiacts trivially on A. Thus I acts trivially on A*
*, and Lemma 4.7
assures us that [A(r), A] = lim[-A(r)i, A].
i
Let _i be the product on Ai. By the above, there is a unique map _ :A ^ A -! *
*A which is
compatible with the maps _i. It is easy to check that this is an associative an*
*d unital product,
and that it is the only one for which the fiare commuting ring maps. It is also*
* easy to check that
_ is commutative if and only if each of the maps Ai-! A commutes with itself, i*
*f and only if each
ficommutes with itself.
Now let B be any ring in R. We may assume that each ximaps to zero in ß*(B), *
*for otherwise
the claimed bijection is between empty sets. As B is a ring, this means that ea*
*ch xiacts trivially
on B, so that [A(r), B] = lim[-A(r)i, B]. We see from Lemma 4.2 that ring maps *
*from Ai to B
i
biject with systems of ring maps gj:R=xj -!B for j < i such that gj commutes wi*
*th gk for_j 6= k.
The claimed description of ring maps A -!B follows easily. *
* |__|
Corollary 4.9.If each R=xiis commutative, then A is the coproduct of the R=xiin*
* Rc. |___|
Remark 4.10. If the sequence (xi) is regular, then it is easy to see that ß*(A)*
* = R*=(x1, x2, . .)..
Note also that ring maps out of R=x were analysed in Proposition 3.15.
We now restate and prove Theorems 2.6 and 2.7. Of course, the former is a sp*
*ecial case of
the latter, but it seems clearest to prove Theorem 2.6 first and then explain t*
*he improvements
necessary for Theorem 2.7.
Theorem 4.11. If A* is an LRQ of R* and 2 is a unit in A* then A* is strongly r*
*ealisable.
Proof.We can use Proposition 4.6 to reduce to the case where A* = R*=I where 2 *
*is invertible
in R* and I is generated by a regular sequence (x1, x2, . .).. We know from Pro*
*position 3.1 that
there is a unique commutative product OEi on R=xi. If C 2 Recand xi= 0 in ß*(C)*
* then in the
notation of Proposition 3.15 we have 2dC(OEi) = 0 and thus dC(OEi) = 0, so the *
*unique unital map
R=xi -!C is a ring map. It follows that R=xi is a strong realisation of R*=xi,*
* and thus that
Ai= R=x1^ . .^.R=xiis a strong realisation of R*=(x1, . .,.xi). Using Propositi*
*on 4.8, we_get a
ring A which is a strong realisation of R*=I. *
* |__|
We next address the case where 2 is not a zero-divisor, but is not invertible*
* either.
Theorem 4.12. Let A* = (S-1R*)=I be a PLRQ of R* which has no 2-torsion. Suppos*
*e also
that eP(I) S-1R* maps to 0 in A*=2, where eP:Rd -!R2d+2=2 is the power operat*
*ion defined
in Section 10. Then A* is strongly realisable.
Proof.After using Proposition 4.6, we may assume that S = ;. Choose a regular s*
*equence (xi)
generating I. As _c(xi) = eP(xi) 2 I (mod 2), we can choose a product OEi on R*
*=xi such that
c(OEi) 2 I. We let A be the "infinite smash productö f the R=xi, as in Proposi*
*tion 4.8, so that
12 N. P. STRICKLAND
ß*(A) = A*. Because c(OEi) maps to zero in ß*(A), we see easily that the map R=*
*xi-! A commutes
with itself. By Proposition 4.8, we conclude that A is commutative.
Let B 2 Recbe an even commutative ring, and that ß*(B) has no 2-torsion. The *
*claim is that
R(A, B) = R*(A*, ß*(B)). The right hand side has at most one element, and if it*
* is empty, then
the left hand side is also. Thus, we may assume that there is a map A* -!ß*(B) *
*of R*-algebras,
and we need to show that there is a unique ring map A -!B.
By Proposition 4.8, we know that ring maps A -!B biject with systems of ring *
*maps R=xi-! B
(which automatically commute as B is commutative). There is a unique unital map*
* f :R=xi-! B,
and Proposition 3.15 tells us that the obstruction to f being a homomorphism sa*
*tisfies 2dB(OEi) =
c(OEi) = 0 2 ß*(B). Because ß*(B) has no 2-torsion, we have dB(OEi) = 0, so the*
*re is_a_unique ring
map R=xi-! B, and thus a unique ring map A -!B as required. *
* |__|
The following result is also useful.
Proposition 4.13.Let A* be a strongly realisable R*-algebra, and let A* -! B* b*
*e a map of
R*-algebras that makes B* into a free module over A*. Then B* is strongly reali*
*sable.
Proof.First, observe that if F and M are A-modules, there is a natural map
Hom A(F, M) -!Hom A*(F*, M*),
which is an isomorphism if F is a wedge of suspensions of A (in other words, a *
*free A-module).W
Choose a homogeneous basis {ei} for B* over A*, where eihas degree di. Define*
* B := i diA,
so that B is a free A-module with a given isomorphism ß*B ' B* of A*-modules. D*
*efine B0 := A
and B1 := B and
`
B2 := di+djA
i,j
`
B3 := di+dj+dkA.
i,j,k
The product map ~: A ^ A -! A gives rise to evident maps OEk: B(k)-! Bk which i*
*n turn
give isomorphisms B*A*k = ß*Bk of A*-modules. The multiplication map B* A* B**
* -! B*
corresponds under the isomorphism Hom A(B2, B) = Hom A*(ß*B2, B*) to a map B2 -*
*!B. After
composing this with OE2, we get a product map ~B :B ^ B -!B. A similar procedur*
*e gives a unit
map A -!B.
We next prove that this product is associative. Each of the two associated pr*
*oducts B(3)-!B
factors as OE3 followed by a map B3 -!B, corresponding to a map B*A*3 -!A*. The*
* two maps
B*A*3 -!A* in question are just the two possible associated products, which are*
* the same because
B* is associative. It follows that B is associative. Similar arguments show tha*
*t B is commutative
and unital.
Now consider an object C 2 R equipped with a map B* -!C* (and thus a map A* -*
*!C*). As
A is a strong realisation of A*, there is a unique map A -!C compatible with th*
*e map A* -!C*.
This makes C into an A-module, and thus gives an isomorphism Hom A(B, C) = Hom *
*A*(B*, C*).
There is thus a unique A-module map B -!C inducing the given map B* -!C*. It fo*
*llows easily_
that B* is a strong realisation of B*. *
* |__|
We will need to consider certain R*-algebras that are not strongly realisable*
*. The following
result assures us that weaker kinds of realisation are not completely uncontrol*
*led.
Proposition 4.14.Let A* be an LRQ of R*, and let B, C 2 Re be rings (not necess*
*arily commu-
tative) such that ß*(B) = A* = ß*(C). Then there is an isomorphism f :B -!C (no*
*t necessarily
a ring map) that is compatible with the unit maps B - R -!C.
Proof.We may as usual assume that S = ;, and write I = (x1, x2, . .).. Let A be*
* the infinite
smash product of the R=xi's, so that ß*(A) = A*. It will be enough to show that*
* there is a unital
isomorphism A -! B. Moreover, any unital map A -! B is automatically an isomorp*
*hism, just
by looking at the homotopy groups.
PRODUCTS ON MU-MODULES 13
There is a unique unital map fi:R=xi-! B. Write Ai= R=x1^ . .^.R=xi, and let *
*gibe the
map
Ai-f1^...^fi-----!B(i)-!B,
where the second maps is the product. Because B is a ring and each xigoes to ze*
*ro in ß*(B),_we
can apply Lemma 4.7 to get a unital map g :A -!B as required. *
* |__|
We conclude this section by investigating R-module maps A -!A for various R-a*
*lgebras A 2 R.
Proposition 4.15.Let {x1, x2, . .}.be a regular sequence in R*, let OEi be a pr*
*oduct on R=xi,
and let A be the infinite smash_product of the rings R=xi. Let Qi:A -! |xi|+1*
*A be obtained
by smashing the Bockstein map fixi:R=xi-! |xi|+1R=xi with the identity map on *
*all the other
R=xj's. Then D(A, A)* is isomorphic as an algebra over A* to the completed exte*
*rior algebra on
the elements Qi.
Proof.It is not hard to see that QiQj = -QjQi,_with a sign coming from an impli*
*cit permutation
of suspension coordinates. We also have fi2i= 0 and thus Q2i= 0. Given any fi*
*nite subset
S = {i1 < . .<.in} of the positive integers, we define
QS := Qi1Qi2. .Q.in:A -! dSA,
P
where dS =P j(|xij| + 1). The claim is that one can make sense of homogeneous *
*infinite sums of
the form SaSQS with aS 2 A*, and that any graded map A -!A of R-modules is un*
*iquely of
that form.
Write An = R=x1 ^ . .^.R=xn, and let in: An -!A be the evident map. It is eas*
*y to check
that QS O in = 0 if max(S) > n, and a simple induction shows that D(An, A)* is *
*a free module
over A* generated by the maps QS O in for which max(S) n. Moreover, Lemma 4.7*
* implies_that
D(A, A)* = lim -D(An, A)*. The claim follows easily. *
* |__|
n
The above result relies more heavily than one would like on the choice of a r*
*egular sequence
generating the ideal ker(R* -!A*). We will use the following construction to ma*
*ke things more
canonical.
Construction 4.16.Let A 2 Re be an even ring, with unit j :R -!A, and let I be *
*the kernel
of j*:R* -!A*. Given a derivation Q: A -! kA, we define a function d(Q): I -!A**
* as follows.
Given x 2 I, we have a cofibration
dR x-!R æx-!R=x fix-! d+1R
as usual. Here x may be a zero-divisor in R*, so we need not have ß*(R=x) = ß*(*
*R)=x. Nonethe-
less, we see easily that there is a unique map fx: R=x -! A such that fx O æx =*
* j. As Q is a
derivation, one checks easily that Q O j = 0, so (Q O fx) O æx = 0, so Q O fx =*
* y O fix for some
y : d+1R -! kA. Because x acts as zero on A, we see that y is unique. We can t*
*hus define
d(Q)(x) := y 2 ßd+1-kA.
Proposition 4.17.Let A 2 Re be such that ß*(A) = R*=I, where I can be generated*
* by a regular
sequence. Let Der(A) be the set of derivations A -! A. Then Construction 4.16 g*
*ives rise to a
natural monomorphism d: Der(A) -!Hom R*(I=I2, A*) (with degrees shifted by one).
Proof.Choose a regular sequence {x1, x2, . .}.generating I. Write An = R=x1^ . *
*.^.R=xn, and
let jn be the map
fx1^...^fxn(n)product
R=x1^ . .^.R=xn --------!A -----! A.
It is easy to see that A is the homotopy colimit of the objects An (although th*
*ere may not be a
ring structure on An for which jn is a homomorphism). We also write An,ifor the*
* smash product
of the R=xj for which j n and j 6= i, and jn,ifor the evident map An,i-!An jn*
*-!A.
14 N. P. STRICKLAND
Consider a derivation Q: A -! kA, and write bi = d(Q)(xi). Because Q is a de*
*rivation, we
see that Q O jn is a sum of n terms, of which the i'th is bitimes the composite
1^fixi|x |+1 jn,i |x |+1
An = An,i^ R=xi----! i An,i--! i A.
P n
Now consider an element x = i=1aixiof I. It is easy to see that there is a *
*unique unital map
f0x:R=x -!An, and that jn O f0x= fx. Now consider the following diagram.
___x___// __æx_//_ __fix_//_d+1
dR R R=x R
ai|| 1 || |f0x| ai||
fflffl| fflffl| fflffl| fflffl|
|xi|An,ixi//_An,i1^æx//_An__// |xi|+1An,i
i 1^fixi
The left hand square commutes because the terms ajxj for j 6= i become zero in *
*ß*(An,i). It
follows that there exists a map R=x -!An making the whole diagram commute. Howe*
*ver, f0xis the
unique map making the middle square commute, so the whole diagram commutes asPd*
*rawn. Thus
jn,iO(fixi^1)Of0x= aiOfix (thinking of aiasPan element of ß*(A)). As QOjnP= i*
*bi.(jn,iO(fixi^1)),
we conclude that Q O fx = Q O jn O f0x= ( iaibi) O fix. Thus d(Q)(x) = iaibi.
This shows that d(Q) is actually a homomorphism I=I2 -! A*. It is easy to che*
*ck that the
whole construction gives a homomorphism d: Der(A) -!Hom A*(I=I2, A*). If d(Q) =*
* 0 then all
the elements bi are zero, so Q O jn = 0. As A is the homotopy colimit of the o*
*bjects_An,_we
conclude from Lemma 4.7 that Q = 0. Thus, d is a monomorphism. *
* |__|
The meaning of the proposition is elucidated by the following elementary lemm*
*a.
Lemma 4.18. If {x1, x2, . .}.is a regular sequence in R*, and I is the ideal th*
*at it generates,
then I=I2 is freely generated over R*=I by the elements xi.
Proof.ItPis clear that I=I2 is generated by the elements xi. Suppose that we h*
*ave a relation
n 2
i=1aixi = 0 in I (not I=I ). We claim that aiP2 I for all i. Indeed, it is cle*
*ar that anxn 2
(x1,P. .,.xn-1) so by regularity we have an = n-1i=1bixi say; in particular, *
*an 2 I. Moreover,
n-1
i=1(ai+ bixn)xi = 0, so by induction we have ai+ bixn 2 I for i < n, and thus *
*ai 2 I as
required. P P P
P Now supposePthat we have a relation iaixi2 I2, say iaixi=P j ibijxixj. W*
*e then have
i(ai- j ibijxj)xi= 0, so by the previous claim we have ai- j ibijxj 2 I, s*
*o ai2 I._This
shows that the elements xigenerate I=I2 freely. *
* |__|
Corollary 4.19.In the situation of Proposition 4.15 the map d: Der(A) -!Hom (I=*
*I2, A*) is an
isomorphism, and D(A, A)* is the completed exterior algebra generated by Der(A).
Proof.It is easy to see that Qiis a derivation and that d(Qi)(xj) = ffiij(Krone*
*cker's delta)._This
shows that d is surjective, and the rest follows. *
* |__|
5.Formal group theory
In this section, we take R = MU, and let F be the usual formal group law over*
* MU*. In places
it will be convenient to use cohomological gradings; we recall the convention A*
** = A-*. We will
write q for the usual map MU* -!BP *, and note that q(w1) = v1 (mod 2).
A well-known construction gives a power operation
P :RdX -!R2d(RP 2x X),
which is natural for spaces X and strictly commutative ring spectra R. A good r*
*eference for such
operations is [7]; in the case of MU, the earliest source is probably [28].
In the case R = S-1MU there is an element ffl 2 R2RP 2such that R*RP 2= R*[ff*
*l]=(2ffl, ffl2). More
generally, the even-dimensional part of R*(RP 2x X) is R*(X)[ffl]=(2ffl, ffl2),*
* and P (x) = x2+ fflPe(x)
PRODUCTS ON MU-MODULES 15
for a uniquely determined operation Rd(X) -!R2d-2(X)=2. We also have the follow*
*ing properties:
P (1)= 1
P (xy)= P (x)P (y)
P (x + y)= P (x) + P (y) + (2 + fflw1)xy
P (x)= x(x +F ffl) if x is the Euler class of a complex line bund*
*le.
To handle the nonadditivity of P , we make the following construction. For an*
*y MU*-algebra
A*, we define
T (A*) := {(r, s) 2 A*=2 x A*[ffl]=(2, ffl2) | s = r2 (mod ffl)*
*}.
Given a, b 2 A* (with |b| = 2|a| - 2) we define [a, b] := (a, a2+ fflb) 2 T (A**
*). We make T (A*) into
a ring by defining
(r, s) + (t,:u)= (r + t, s + u + fflw1rt)
(r, s).(t,:u)= (rt, su)
or equivalently
[a, b] + [c,:d]= [a + c, b + d + w1ac]
[a, b].[c,:d]= [ac, a2d + bc2].
Note that 2[a, b] = [0, w1a2] and 4[a, b] = 0, so 4T (A*) = 0. If we define Q(*
*a) := (a, P (a)) =
[a, eP(a)], then Q gives a ring map MU*X -!T (MU*X).
Definition 5.1.Suppose that A* is a PLRQ of MU*, and let f :MU* -!A* be the_uni*
*t_map.
We say that A* has an induced power operation (IPO) if there is a ring map Q :A*
** -! T (A*)
making the following diagram commute:
Q *
MU* _____//T (MU )
f || T(f)||
fflffl| fflffl|
A*______Q//_T (A*).
Because A* is an LRQ, we know that such a map is unique if it exists.
If A* = S-1MU* then we know that S-1MU can be constructed as a strictly commu*
*tative
MU-algebra and thus an E1 ring spectrum, and the power operation coming from t*
*his E1
structure clearly gives an IPO on S-1MU*. For a more elementary proof, it suff*
*ices to show
that when x 2 S the image of Q(x) in T (S-1MU*) is invertible. However, the ele*
*ment (x, x2) is
trivially invertible in T (S-1MU) and Q(x) differs from this by a nilpotent ele*
*ment, so it too is
invertible.
It is now easy to reduce the following result to Theorem 2.7.
Proposition 5.2.Let A* be a PLRQ of MU* which has no 2-torsion and admits an IP*
*O. Then_
A* is strongly realisable. *
* |__|
We now give a formal group theoretic criterion for the existence of an IPO.
Definition 5.3.Let F be a formal group law over a ring A*. Given an algebra B* *
*over A* and
an element x 2 B2, we define Z(x) := (x, x(x +F ffl)) 2 T (B*). (We need x to b*
*e topologically
nilpotent in a suitable sense to interpret this, but we leave the details to th*
*e reader.) Thus, if X
is a space, A* = MU*, B* = MU*X and x is the Euler class of a complex line bund*
*le over X
then Z(x) = Q(x).
Proposition 5.4.Let_A* be a LRQ of MU*, and let F be the obvious formal group l*
*aw over A*.
Then a ring map Q :A* -!T (A*) is an IPO if and only if we have
Z(x) +__Q*FZ(y) = Z(x +F y) 2 T (A*[[x, y]]).
16 N. P. STRICKLAND
Proof.Let F 0be the universal FGL over MU* and put Z0(x) = (x, x(x+F0ffl)). Let*
*_f_:MU* -!A*
be the unit map, so that F = f*F 0. Using the universality of F 0, we see that *
*Q is an IPO if and
only if we have
X +__Q*f*F0Y = X +T(f)*Q*F0Y 2 T (A*)[[X,.Y ]]
The left hand side is of course X +__Q*FY . There is an evident map
T (A*)[[X,-Y!]]T (A*[[x, y]]),
*
* __
sending X to Z(x) and Y to Z(y), and one can check that this is injective. Thus*
*, Q is an IPO if
and only if
Z(x) +__Q*FZ(y) = Z(x) +T(f)*Q*F0Z(y) 2 T (A*[[x, y]]).
The right hand side here is T (f)(Z0(x) +Q*F0Z0(y)) and Z(x +F y) = T (f)(Z0(x *
*+F0y)) so the
proposition will follow once we prove that Z0(x +F0y) = Z0(x) +Q*F0Z0(y) 2 T (M*
*U*[[x, y]]). To
do this, we use the usual isomorphism MU*[[x, y]]= MU*(CP 1 x CP 1), so that x,*
* y and x +F0y
are Euler classes, so Q(x) = Z0(x) and Q(y) = Z0(y) and Q(x +F0y) = Z0(x +F0y).*
* As Q is a
natural multiplicative operation we also have Q(x +F0y) = Q(x) +Q*F0Q(y) = Z(x)*
* +Q*F0Z(y),_
which gives the desired equation. *
* |__|
We now use this to show that there is an IPO on kU*. In this case the real r*
*eason for the
IPO is that the Todd genus gives an H1 map MU -!kU, but we give an independent *
*proof as a
warm-up for the case of BP *.
Proposition 5.5.Let f :MU* -! kU*_:=_Z[u] be the Todd genus. Then there is an *
*induced
power operation on kU*, given by Q(u) = [u, u3]. Thus, kU* and KU* are strongly*
* realisable.
Proof.The FGL over kU* coming from f is just the multiplicative FGL x +F y = x *
*+ y + uxy, so
Z(x) = (x, x2+xffl+ux2ffl) = [x, x+ux2]. If we put U = [u, u3] then X+__Q*FY = *
*X+Y +UXY . We
thus need only verify that Z(x) + Z(y) + U Z(x)Z(y) = Z(x + y + uxy). This is a*
* straightforward
calculation; some steps are as follows.
Z(x) + Z(y)= [x + y, x + y + u(x2+ xy + y2)]
Z(x)Z(y)= [xy, xy(x + y)]
U Z(x)Z(y)= [uxy, u2xy(x + y) + u3x2y2]
Z(x + y + uxy)= [x + y + uxy, x + y + u(x2+ xy + y2) + u3x2y2].
|___|
We now turn to the case of BP *. For the moment we prove only that an IPO exi*
*sts; in the
next section we will calculate it.
Proposition 5.6.There is an IPO on BP *, so BP *is strongly realisable.
This is proved after Lemma 5.9.
Definition 5.7.For the rest of this section, we will write
X k k
z := z(x) := v21x2 2 F2[v1][[x]].
k 0
Note that
z2 = z + v1x
so
z=(v1x) = 1=(1 + z).
Lemma 5.8. We have
x +F ffl = x + (1 + z)ffl in BP *[ffl][[x]]=(2ffl, ffl2*
*).
PRODUCTS ON MU-MODULES 17
Proof.Working rationally and modulo ffl2, we have
logF(ffl) = ffl
so
x +F ffl = expF(logF(x) + ffl) = x + exp0F(logF(x))ffl = x + ffl= lo*
*g0F(x).
Note that log0F(x) is integral and its constant term is 1, so the above equatio*
*n is between integral
terms and we can sensibly reduce it modulo 2.
We next recall the formula for logF(x) given in [25, Section 4.3]. We conside*
*r sequences I =
(i1, . .,.ir) with l 0 and ij > 0 for each j. We write |I| := r and kIk := i1*
*+ . .+.ir. We also
write
vI := vm1i1.v.m.rir
where
P
mj := 2 k0Q(a1j)XjW , and
Q(a1j)XjW = [a1j, eP(a1j)][xj, 0][0, w] = [0, a21jx2jw].
18 N. P. STRICKLAND
This expression is to be interpreted in T (BP *[[x,)y]], so we need to interpre*
*t a1jin BP *=2. Thus
Lemma 5.8 tells us that a10= 1 and a1,2k= v2k1and all other a1j's are zero. Thus
X k+1
X +QF W = X + W + [0, (v1x)2 w] =
k 0 2 0 13
X k+1
4 x, w @1 + (v1x)2 A5= [x, w(1 + z2)] = [x,*
* y]
k 0
as claimed.
For the last statement, Lemma 5.8 gives
Z(x) = (x, x(x +F ffl)) = (x, x2+ x(1 + z)ffl) = [x, x(1 + z)].
By the previous paragraph, this can be written as [x, 0] +QF [0, x(1 + z)=(1 + *
*z)2]_= [x, 0] +QF
[0, z=v1]. *
*|__|
__
Proof of PropositionT5.6.o show that Q exists, it is enough to show that the fo*
*rmal group law
on T (BP *) obtained from the map MU* Q-!T (MU*) T(q)---!T (BP *) is 2-typical.*
* Let p be an odd
prime, so the associated cyclotomic polynomial is p(t) = 1+t+. .+.tp-1. We nee*
*d to show that
X +QF X +QF . .+.QF p-1X = 0 in B* := T (BP *)[ ][[X]]= p( ).
(This is just the definition of 2-typicality for formal groups over rings which*
*Qmay have torsion.)
Consider the ring C* := T (BP *[!][[x]]= p(!)). As Pp(!) = 0 we have tp- 1 = p*
*-1j=0(t - !j), and
by looking at the coefficient of tp-2 we find that 0 i<[0, v1] ifn = 0
Q (vn) = >[v1, v2] ifn = 1
:[vn, v1v2
n + un]ifn > 1
Moreover, we have un = vn+1 (mod v21).
This is proved after Corollary 6.4. We will reuse the notation of Definition *
*5.7.
Lemma 6.3. We have expF(2x) = 2z=v1 in BP *[[x]]=4.
Proof.Using Ravenel's formulae as in the proof of Lemma 5.8, we have
X kIk kIk
logF(2x)=2 = 22 -|I|-1vIx2 .
I
When k 0 we have 2k k + 1, with equality only when k = 0 or k = 1. It follo*
*ws easily that
logF(2x)=2 = x + v1x2 (mod 2).
By inverting this, we find that
X k k
expF(2x)=2 = v21-1x2 = z=v1 (mod 2),
k 0
and thus that expF(2x) = 2z=v1 (mod 4). *
* |___|
Because T (BP *) is a torsion ring, the formal group law QF has no expseries.*
* Nonetheless,
expF(2X) is a power series over BP *, so we can apply Q to the coefficients to *
*get a power series
over T (BP *) which we call expQF(2X). This makes perfect sense even though exp*
*QF(X) does
not.
Corollary 6.4.In T (BP *)[[X]], we have
X k+1 k
expQF(2X) = [0, v21 -1]X2 .
k 0
By taking X = Z(x) 2 T (BP *[[x]]), we get
2 3
X j j
expQF(2Z(x)) = 40, v21-1x2 5= [0, z=v1+ x].
j>0
20 N. P. STRICKLAND
Proof.Because 4 = 0 in T (BP *), it follows immediately from the lemma that exp*
*QF(2X) =
P __ 2k-1 2k __ e __ 2k-1 2k+*
*1-1
k 2Q (v1) X . Using Q(v1) = [v1, P(v1)], we see that 2Q (v1) =k[0,+v11k*
* ], andkthe+1k+1
first claim follows. If we now put X = Z(x) = [x, x(1+z)] then [0, v21 -1]X2 =*
* [0,_v21_ -1x2 ],
and the second claim follows. *
* |__|
Proof_of_PropositionL6.2.et pk denote the image of eP(vk) 2 MU2k+2-2=2 in BP *=*
*2 and write
Vk = Q(vk) = [vk, pk] 2 T (BP *). Recall that the Hazewinkel generators vk are *
*characterised by
the formula
XF k
[2]F(x) = expF(2x) +F vkx2 2 BP *[[x]].
k>0
__
By applying the ring map Q and putting X = Z(x) we obtain
QFX k
[2]QF (Z(x)) = expQF(2Z(x)) +QF VkZ(x)2 2 T (BP *[[x]]).
k>0
The first term can be evaluated using Corollary 6.4. For the remaining terms, w*
*e have
k 2k 2k 2k+1
VkZ(x)2 = [vk, pk][x , 0] = [vkx , pkx ].
We can use Lemma 5.9 to rewrite this as " #
k 2k X 2k 2l-2 2k+1
VkZ(x)2 = [vkx , 0] +QF 0, (v1vkx ) pkx
" l>0 #
k X 2l-2 2k+l
= [vkx2 , 0] +QF 0, (v1vk) pkx .
l>0
After using the formula [0, b] +QF [0, c] = [0, b + c] to collect terms, we fin*
*d that
2 3
X l l X l k+l QFX k
(4) [2]QF (Z(x)) = 40, v21-1x2 + (v1vk)2 -2pkx2+Q5F [vkx2 , 0].
l>0 k,l>0 k>0
On the other hand, we know that
[2]QF (Z(x))= Z([2]F(x))
_ F !
X k
= Z expF(2x) +F vkx2
k>0
QFX k
= Z(expF(2x)) +QF Z(vkx2 ).
k>0
The first term is zero because expF(2x) is divisible by 2. For the remaining te*
*rms, Lemma 5.9
gives
k 2k X 2j-1 2j 2k+j
Z(vkx2 ) = [vkx , 0] +QF [0, v1 vk x ].
j 0
Thus, we have
2 3
X X l l k+l QFX k
(5) [2]QF (Z(x)) = 40, v21-1v2kx2 5+QF [vkx2 , 0].
k>0l 0 k>0
By comparing this with equation (4) and equating coefficients of x2n+1, we find*
* that
n+1-1 Xn 2j-2 Xn 2j-1 2j
v21 + (v1vn+1-j) pn+1-j= v1 vn+1-j.
j=1 j=0
PRODUCTS ON MU-MODULES 21
After some rearrangement and reindexing, this becomes
n+1-1 n-1X 2(2j-1) 2
pn + v1v2n= v21 + vn+1+ (v1vn-j) (pn-j+ v1vn-j).
j=1
In particular, we have p1 = v2. We now define
8
>v2 ifn = 1
:v1v2
n + un ifn > 1
The claim of the proposition is just that pn = p0nfor all n 0. Using the recu*
*rrence relation given
in definition 6.1, one can check that for all n > 0 we have
n+1-1 n-1X 2(2j-1)0 2
p0n+ v1v2n= v21 + vn+1+ (v1vn-j) (pn-j+ v1vn-j).
j=1
In particular, we have p01= v2 = p1, and it follows inductively that pn = p0nfo*
*r all n > 0. We also
have
Q(v0) = Q(1) + Q(1) = [1, 0] + [1, 0] = [0, v1]
so p0 = v1 = p00. *
* |___|
Remark 6.5. The first few cases are
p0= v1
p1= v2
p2= v41v2+ v1v22+ v3
p3= v121v2+ v61v32+ v21v22v3+ v1v23+ v4.
In particular, we find that p3 62 (vk | k 3), which shows that there is no co*
*mmutative product on
BP <2>, considered as an object of D. This problem does not go away if we repla*
*ce the Hazewinkel
generator vk by the correspondingkAraki generator, or the bordism class wk of a*
* smooth quadric
hypersurface in CP 2. However, it is possible to choose a more exotic sequence *
*of generators for
which the problem does go away, as indicated by the next result.
Proposition 6.6.Fix an integer n > 0. There is an ideal J BP *such that
1.The evident map
Z(2)[v1, . .,.vn] -!BP *-!BP *=J
is an isomorphism.
2.eP(J) J (mod 2).
3.In + J = In + (vk | k > n) = (vk | k 6= n).
The proof will construct an ideal explicitly, but it is not the only one with t*
*he stated properties. If
n = 1 we can take J = (vk | k > n), but for n > 1 this violates condition (2).
Remark 6.7. The subring Z(2)[v1, . .,.vn] of BP *is the same as the subring gen*
*erated by all
elements of degree at most 2n+1- 2; it is thus defined independently of the cho*
*ice of generators
for BP*.
Proof.First consider the case n = 1, and write J = (vk | k > 1). By inspecting *
*definition 6.1, we
see that un 2 J for all n > 1, and thus Proposition 6.2 tells us that eP(J) J*
* (mod 2). We may
thus assume that n > 1. Write B* = Z(2)[v1, . .,.vn], thought of as a subring o*
*f BP *. We will
recursively define a sequence of elements xk 2 BP *for k > n such that
(a)xk 2 vk + v21B*
(b)eP(xk-1) 2 (xn+1, . .,.xk) (mod 2) if k > n + 1.
22 N. P. STRICKLAND
It is clear that we can then take J = (xk | k > n). We start by putting xn+1 = *
*vn+1. Suppose
that we have defined xn+1, . .,.xk with the stated properties. There is an evid*
*ent map
F[v1, . .,.vn, vk+1] f-!BP*=(2, xn+1, . .,.xk),
which is an isomorphism in degree 2(2k+1 - 1) = |vk+1|. Let _pkbe the image of*
* Pe(xk) in
BP*=(2, xn+1, . .,.xk), and write __xk+1= f-1(_pk). We can lift this to get an*
* element xk+1 of
Z(2)[v1, . .,.vn, vk+1] such that __xk+1= xk+1 (mod 2) and every coefficient in*
* xk+1 is 0 or 1. It
is easy to see that condition (b) is satisfied, and that xk+1 2 vk+1+ B*. Howev*
*er, we still need
to show that xk+1- vk+1 is divisible by v21. By assumption we have xk = vk + v*
*21b for some
b 2 B*. Recall from Proposition 6.2 that eP(vk) = vk+1+v1v2k(mod 2, v21). It fo*
*llows after a small
calculation that eP(xk) = vk+1+ v1v2k(mod 2, v21) also. Moreover, we have v2k= *
*v41b2 (mod 2, xk),_
so _pk= vk+1 (mod 2, v21). It follows easily that xk+1= vk+1 (mod v21), as requ*
*ired. |__|
We give one further calculation, closely related to Proposition 6.2.
Proposition 6.8.Recall that Ik := (w0,i. .,.wk-1) < MU*, where wi is the bordis*
*m class of a
smooth quadric hypersurface in CP 2. We have eP(Ik-1) Ik, and eP(wk-1) = wk (*
*mod Ik).
Proof.If k = 1 we have I0 = 0 and w0 = 2, so P (w0) = P (1) + P (1) + (2 + w1ff*
*l) = w1ffl
(mod 2), as required. Thus, we may assume that k > 1, and it follows easily fro*
*m the formulae
for P (x + y) and P (xy) that P induces a ring map MU* -! B* = (MU*=Ik)[ffl]=ff*
*l2. Note that
[2]F(x) = wkx2k+ O(x2k+1) over B*. Write X = x(x +F ffl) 2 MU*[ffl][[x]]=(Ik, f*
*fl2). Arguing in the
usual way, we see that
k 2k+1
[2]P*F(X) = [2]F(x)([2]F(x) +F ffl) = fflwkx2 + O(x ).
It follows easily that we must have
k-1 2k-1+1
[2]P*F(X) = fflwkX2 + O(X ).
It follows that P (wi) = 0 2 B* for i < k - 1, and that P (wk-1) = fflwk 2 B*, *
*as required. |___|
7. Applications to MU
Proof of PropositionT2.9.he claims involving kU and KU follow from Proposition *
*5.5, and those
for BP follow from Proposition 5.6. The claim H follows from Theorem 2.7, as t*
*he condition
eP(I) I (mod 2) is trivially satisfied for dimensional reasons. The claim for*
* HF can be proved
in the same way as Theorem 4.12 after noting that all the obstruction groups ar*
*e trivial. |___|
Proof of Proposition 2.10.Choose an ideal J as in Proposition 6.6 and set BP 0*= BP*=J._
Everything then follows from Theorem 2.7. *
* |__|
We now take R = MU(2)and turn to the proof of Theorem 2.13. As previously, w*
*e let
k
wk 2 ß2k+1-2R denote the bordism class of the quadric hypersurface W2k in CP 2.*
* Recall that
the image of wk in BP* is vk modulo Ik = (v0, . .,.vk-1), and thus P (n)* = BP**
*=(w0, . .,.wn-1).
We next choose a product OEk on MU=wk for each k. For k = 0 we choose there *
*are two
possible products, and we choose one of them randomly. (It is possible to speci*
*fy one of them
precisely using Baas-Sullivan theory, but that would lead us too far afield.) F*
*or k > 0, we recall
from Proposition 6.8 that eP(wk) = wk+1 (mod Ik+1). It follows easily that ther*
*e is a product OEk
such that c(OEk) = wk+1 (mod w1, . .,.wk), and that this is unique up to a term*
* u O (fi ^ fi) with
u 2 (w1, . .,.wk). From now on, we take OEk to be a product with this property.*
* It is easy to see
that the resulting product MU=w0^ . .^.MU=wn-1 is independent of the choice of *
*OEk's (except
for OE0).
Definition 7.1.We write
MU=In = MU=w0^ . .^.MU=wn-1,
made into a ring as discussed above. For i < n, we define
i+1-1
Qi:MU=In -! 2 MU=In
PRODUCTS ON MU-MODULES 23
__ i+1
by smashing the Bockstein map fi:MU=wi -! 2 -1MU=wi with the identity on the *
*other
factors. We also define
P (n):= BP ^ MU=In
B(n):= w-1nP (n)
k(n):= BP 0^ MU=In
K(n) := w-1nk(n).
It is clear that ß*(MU=In) = MU*=In and ß*(P (n)) = P (n)* and ß*(B(n)) = B(n)**
*. Condi-
tion (2) in Proposition 2.10 assures us that ß*k(n) = k(n)* and ß*K(n) = K(n)* *
*as well. As
BP and BP 0are commutative, it is easy to see that P (n), B(n), k(n) and K(n*
*) are central
algebras over BP , v-1nBP , BP 0and E(n)0respectively. The derivations Qion *
*MU=In clearly
induce compatible derivations on P (n), B(n), k(n) and K(n).
Proposition 7.2.The product OE on MU=In satisfies
OE - OE O ø = wnOE O (Qn-1^ Qn-1).
Similarly for P (n), B(n), k(n) and K(n).
Proof.This follows easily from the fact that c(OEk-1) = wk (mod Ik), given by P*
*roposition 6.8. |___|
Proposition 7.3.Let A be a central BP -algebra such that ß0(A) = {0, 1}, ß2n+1-*
*2(A) = {0, vn}
and ßk(A) = 0 for 0 < k < 2n+1-2. Then either there is a unique map P (n) -!A o*
*f BP -algebras,
or there is a unique map P (n) -!Aop (but not both). Analogous statements hold *
*for B(n), k(n)
and K(n) with BP replaced by v-1nBP , BP 0and E(n)0respectively.
Proof.We treat only the case of P (n); the other cases are essentially identica*
*l. Any ring map
MU=In -!A commutes with the given map BP -!A, because the latter is central. It*
* follows that
maps P (n) -!A of BP -algebras biject with maps MU=In -!A of rings, which bijec*
*t with systems
of commuting ring maps MU=wi-! A for 0 i < n. For i < n - 1 we have ß2|wi|+2(*
*A) = 0, so
Proposition 3.15 tells us that the unique unital map fi:MU=wi-! A is a ring map*
*. This remains
the case if we replace the product _ on A by _ O ø, or in other words replace A*
* by Aop. There is
an obstruction dA(OEn-1) 2 ß2n+1-2(A) = {0, vn} which may prevent fn-1 from bei*
*ng a ring map.
If it is nonzero, we have
dA(OEn-1O ø) = dA(OEn-1+ eP(wn-1) O (fi ^ fi)) = dA(OEn-1) + vn = 0
This shows that fn-1:MU=wn-1 -!Aop is a ring homomorphism. After replacing A by*
* Aop if
necessary, we may thus assume that all the fi:MU=wi-! A are ring maps.
The obstruction to fi commuting with fj lies in ß|wi|+|wj|+2(A). If i and j a*
*re different then
at least one is strictly less than n - 1; it follows that |wi| + |wj| + 2 < 2n+*
*1- 2 and thus that the
obstruction group is zero. Thus fi commutes with fj when i 6= j, and we get a u*
*nique_induced
map MU=In -!A, as required. |*
*__|
8.Point-set level foundations
In order to analyse the commutativity obstruction _c(x) more closely and rela*
*te them to power
operations, we need to recall some internal details of the EKMM category.
EKMM use the word "spectrum" in the sense defined by Lewis and May [15], rath*
*er than the
sense we use elsewhere in this paper. They construct a category LS of "L-spectr*
*a". This depends
on a universe U, but the functor L(U, V) nL(U)(-) gives a canonical equivalence*
* of categories
from L-spectra over U to L-spectra over V, so the dependence is only superficia*
*l. (Here L(U, V)
is the space of linear isometries from U to V.) We therefore take U = R1 . EKMM*
* show that LS
has a commutative and associative smash product ^L, which is not unital. Howeve*
*r, there is a
sort of "pre-unitö bject S, with a natural map S ^L X -!X. They then define th*
*e subcategory
M := MS = {X | S ^L X = X} of "S-modules", and prove that S ^L S = S so that S *
*^L X is an
S-module for any X. We write ^ for the restriction of ^L to M.
24 N. P. STRICKLAND
We next give a brief outline of the properties of M. Let T be the category o*
*f based spaces
(all spaces are assumed to be compactly generated and weakly Hausdorff). We wri*
*te 0 for the
one-point space, or for the basepoint in any based space, or for the trivial ma*
*p between based
spaces.
We give T the usual Quillen model structure for which the fibrations are Serr*
*e_fibrations. We
write hT for the category with Hom sets ß0F (A, B) = T(A, B)=homotopy, and hT f*
*or the category
obtained_by inverting the weak equivalences. We refer to hT as the strong homot*
*opy category of
T, and hT as the weak homotopy category.
The category M is a topological category: the Hom sets M(X, Y ) are based spa*
*ces, and there
are continuous composition maps
M(X, Y ) ^ M(Y, Z) -!M(X, Z).
We again have a strong homotopy category hM, with hM(X, Y ) = ß0M(X, Y_);_when *
*we have
defined homotopy groups, we will also define a weak homotopy category hM in the*
* obvious way.
M is a closed symmetric monoidal category, with smash product and function ob*
*jects again
written as X ^ Y and F (X, Y ). Both of these constructions are continuous fu*
*nctors of both
arguments. The unit of the smash product is S.
There is a functor 1 :T -!M, such that
1 S0= S
1 (A ^ B)= 1 A ^ 1 B
M( 1 A ^ X, Y )= T(A, M(X, Y ))
M( 1 A, 1 B)= T(A, B).
(For the last of these, see [8].)
The last equation shows that 1 is a full and faithful embedding of T in M, *
*so that all of
unstable homotopy theory is embedded in the strong homotopy category hM. In par*
*ticular, hM
is very far from Boardman's_stable homotopy category B. However, it turns out t*
*hat the weak
homotopy category hM is equivalent to B.
The definition of this weak homotopy category involves certain öc fibrant sph*
*ere objects" which
we now discuss. It will be convenient for us to give a slightly more flexible c*
*onstruction than that
used in [9], so as to elucidate certain questions of naturality. Let U be a un*
*iverse. There is a
natural way to make the Lewis-May spectrum 1 L(U, R1 )+ into a L-spectrum, usi*
*ng the action
of L(1) = L(R1 , R1 ) on L(U, R1 ) as well as on the suspension coordinates. On*
*e way to see this
is to observe that 1 L(U, R1 )+ = L(U, R1 ) n S0, where the S0 on the right ha*
*nd side refers to
the sphere spectrum indexed on the universe U. We then define S(U) = S ^L 1 L(*
*U, R1 )+. This
gives a contravariant functor S :{Universes} -!M, and it is not hard to check t*
*hat S(U)^S(V) =
S(U V).
Moreover, for any finite-dimensional subspace U < U, there is a natural subob*
*ject S(U, U)
S(U) and a canonical isomorphisms
S(U, U) ^ S(U, V=)S(U V, U V )
US(U, U V )= S(U, V ).
This indicates that the objects S(U, U) are in some sense stable. They can be d*
*efined as follows:
take the Lewis-May spectrum 1US0 indexed on U, and then take the twisted half *
*smash product
with the space L(U, R1 ) to get a Lewis-May spectrum indexed on R1 which is eas*
*ily seen to be
an L-spectrum in a natural way. We then apply S ^L (-) to get S(U, U).
For any n > 0 and d 0 we write
L(n) := L((R1 )n, R1 )
S(n):= S((R1 )n) = S(1)(n)
Sd(1):= dS(R1 )
S-d(1):= S(R1 , Rd)
PRODUCTS ON MU-MODULES 25
We will also allow ourselves to write Sd(n) for kS((R1 )n, V ) where V is a su*
*bspace of (R1 )n
of dimension k - d and k and V are clear from the context.
Any object of the form SU ^S(V, V ) is non-canonically isomorphic to Sd(1), w*
*here d = dim(U)-
dim(V ), but when one is interested in the naturality or otherwise of various c*
*onstructions it is
often a good idea to forget this fact. There are isomorphisms Sn(1) ^ Sm (1) '*
* Sn+m (1) that
become canonical and coherent in the homotopy category. The homotopy groups of*
* an object
X 2 M are defined by
ßn(X) := hM(Sn(1), X).
We say that a map f :X -!Y is a weak equivalence_if it induces an isomorphism ß*
**(X) -!ß*(Y ),
and we define the weak homotopy category hM by inverting weak equivalences. We *
*define a cell
object to be an_object of M that is built from the sphere objects Sn(1) in the *
*usual sort of way;
the category hM is then equivalent to the category of cell objects and homotopy*
* classes of maps.
Remark 8.1. In subsequent sections we will consider various spaces of the form *
*M(S(1), X) =
1 FL(S, X). This is weakly equivalent to 1 X but not homeomorphic to it; the*
* functor
1 :MS -! T is not representable and has rather poor behaviour. For this and m*
*any related
reasons it is preferable to replace X by FL(S, X) and thus work with EKMM's "mi*
*rror image"
category MS = {Y | FL(S, Y ) = Y } rather than the equivalent category MS. Ho*
*wever, our
account of these considerations is still in preparation so we have used MS in t*
*he present work.
Now let R be a commutative ring object in M, in other words an object equippe*
*d with maps S j-!
R -~ R^R making the relevant diagrams geometrically (rather than homotopically)*
* commutative.
(The term "ring" is something of a misnomer, as there is no addition until we p*
*ass to homotopy.)
We let MR denote the category of module objects over R in the evident sense. T*
*his is again
a topological model category with a closed symmetric monoidal structure. The_b*
*asic_cofibrant
objects are the free modules Sd(1) ^ R for d 2 Z. The weak homotopy category hM*
*R obtained
by inverting weak equivalences is also known as the derived category of R, and *
*written D = DR;
it is equivalent to the strong homotopy category of cell R-modules. It is not h*
*ard to see that D
is a monogenic stable homotopy category in the sense of [11]; in particular, it*
* is a triangulated
category with a compatible closed symmetric monoidal structure.
9.Strictly unital products
In the previous sections we worked in the derived category D of (strict) R-mo*
*dules. In this
section we sharpen the picture slightly by working with_modules with strict uni*
*ts. These are
not cell R-modules, so we need to distinguish between hMR(X, Y ) := D(X, Y ) = *
*[X, Y ] and
hMR(X, Y ) := ß0MR(X, Y ) = MR(X, Y )=homotopy. Note that the latter need not h*
*ave a group
structure (let alone an Abelian one). However, most of the usual tools of unst*
*able homotopy
theory are available in hMR, because MR is a topological category enriched over*
* pointed spaces.
In particular, we will need to use Puppe sequences.
As previously, we let x be a regular element in ßd(R), so d is even. We regar*
*d x as an R-module
map Sd(1) ^ R -!R, and we write R=x for the cofibre. There is thus a pushout di*
*agram
Sd(1) ^ R____//I ^ Sd(1) ^ R
x || ||
fflffl| fflffl|
R ______æ____//R=x.
As R is not a cell R-module, the same is true of R=x. However, the map æ: R -!*
* R=x is a
q-cofibration. One can also see that S0(1) ^ R=x is a cell R-module which is th*
*e cofibre in D of
the map x: dR -! R, so it has the homotopy type referred to as R=x in the prev*
*ious section.
Moreover, the map S0(1) ^ R=x -!R=x is a weak equivalence. It follows that our *
*new R=x has
the same weak homotopy type as in previous sections.
26 N. P. STRICKLAND
Let W be defined by the following pushout diagram:
æ
R _____//_R=x
æ|| |i0|
fflffl| fflffl|
R=x _i1__//_W
There is a unique map r: W -!R=x such that ri0 = 1 = ri1, and there is an evide*
*nt cofibration
S2d+1(2) ^ R -!W -!(R=x)(2).
Here (
2d+1S(R1 R1 ) ifd 0
S2d+1(2) = Sd(1) ^ Sd(1) =
S(R1 R1 , R|d| R|d|)ifd < 0.
We define a strictly unital product on R=x to be a map OE: (R=x)(2)-! R=x_of R-*
*modules such
that OE|W = r. Let P be the space of strictly unital products, and let P be the*
* set of products on
R=x in the sense of section 3.
__
Proposition 9.1.The evident map ß0(P ) -!P is a bijection.
Proof.The cofibration
S2d+1(2) ^ R -i!W -j!(R=x)(2)
*
gives a fibration MR((R=x)(2), R=x) -j! MR(W, R=x) of spaces. The usual theory*
* of Puppe
sequences and fibrations tells us that the image of j* is the union of those co*
*mponents in
ß0MR((R=x)(2), R=x) that map to zero in ß0MR(S2d+1(2) ^ R, R=x) = ß2d+1(R=x) = *
*0, so j*
is surjective. In particular, we find that P = (j*)-1{r} is nonempty. Similar c*
*onsiderations then
show that the H-space H = MR(S2d+2(1) ^ R, R=x) acts on P , and that for any OE*
* 2 P the action
map h 7! h.OE gives a weak equivalence H ' P . This shows that ß0(H) = ß2d+2(R=*
*x) acts freely
and transitively on_ß0(P ). This is easily seen to be compatible with our free *
*and transitive_action
of ß2d+2(R=x) on P (Lemma 3.7), and the claim follows. *
* |__|
Remark 9.2. These ideas also give another proof of associativity. Let Y be the *
*union of all cells
except the top one in (R=x)(3), so there is a cofibration S3d+2(3)^R -!Y -! (R=*
*x)(3). Let OE be a
product on R=x; by the proposition, we may assume that it is strictly unital. I*
*t is easy to see that
OE O (OE ^ 1) and OE O (1 ^ OE) have the same restriction to Y (on the nose). I*
*t follows using the Puppe
sequence that they only differ (up to homotopy) by the action of the group ß3d+*
*3(R=x) = 0. Thus,
OE is automatically associative up to homotopy.
We end this section with a more explicit description of the element _c(x) 2 ß*
*2d+d(R)=(2, x).
Define X := M(S2d(2), R=x); this is a space with ßkX = ß2d+k(R)=x. The twist m*
*ap ø of
S2d(2) = Sd(1) ^ Sd(1) gives a self-map of X, which we also call ø. Let y be th*
*e map
Sd(1) ^ Sd(1) x^x--!R ^ R mult---!R æ-!R=x,
considered as a point of X. As R is commutative, this is fixed by ø.
Next, let fl :I ^ Sd(1) -!R=x be the obvious nullhomotopy of x, and consider *
*the map
I ^ Sd(1) ^ Sd(1) fl^x--!R=x ^ R mult---!R=x.
This is adjoint to a path ffi :I -!X with ffi(0) = 0 and ffi(1) = y. We could d*
*o a similar thing using
x ^ fl to get another map ffi0:I -! X, but it is easy to see that ffi0= ø O ffi*
*. We now define a map
OE0:@(I2) -!X by
OE0(s,=0)0
OE0(0,=t)0
OE0(s,=1)ffi(s)
OE0(1,=t)øffi(t).
PRODUCTS ON MU-MODULES 27
We can use the pushout description of R=x to get a pushout description of (R=x)*
*(2). Using this,
we find that strictly unital products are just the same as maps OE: I2 -!X that*
* extend OE0.
Let OE be such an extension. Let Ø: I2 -!I2 be the twist map; we find that OE*
*0:= ø O OE O Ø also
extends OE0 and corresponds to the opposite product on R=x. Let U be the space *
*({ 1} x I2)= ~,
where (1, s, t) ~ (-1, s, t) if (s, t) 2 @(I2); clearly this is homeomorphic to*
* S2. Define _ :U -!X
by _(1, s, t) = OE(s, t) and _(-1, s, t) = OE0(s, t) = øOE(t, s). It is not har*
*d to see that the class in
ß2(X) = ß2d+2(R)=x corresponding to _ is just c(OE), and thus that the image in*
* ß2d+2(R)=(2, x)
is _c(x).
Another way to think about this is to define a map ø :U -! U by ø(r, s, t) = *
*(-r, t, s), and
to think of I2 as the image of 1 x I2 in U. We can then say that _ is the uniqu*
*e ø-equivariant
extension of OE.
10.Power operations
In this section, we identify the commutativity obstruction _c(x) of Propositi*
*on 3.1 with a kind
of power operation. This is parallel to a result of Mironov in Baas-Sullivan th*
*eory, although the
proofs are independent. We assume for simplicity that d := |x| 0.
10.1. The definition of the power operation. Because R* is concentrated in even*
* degrees, we
know that the Atiyah-Hirzebruch spectral sequence converging to R*CP 1 collapse*
*s and thus that
R is complex orientable. We choose a complex orientation once and for all, taki*
*ng the obvious one
if R is (a localisation of) MU. This gives Thom classes for all complex bundles.
We write Rev(X) for the even-degree part of R*(X), so that Rev(RP 2) = R*[ffl*
*]=(2ffl, ffl2). (In
the interesting applications the ring R* has no 2-torsion and so R*RP 2has no o*
*dd-degree part.)
We will need notation for various twist maps. We write ! for the twist map of*
* R2d= Rdx Rd,
or for anything derived from that by an obvious functor. Similarly, we write s *
*for the twist map
of (R1 )2, and oe = S(s) for that of S(2) = S(1) ^ S(1) = S((R1 )2). We can thu*
*s factor the twist
map ø of S2d(2) as ø = !oe = oe!.
We will need to consider the bundle V d= R2dxC2 S2 over S2=C2 = RP 2. Here C2*
* is acting
on R2dby !, and antipodally on S2; the Thom space is S2+^C2S2d. As d is even, w*
*e can regard
V das C R V d=2, so we have a Thom class in eR2d(S2+^C2S2d) which generates eR*
*ev(S2+^C2S2d)
as a free module over RevRP2 = R*[ffl]=(2ffl, ffl2).
Suppose that x 2 ßd(R). Recall that x is represented by a map x: Sd(1) = Sd ^*
* S(1) -!R. By
smashing this with itself and using the product structure of R we obtain a map *
*y :S2d(2) -!R.
As R is commutative we have yø = y.
Because S(U) is a continuous contravariant functor of U, we have a map L(2)+ *
*^ S(1) `-!S(2)
and thus a map L(2)+ ^ S2d^ S(1) y`-!R. If we let s: (R1 )2 -!(R1 )2 be the twi*
*st map and let
C2 act on L(2) by g 7! g O s then L(2) is a model for EC2 and thus L(2)=C2 ' RP*
* 1. As yø = y
we see that our map factors through (L(2)+ ^C2 S2d) ^ S(1) ' (RP 1)V d^ S(1). F*
*or any CW
complex A, the spectrum A ^ S(1) is a cofibrant approximation to 1 A, so we ca*
*n regard this
map as an element of R0(RP 1)V d. By restricting to RP 2and using the Thom isom*
*orphism, we
get an element of R-2dRP 2; we define P (x) to be this element. We also recall *
*that Rev(RP 2) =
R*[ffl]=(2ffl, ffl2) and define eP(x) to be the coefficient of ffl in P (x), so*
* eP(x) 2 R-2d-2=2 = ß2d+2(R)=2.
If A is a CW complex with only even-dimensional cells then we can replace R by *
*F (A+, R) to
get power operations P :R-dA -!R-2d(RP 2x A) and eP:R-dA -!R-2d-2(A)=2. It is n*
*ot hard
to check that this is the same as the more classical definition given in [7] an*
*d thus to deduce the
properties listed at the beginning of Section 5.
We also need a brief remark about the process of restriction to RP 2. The sp*
*ace of maps
~: S2 -!L(2) such that ~(-u) = ~(u) O s is easily seen to be contractible. Choo*
*se such a map ~.
We then have (RP 2)V d= S2+^C2S2d, and P (x) is represented by the composite
(S2+^C2S2d) ^ S(1) ~^1^1----!(L(2)+ ^C2S2d) ^ S(1) y`-!R.
We call this map fi0.
28 N. P. STRICKLAND
10.2. A small modification. Let M be the monoid L((R1 )2, (R1 )2). This acts co*
*ntravariantly
on S(2), giving a map
(M+ ^C2S2d) ^ S(2) -!S2d(2) y-!R.
Here we use the action of C2 on M given by g 7! g O s. There is also a homotopi*
*cally unique map
~: S2 -!M such that ~(-u) = ~(u) O s for all u 2 S2. By combining this with the*
* above map,
we get a map
(S2+^C2S2d) ^ S(2) -!S2d(2) y-!R.
We call this map fi1. Recall that in the homotopy category theredis a canonica*
*l isomorphism
S(1) ' S(2), so fi1 again represents an element of R0(RP 2)V = R-2dRP 2. We c*
*laim that
this is the same as P (x). To see this, choose an isomorphism v :(R1 )2 ' R1 ,*
* giving a map
v*:M -! L(2) and a map S(v): S(1) -!S(2). Take v* O ~: S2 -!L(2) as our choice *
*of ~, and
use S(v) as a representative of the canonical equivalence S(1) ' S(2) in the ho*
*motopy category;
under these identifications, fi1 becomes fi0. We leave the rest of the details *
*to the reader.
10.3. An adjunction. Define Y := M(S2d(2), R). The twist maps !, oe and ø induc*
*e commuting
involutions of Y with ø = !oe. We can think of y :S2d(2) -! R as a point of Y ,*
* which is fixed
under ø. The contravariant action of M on S(2) gives a covariant action on Y , *
*which commutes
with !. Using this and our map ~: S2 -!M we can define a map fi2:S2+-!Y by fi2(*
*u) = ~(u).y.
If we let C2 act on Y by ! then one finds that this is equivariant. We can thin*
*k of fi2 as an adjoint
of fi1 and thus a representative of P (x).
10.4. Mapping into R=x. We are really only interested in the image of eP(x) in *
*ß2d+2(R)=(2, x).
To understand this, we reintroduce the space X := M(S2d(2), R=x) as in Section *
*9. The unit map
æ: R -!R=x induces an equivariant map æ*:Y -! X. We define fi3 = æ*O fi2:S2+-!X.
Note that ß0Y = ß2dR, and fi2 lands in the component corresponding to x2, and*
* æ(x2) = 0, so
fi3 lands in the base component. Moreover, we have ß1X = ß2d+1(R)=x = 0 and C2 *
*acts freely
on S2+so by equivariant obstruction theory we can extend fi3 over the cofibre o*
*f the inclusion
S1+-!S2+to get a map fi4 say. This cofibre is equivariantly equivalent to C2+ ^*
* S2 and
[C2+ ^ S2, X]C2 ' ß2(X) = ß2d+2(R)=x.
It is not hard to see that the element of ß2d+2(R)=(2, x) coming from fi4 is ju*
*st the image of eP(x).
10.5. The abstract argument. We now set up an abstract situation in which we ha*
*ve a space
X and we can define two elements ff, fi 2 ß2(X)=2 and prove that they are equal*
*; later we apply
this to show that _c(x) = eP(x) 2 ß2d+2(R)=(2, x). While this involves some rep*
*etition of previous
constructions, we believe that it makes the argument clearer.
Let M be a 2-connected topological monoid, containing an involution oe. Let C*
* = {1, !} be the
group of order two, and define ø = oe! 2 C x M. Let X be a space with basepoint*
* 0 and another
distinguished point y in the base component. Suppose that C x M acts on X, the *
*whole group
fixes 0, and ø fixes y. Suppose also that ! ' 1: X -!X and ß1(X) = 0.
Definition 10.1.Write V = ' C22. Given j, k 0 we can let V act on Rj+*
*k= Rj Rk by
! = (-1) (-1) and oe = 1 (-1) so ø = (-1) 1. We write Rj,kfor this repres*
*entation of V ,
and Sj,kfor the sphere in Rj,k, so that Sj,k= Sj+k-1nonequivariantly.
Definition 10.2.Define a ø-equivariant map ff0:S0,1= {-1, 1} -! X by ff0(-1) = *
*0 and
ff0(1) = y. Using the evident ø-equivariant CW structure on S2,1and the fact th*
*at ß1(X) = 0
we find that there is an equivariant extension of ff over S2,1, which is unique*
* modulo 1 + ø*.
Nonequivariantly we have S2,1= S2 and ø ' 1: S2 -! S2 so we get a homotopy clas*
*s of maps
S2 -!X, which is unique modulo 2. We write ff for the corresponding element of *
*ß2(X)=2.
Definition 10.3.Define ~0:S0,1= {-1, 1} -! M by ~0(-1) = 1 and ~0(1) = oe; this*
* is equi-
variant with respect to the evident right action of ! on M. As oe acts freely o*
*n S2,1and M is
2-connected, we see that there is a !-equivariant extension ~: S2,1-!M, which i*
*s unique up to
equivariant homotopy.
PRODUCTS ON MU-MODULES 29
Definition 10.4.Define fi :S2,1-!X by fi(u) = ~(u).y. As ~ is oe-equivariant an*
*d y is fixed by
ø = !oe and ! commutes with M we find that fioe = !fi :S2,1-!X. We next claim t*
*hat fi can be
extended over the cofibre of the inclusion of S2,0in S2,1in such a way that we *
*still have fioe = !fi.
This follows easily from the fact that oe acts freely on S2,0and y lies in the *
*base component of X
and ß1(X) = 0. The cofibre in question can be identified oe-equivariantly with *
*S2 ^ {1, oe}+. By
composing with the inclusion S2 -!S2 ^ {1, oe}+ we get an element of ß2(X). Thi*
*s can be seen
to be unique modulo 1 + !* but by hypothesis ! ' 1: X -!X so we get a well-defi*
*ned element of
ß2(X)=2, which we also call fi.
Proposition 10.5.We have ff = fi 2 ß2(X)=2.
Proof.Consider the following picture of R2,1.
The axes are set up so that N = (0, 0, 1) and S = (0, 0, -1), so
oe(x, y,=z)(-x, -y, -z)
!(x, y, z)= (x, y, -z)
ø(x, y, z)= (-x, -y, z).
We write H+ and H- for the upper and lower hemispheres and D for the unit disc *
*in the plane
z = 0. Thus H+ [ H- = S2,1and H+ \ H- = S2,0, so H+ [ H- [ D can be identified *
*with the
cofibre of the inclusion S2,0-!S2,1. Note also that H+ [ D is ø-equivariantly h*
*omeomorphic to
S2,1(by radial projection from the ø-fixed point (0, 0, 1=2), say).
Let D0be the closed disc of radius 1=2 centred at O and let A be the closure *
*of D \ D0. Define
ff0:S2,1[ D0-! X by ff0= y on S2,1and ff0= 0 on D0. We see by obstruction theor*
*y that ff0can
be extended ø-equivariantly over the whole of S2,1[ D. Moreover, if we identify*
* H+ [ D with S2,1
as before then the restriction of ff0to H+ [ D represents the same homotopy cla*
*ss ff as considered
in Definition 10.2, as one sees directly from the definition.
Next, note that S2,1[A retracts oe-equivariantly onto S2,1, so we can extend *
*our map ~: S2,1-!
M over S2,1[ A equivariantly. As M is 1-connected, we can extend it further ove*
*r the whole of
S2,1[ D, except that we have no equivariance on D0.
30 N. P. STRICKLAND
Now define fi0:S2,1[ D -! X by fi0(u) = ~(u).ff0(u). We claim that fi0oe = !f*
*i0. Away from
D0this follows easily from the equivariance of ~ and ff0, and on D0it holds bec*
*ause both sides are
zero. Using this and our identification of S2,1[ D with the cofibre of S2,0-!S2*
*,1we see that the
restriction of fi0to H+ [ D represents the class fi in Definition 10.4.
Now observe that S2,1[D is 2-dimensional and M is 2-connected, so our map ~: *
*S2,1[D -!M
is nonequivariantly homotopic to the constant map with value 1. This implies th*
*at ff0is_homotopic_
to fi0, so ff = fi 2 ß2(X)=2 as claimed. *
* |__|
10.6. The proof that _c(x) = eP(x). We now prove that _c(x) = eP(x). We take X *
*:= M(S2d(2), R=x)
and M := L((R1 )2, (R1 )2) as before, and define involutions !, oe and ø as in *
*Section 10.1. We also
define y as in Section 10.3. It is then clear that the map fi4 of Section 10.4 *
*represents the class fi
of Definition 10.4, so that fi = eP(x) 2 ß2d+2(R)=(2, x). Now consider the cons*
*tructions at the end
of Section 9. It is not hard to see that the space U defined there is ø-equivar*
*iantly homeomorphic
to S2,1, with the two fixed points being (0, 0, 1) and (1, 1, 1). As the map _ *
*:U -!X is equivariant
and _(0, 0, 1) = 0 and _(1, 1, 1) = y, we see that _ represents the class ff of*
* Definition 10.2, so
_c(x) = fi 2 ß e _
2d+2(R)=(2, x). It now follows from Proposition 10.5 that P(x) = *
*c(x) (mod 2, x), as
claimed.
References
[1]J. F. Adams. A variant of E. H. Brown's representability theorem. Topology, *
*10:185-198, 1971.
[2]J. F. Adams. Stable Homotopy and Generalised Homology. University of Chicago*
* Press, Chicago, 1974.
[3]M. Ando. Isogenies of formal group laws and power operations in the cohomolo*
*gy theories En. Duke Mathe-
matical Journal, 79(2):423-485, 1995.
[4]N. Baas. On bordism theory of manifolds with singularities. Math. Scand., 33*
*:279-302, 1973.
[5]E. H. Brown. Cohomology theories. Annals of Mathematics, 75:467-484, 1962.
[6]E. H. Brown and F. P. Peterson. A spectrum whose Zpcohomology is the algebra*
* of reduced p-th powers.
Topology, 5:149-154, 1966.
[7]R. R. Bruner, J. P. May, J. E. McClure, and M. Steinberger. H1 Ring Spectra *
*and their Applications, volume
1176 of Lecture Notes in Mathematics. Springer-Verlag, 1986.
[8]A. D. Elmendorf. Stabilisation as a CW approximation. Available from http://*
*hopf.math.purdue.edu, 1997.
[9]A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May. Rings, Modules and A*
*lgebras in Stable Homotopy
Theory, volume 47 of Amer. Math. Soc. Surveys and Monographs. American Mathe*
*matical Society, 1996.
[10]J. Franke. On the construction of elliptic cohomology. Math. Nachr., 158:43*
*-65, 1992.
[11]M. Hovey, J. H. Palmieri, and N. P. Strickland. Axiomatic stable homotopy t*
*heory. Mem. Amer. Math. Soc.,
128(610):x+114, 1997.
[12]M. Hovey and N. P. Strickland. Morava K-theories and localisation. To appea*
*r in the Memoirs of the American
Mathematical Society, 1995.
[13]P. S. Landweber. Homological properties of comodules over MU*(MU) and BP*(B*
*P). American Journal of
Mathematics, 98:591-610, 1976.
[14]L. G. Lewis. Is there a convenient category of spectra? Journal of Pure And*
* Applied Algebra, 73:233-246,
1991.
[15]L. G. Lewis, J. P. May, and M. S. (with contributions by Jim E. McClure). E*
*quivariant Stable Homotopy
Theory, volume 1213 of Lecture Notes in Mathematics. Springer-Verlag, New Yo*
*rk, 1986.
[16]H. R. Margolis. Spectra and the Steenrod Algebra. North-Holland, 1983.
[17]H. R. Miller. Finite localizations. Boletin de la Sociedad Matematica Mexic*
*ana, 37:383-390, 1992. This is a
special volume in memory of Jos'e Adem, and is really a book. The editor is *
*Enrique Ram'irez de Arellano.
[18]O. K. Mironov. Existence of multiplicative structures in the theory of cobo*
*rdism with singularities. Izv. Akad.
Nauk SSSR Ser. Mat., 39(5):1065-1092,1219, 1975.
[19]O. K. Mironov. Multiplicativity in the theory of cobordism with singulariti*
*es and the Steenrod-Tom Dieck
operations. Math USSR Izvestiya, 13(1):89-107, 1979.
[20]J. Morava. A product for the odd primary bordism of manifolds with singular*
*ities. Topology, 18:177-186, 1979.
[21]C. Nassau. On the structure of P(n)*(P(n)) for p = 2. preprint, 1996.
[22]C. Nassau. Eine nichtgeometrische Konstruktion der Spektren P(n), Multiplik*
*ativen und antimultiplikativen
Automorphismen von K(n). PhD thesis, Johann Wolfgang Goethe-Universität Fran*
*kfurt, October 1995.
[23]D. G. Quillen. On the formal group laws of unoriented and complex cobordism*
*. Bulletin of the American
Mathematical Society, 75:1293-1298, 1969.
[24]D. G. Quillen. Elementary proofs of some results of cobordism theory using *
*Steenrod operations. Advances in
Mathematics, 7:29-56, 1971.
[25]D. C. Ravenel. Complex Cobordism and Stable Homotopy Groups of Spheres. Aca*
*demic Press, 1986.
PRODUCTS ON MU-MODULES 31
[26]D. C. Ravenel. Nilpotence and Periodicity in Stable Homotopy Theory, volume*
* 128 of Annals of Mathematics
Studies. Princeton University Press, 1992.
[27]N. Shimada and N. Yagita. Multiplications in the complex bordism theory wit*
*h singularities. Publications of
Research Institute of Mathematical Sciences, Kyoto University, 12:259-293, 1*
*976.
[28]T. tom Dieck. Steenrod-Operationen in Kobordismen-Theorien. Math. Z., 107:3*
*80-401, 1968.
[29]W. S. Wilson. Brown-Peterson Homology: An Introduction and Sampler, volume *
*48 of Regional Conference
Series in Mathematics. American Mathematical Society, 1982.
[30]J. J. Wolbert. Classifying modules over K-theory spectra. J. Pure Appl. Alg*
*ebra, 124(1-3):289-323, 1998.
[31]U. Würgler. Cohomology theory of unitary manifolds with singularities and f*
*ormal group laws. Mathematische
Zeitschrift, 150:239-60, 1976.
[32]U. Würgler. On products in a family of cohomology theories associated to th*
*e invariant prime ideals of i*(BP).
Commentarii Mathematici Helvetici, 52:457-81, 1977.
[33]U. Würgler. Commutative ring-spectra in characteristic 2. Commentarii Mathe*
*matici Helvetici, 61:33-45,
1986.
[34]Z.-i. Yosimura. Universal coefficient sequences for cohomology theories of *
*cw-spectra. Osaka J. Math.,
12(2):305-323, 1975.
Trinity College, Cambridge CB2 1TQ, England
E-mail address: neil@dpmms.cam.ac.uk