CORES OF SPACES, SPECTRA, AND E1 RING SPECTRA
P. HU, I. KRIZ, AND J.P. MAY
Abstract.In a paper that has attracted little notice, Priddy showed that
the Brown-Peterson spectrum at a prime p can be constructed from the p-
local sphere spectrum S by successively killing its odd dimensional homo*
*topy
groups. This seems to be an isolated curiosity, but it is not. For any s*
*pace
or spectrum Y that is p-local and (n0- 1)-connected and has in0(Y ) cycl*
*ic,
there is a p-local, (n0- 1)-connected ün clear" CW complex or CW spectrum
X and a map f : X ! Y that induces an isomorphism on in0and a monomor-
phism on all homotopy groups. Nuclear complexes are atomic: a self-map t*
*hat
induces an isomorphism on in0 must be an equivalence. The construction of
X from Y is neither functorial nor even unique up to equivalence, but it*
* is
there. Applied to the localization of MU at p, the construction yields B*
*P.
In 1999, the third author gave an April Fool's talk on how to prove that BP i*
*s an
E1 ring spectrum or, in modern language, a commutative S-algebra. As explained
in [20], he gave a quite different April Fool's talk on the same subject two ye*
*ars
earlier. His new idea was to exploit the remarkable paper of Stewart Priddy [23*
*],
in which Priddy constructed BP by killing the odd degree homotopy groups of the
sphere spectrum. The hope was that by mimicking Priddy's construction in the
category of commutative S-algebras, one might arrive at a construction of BP as
a commutative S-algebra. As the first two authors discovered, that argument fai*
*ls.
However, the ideas are still interesting. As we shall explain, Priddy's constru*
*ction of
BP is not an accidental fluke but rather a special case of a very general const*
*ruction.
The elementary space and spectrum level construction is given in Section 1. The
more sophisticated E1 ring spectrum analogue and its specialization to MU are
discussed in Section 2.
It is a pleasure to thank Nick Kuhn and Fred Cohen for very illuminating e-ma*
*ils.
In particular, Example 1.10 is due to Cohen.
1. Cores of spaces and spectra
To set context, we begin by recalling some standard properties of spaces and
spectra that still have not been fully explored. The following successively str*
*onger
conditions are studied in Wilkerson [24]. We assume that all spaces and spectra
are p-local for a fixed prime p and of the homotopy types of p-local CW objects*
* of
finite type. Thus we require that each ßn(Y ) be a finitely generated Z(p)-modu*
*le.
Spaces are to be based and simply connected and spectra are to be bounded below.
Unless otherwise specified, cohomology is to be taken with mod p coefficients.
Definition 1.1. Let X be a space or a spectrum, as above.
(i)X is indecomposable if X admits no non-trivial product decomposition.
____________
Date: July 4, 2001.
1991 Mathematics Subject Classification. Primary 55P15, 55P42, 55P43; Second*
*ary 55S12.
The authors were partially supported by the NSF.
1
2 P. HU, I. KRIZ, AND J.P. MAY
(ii)X is irreducible if it admits no non-trivial retracts.
(iii)X is H*-prime if, for a map f : X -! X, either f* : H*(X) -! H*(X) is
an isomorphism or f* : Hq(X) -! Hq(X) is nilpotent for each q > 0.
The term "indecomposable" is suggested by the analogy with module theory.
Clearly an H*-prime space is irreducible and an irreducible space is indecompos-
able. Additional hypotheses on X which ensure that irreducible implies H*-prime
are given in [24, 3.4, 3.5]. Since retracts of spectra split in the stable cat*
*egory,
where finite wedges are equivalent to finite products, indecomposable spectra a*
*re
irreducible. An elementary space level analogue is that if A is a retract of a *
*co-H-
space X, with A -! X a cofibration, then X is equivalent to A _ X=A.
Definition 1.2. Assume further that X is (n0 - 1)-connected, where we assume
henceforward that n0 2 in the case of spaces, and that ßn0(X) is a cyclic mod*
*ule
over Z(p).
(i)X is atomic if a map f : X -! X that induces an isomorphism on ßn0(X)
is an equivalence.
(ii)X is H*-monogenic if H*(X) is a cyclic algebra (in the case of spaces) or
module (in the case of spectra) over the Steenrod algebra.
If X is H*-monogenic, then X is atomic. If X is atomic, then X is indecompos-
able. If the entire image of the Hurewicz homomorphism of X is concentrated in
the Hurewicz dimension, then X is atomic. We are interested in an especially ri*
*gid
type of atomic space or spectrum. In general, when studying atomic spaces or sp*
*ec-
tra, it seems most natural to work with p-complete rather than just p-local obj*
*ects,
but this paper is concerned with cellular constructions, which work more natura*
*lly
in the p-local context. Thus we consider p-local CW objects (spaces or spectra)*
* of
finite type, and we agree to call such CW objects öc mplexes" throughout. Let Sn
denote a p-local n-sphere. Such spheres are the domains of the attaching maps of
our complexes. In the case of spaces, we require attaching maps to be based. Th*
*us,
if X is a complex, then its (n + 1)-skeleton is the cofiber of a map jn : Jn -!*
* Xn,
where Jn is a wedge of finitely many copies of Sn.
Priddy used the term "irreducible" for a version of the following concept, bu*
*t it
seems more sensible to reserve that term for the standard notion defined above.
Definition 1.3. A complex is nuclear of Hurewicz dimension n0 if its (n0 - 1)-
skeleton is trivial, its n0-skeleton is Sn0, it has finitely many n-cells for e*
*ach n > n0,
and
(1.4) Ker(jn* : ßn(Jn) -! ßn(Xn)) p . ßn(Jn)
for each n n0. When n = n0, this implies that Jn0 is either * or Sn0. Thus the
attaching maps of X are detected by mod p homotopy. If j : Sn0 -! X is the
inclusion, it induces an epimorphism j* : ßn0(Sn0) -! ßn0(X).
The following result is based on a proposition of Priddy [23, x1].
Proposition 1.5. A nuclear complex is atomic.
Proof.Let X be nuclear and let f : X -! X be a map that induces an isomorphism
on ßn0. We must prove that f is a homotopy equivalence. We may assume that
f is cellular, and we prove that f restricts to a homotopy equivalence Xn -! Xn
for all n. Thus assume inductively that f : Xn -! Xn is a homotopy equivalence.
CORES OF SPACES, SPECTRA, AND E1 RING SPECTRA 3
This holds trivially if n < n0 and is easily checked if n = n0. We deduce that
f : Xn+1 -! Xn+1 is a homotopy equivalence. Take homology with coefficients in
Z(p). It suffices to prove that
f* : Hq(Xn+1) -! Hq(Xn+1)
is an isomorphism for q = n and q = n+1. It is easy to check (using the Freuden*
*thal
suspension theorem) that f induces a map from the cofiber sequence
Jn -! Xn -! Xn+1
to itself. There results a commutative diagram
0_____//Hn+1(Xn+1)_____//Hn(Jn)____//Hn(Xn)____//_Hn(Xn+1)____//0
|f*| |f*| ~=f*|| |f*|
fflffl| fflffl| fflffl| fflffl|
0_____//Hn+1(Xn+1)_____//Hn(Jn)____//Hn(Xn)____//_Hn(Xn+1)____//0
with exact rows. By the five lemma, it suffices to prove that
f* : Hn(Jn) -! Hn(Jn)
is an isomorphism. By the Hurewicz theorem, it suffices to prove that
f* : ßn(Jn) -! ßn(Jn)
is an isomorphism. We have a commutative diagram with exact rows
ßn(Jn)_____//ßn(Xn)____//ßn(Xn+1)____//0
f*|| ~=f*|| f*||
fflffl| fflffl| fflffl|
ßn(Jn)_____//ßn(Xn)____//ßn(Xn+1)____//0.
The right arrow f* is an epimorphism by the diagram and is therefore an isomor-
phism since any epimorphic endomorphism of a finitely generated module over a
PID is an isomorphism. It follows that the right arrow is an isomorphism in the
commutative diagram
0_____//kerjn*_i_//_ßn(Jn)___//imjn*____//0
| | |~
| f*| |=
fflffl|i fflffl| fflffl|
0_____//kerjn*___//_ßn(Jn)___//imjn*____//0
In view of (1.4), the inclusion i becomes 0 after tensoring over Z=p. Therefore
f* Z=p is an isomorphism. This implies that f* is an isomorphism.
The following construction is a generalization of Priddy's construction [23] *
*of
the Brown-Peterson spectrum by killing the odd dimensional homotopy groups of
the (p-local) sphere spectrum. He was motivated by the fact that the homotopy
groups of MU are concentrated in even degrees, but MU played no role in his
actual construction. We change the point of view. We consider a preassigned spa*
*ce
or spectrum Y under a sphere Sn0, and we kill the homotopy groups of the kernel
of the given map Sn0 -! Y .
4 P. HU, I. KRIZ, AND J.P. MAY
Construction 1.6. Let Y be n0 - 1 connected with a given map j : Sn0 -! Y .
We construct a nuclear complex X together with a map f : X -! Y under Sn0
that induces a monomorphism on all homotopy groups. We start with Xn0 = Sn0
and fn0 = j : Xn0 -! Y . Assume inductively that we have constructed Xn
and a map fn : Xn -! Y that induces a monomorphism on homotopy groups in
dimension less than n. Choose a minimal (finite) set of generators for the kern*
*el
of fn* : ßn(Xn) -! ßn(Y ), let Jn be the wedge of a copy of Sn for each chosen
generator, and let jn : Jn -! Xn represent the chosen generators. Define Xn+1 to
be the cofiber of jn. Choose a null homotopy hn of the composite fn O jn and us*
*e it
to extend fn to a map fn+1 : Xn+1 -! Y . The cofibration Xn -! Xn+1 induces
an isomorphism on ßi for i < n and an epimorphism on ßn, and fn+1 induces a
monomorphism on ßifor i n. On passage to colimits, we obtain f : X -! Y that
induces a monomorphism on all homotopy groups. The minimality of our chosen
sets of generators ensures that (1.4) holds.
Note that spheres are obviously nuclear and a two cell complex (with cells in
different dimensions) is either nuclear or a wedge (trivial attaching map). The
obvious mechanism for a finite complex not to be nuclear is to have at least on*
*e cell
with a trivial attaching map. Some later cell might attach both to this one and*
* to
another cell lower down. Construction 1.6 then gives a space with smaller homot*
*opic
groups and no non-trivial attaching maps. For example, with Y = Sn0xSn, n > n0,
the construction just gives Sn0.
The construction is most interesting when ßn0(Y ) is cyclic. Here it shows th*
*at
Y has a core, in the sense of the following definition.
Definition 1.7. Let Y be n0 - 1 connected with ßn0(Y ) cyclic. A core of Y is a
nuclear complex X together with a map f : X -! Y that induces an isomorphism
on ßn0 and a monomorphism on all homotopy groups.
The homotopy groups of the fiber F f are then ßq(F f) ~=ßq+1(Y )=f*ßq+1(X),
and the Hurewicz dimension of F f is at least n0. However, F f need not have a
cyclic bottom homotopy group. For spectra, we can use cofibers rather than fibe*
*rs,
and this leads to the following inductive construction.
Construction 1.8. Let Y = Y0 be an (n0- 1)-connected spectrum. We construct
a nuclear decomposition of Y . Choose a (finite) minimal set of generators for
ßn0(Y ). Construct a nuclear complex X0 and a map f0 : X0 -! Y0 from a
representative map j : Sn0 -! Y0 for one of these generators. Let Y1 be the cof*
*iber
of f0. The remaining generators of ßn0(Y ) give generators of ßn0(Y1), and we c*
*an
repeat the construction starting with Y1. Iterating, we kill ßn0 after finitely*
* many
steps, and we then continue by killing ßn0+1 similarly. Iterating, we obtain nu*
*clear
complexes Xiand maps fi: Xi- ! Yiwith cofibers Yi+1such that each fiinduces
a monomorphism on all homotopy groups. The homotopy groups of Yi+1 are the
quotients of the homotopy groups of Yi by the images of the homotopy groups of
Xi, and the colimit of the sequence of cofibrations Yi- ! Yi+1is trivial.
These constructions are related to early work of Freyd, Margolis, and Wilkers*
*on
[16, 19, 24] on cancellation and unique decomposition of finite p-local or p-co*
*mplete
spectra and spaces. In the case of spaces, little seems to be known about which
spaces admit factorizations as products of indecomposable spaces. The notion of*
* an
atomic space, which is due to Cohen, arose in connection with the work of Selic*
*k,
CORES OF SPACES, SPECTRA, AND E1 RING SPECTRA 5
Cohen, Neisendorfer, and Moore on exponents of homotopy groups. Some of the
relevant papers are [1, 6, 7, 8, 9, 11, 12, 13, 14, 25, 26]. These papers iden*
*tify
many particular spaces that arise in applications as being atomic, examine the
relationship between atomicity of spaces and atomicity of their loop spaces, and
study the relationship of atomicity of spaces to the structure of their monoids*
* of
self-maps. There are many open questions about these concepts. For example,
Cohen points out that it is not known whether or not the suspension spectra of
K(Z=2, n) or of the p - 1 wedge summands of K(Z=p, n), p odd, are atomic. Our
constructions raise many new questions. Here are a few general ones.
Questions 1.9. Assume that Y is (n0 - 1)-conected and ßn0(Y ) is cyclic.
(i)For which Y is the core of Y unique?
(ii)Can one classify the cores of Y ?
(ii)Can one explicitly identify cores of some interesting spaces?
(iii)Is every atomic space equivalent to a nuclear complex?
The ideas so far are due to the third author, who tried hard to prove that
the core of Y is unique. The first and second authors provided a spectrum level
counterexample: see Example 1.17 below. The point is that, in Construction 1.6,
there are many choices for the homotopy class of fn+1, which differ by elements*
* in
Im([ Jn, Y ] -! [Xn+1, Y ]). Changing the choice can change the kernel that one
is killing at the next stage, and different choices can lead to very different *
*cores.
We also expect the answer to question (iii) to be no: it seems likely that nu*
*clear
complexes give a special class of atomic spaces. This is strongly suggested by *
*the
following example, which is due to Fred Cohen.
Example 1.10. A map f : X - ! Y between atomic spaces that induces an
isomorphism on the bottom homotopy group and a monomorphism on all homotopy
groups need not be an equivalence. For a counterexample, take Y = 2S5 at the
prime 2. Clearly ß6(Y ) = ß8(S5) ~=Z=8 with generator . The class 4 is detect*
*ed
in cohomology by a map g : Y - ! K(Z=2, 6), so that g induces an epimorphism
on homotopy groups. Let f : X -! Y be the homotopy fiber of g. Then f induces
an isomorphism on ß3 and a monomorphism on all homotopy groups. The space Y
is atomic by [11]. Cohen [unpublished] has shown that X is also atomic.
Definition 1.11. Say that an atomic space Y is minimal if a map f : X -! Y from
an atomic space X into Y that induces an isomorphism on the bottom homotopy
group and a monomorphism on all homotopy groups is necessarily an equivalence.
Clearly, if Y is a minimal atomic space, then a core f : X -! Y is an equival*
*ence.
Thus minimal atomic spaces are equivalent to nuclear complexes.
Question 1.12. Is every nuclear complex a minimal atomic space?
Assume that Y is (n0 - 1)-connected and that ßn0(Y ) is cyclic in the followi*
*ng
two lemmas. The first is immediate from the definitions. The second shows that
the core of Y is unique under very restrictive hypotheses on Y .
Lemma 1.13. If g : Y - ! Z is a map that induces an isomorphism on ßn0 and
a monomorphism on all homotopy groups and if f : X -! Y is a core of Y , then
g O f : X -! Z is a core of Z.
Lemma 1.14. If the homotopy groups and p-local cohomology groups of Y are
concentrated in even degrees, then the core of Y is a retract of Y and is uniqu*
*e.
6 P. HU, I. KRIZ, AND J.P. MAY
Proof.Let f : X -! Y be a core. Then Hn+1(Y ; ßn(X)) = 0 for all n since the
homotopy groups of X and the cohomology groups of Y are both concentrated
in even degrees. By obstruction theory, there is a map g : Y - ! X under Sn0.
The composite g O f : X -! X is an equivalence by Proposition 1.5. Thus X
is a retract of Y , hence also has p-local cohomology groups concentrated in ev*
*en
degrees. If f0 : X0- ! Y is another core, there are no obstructions to construc*
*ting
maps i : X -! X0 and j : X0- ! X under Sn0. The composites j O i and i O j are
equivalences by Proposition 1.5, hence X and X0 are equivalent.
Since BP is irreducible, these lemmas imply a version of Priddy's result [23].
Proposition 1.15. BP is the core of MU.
Remark 1.16. In Lemma 1.14, we are only claiming that X is unique up to equiv-
alence, not that a map f : X -! Y that identifies X as the core of Y is unique.
For example, BP is the core of BP ^ BP , as is displayed by both the left and t*
*he
right unit maps BP -! BP ^ BP .
The conclusion of Lemma 1.14 fails if we drop the hypothesis about cohomology.
Example 1.17. The units of BP and HZ(p)induce maps
BP _____//BP ^ HZ(p)oo_HZ(p),_
both of which induce monomorphisms on all homotopy groups and an isomorphism
on ß0. Since BP and HZ(p)are each their own cores, it follows from Lemma 1.13
that both are cores of BP ^ HZ(p).
The following analogue of Proposition 1.15 must be true, but even this does n*
*ot
seem to be quite trivial.
Conjecture 1.18. For p odd, BP <1> is the core of ku.
Certainly the first non-zero positive dimensional homotopy group of a core X *
*is
ß2p-2(X), since the first cell that we attach to S kills ß2p-3(S).
Warning 1.19. We can construct cores similarly for integral or rational spaces *
*or
spectra, rather than just for p-local ones. However, these constuctions will no*
*t be
compatible with the p-local construction. For example, the rational core of MU *
*is
S rather than BP , and it is unclear what the integral core of MU is. Whatever *
*it
is, it is unique by the proof of Lemma 1.14. Comparison with the versions of BP
and MU in Boardman's papers [2, 3] may be of interest.
2. Cores of E1 ring spectra
The ideas of the previous section can be adapted to a variety of frameworks
in which one has a notion of CW objects. We shall illustrate this by presenting
the construction used in the third author's failed attempt to prove that BP is *
*an
E1 ring spectrum. The construction surely gives rise to new E1 ring spectra,
but it is hard to analyze what they look like. We work in the context of [15],
replacing E1 ring spectra by weakly equivalent commutative S-algebras. Again,
we work p-locally. Recall from [15, VIII.2.2] that localizations of commutative
S-algebras are commutative algebras over the p-local sphere spectrum, which we
denote henceforward by S. We let Sn be a p-local cofibrant n-sphere in the cate*
*gory
of S-modules.
CORES OF SPACES, SPECTRA, AND E1 RING SPECTRA 7
Everything done for spectra in x1 could equally well and perhaps more sensibly
have been done in the category M of S-modules. We let C denote the category of
commutative S-algebras; our constructions below have evident analogues for non-
commutative S-algebras (or A1 -ring spectra). Note that S is cofibrant in C but*
* not
in M ; [15] explains how to deal with such homotopical details. We have a forge*
*tful
functor C -! M with left adjoint free functor P : M -! C . Let : X -! PX
denote the unit of the adjunction. Let CX denote the cone on an S-module X and
let ' : X -! CX be the canonical inclusion.
Construction 2.1. Let R be a connective cofibrant commutative S-algebra whose
unit S -! R induces an isomorphism on ß0 and whose homotopy groups are finitely
generated Z(p)-modules. We construct a map of S-algebras g : Q -! R, which
we call a core of R as a commutative S-algebra, by inductively killing homotopy
groups. Let Q0 = S and let g0 : Q0 -! R be the unit of R. Assume that we have
constructed an S-algebra Qn and a map of S-algebras gn : Qn -! R. Let Kn be
the wedge of one copy of Sn for each element in a chosen (finite) minimal set of
generators for the kernel of gn* : ßn(Qn) -! ßn(R). Let kn : Kn -! Qn be a map
of S-modules that realizes these generators. By minimality,
(2.2) Ker(kn* : ßn(Kn) -! ßn(Qn)) p . ßn(Kn).
The induced map ~kn: PKn -! Qn of S-algebras gives Qn a structure of PKn-
algebra. Define
Qn+1 = PCKn ^PKn Qn.
Thus, as in [15, II.3.7], Qn+1 is the pushout of the diagram
~kn
(2.3) PCKn ooP'_PKn _____//Qn
in the category of commutative S-algebras. By construction, gn O kn is null ho-
motopic. Choose a null homotopy hn : CKn -! R and let ~hn: PCKn -! R
be the induced map of S-algebras. By the universal property of pushouts, there
results a map gn+1 : Qn+1 -! R of S-algebras that restricts to gn on Qn. Define
Q = colimQn and let g : Q -! R be the map of S-algebras obtained by passage
to colimits from the gn. By construction, the induced map of homotopy groups
g* : ß*(Q) -! ß*(R)
is a monomorphism of Z(p)-algebras.
Example 2.4. Arguing exactly as in Example 1.17, but with BP there replaced
by MU since we do not know that BP is a commutative S-algebra, we obtain an
explicit counterexample that shows that cores of commutative S-algebras are not
unique. At p = 2, we will identify a core of MU in Theorem 2.12 below.
Recall that colimits of sequences of S-algebras are computed as the colimits *
*of
the underlying sequences of S-modules [15, VII.3.10]. By construction, Q is a c*
*ell
S-algebra and is thus cofibrant. The analogue of Lemma 1.13 is obvious.
Lemma 2.5. Let h : R -! T be a map of commutative S-algebras such that
h* : ß*(R) -! ß*(T ) is a monomorphism. If g : Q -! R is a core of R, then h O g
is a core of T .
8 P. HU, I. KRIZ, AND J.P. MAY
If the homotopy groups of R are concentrated in even degrees, then so are the
homotopy groups of Q. In particular, we are then killing all of the odd dimensi*
*onal
homotopy groups of S in our inductive construction of Q. In a small range of
dimensions the homotopy groups of Qn+1 agree with those of the cofiber of ~kn, *
*as
we see, for example, from the spectral sequence
(2.6) E2p,q= Torß*(PKn)p,q(ß*(S), ß*(Qn)) =) ßp+q(Qn+1)
of [15, IV.4.1]. Cores of commutative S-algebras are nuclear, in the following *
*sense.
Definition 2.7. A commutative S-algebra Q is nuclear if Q = colimQn where
Q0 = S and, inductively, Qn+1 is the pushout of a diagram of the form (2.3), wh*
*ere
Kn is a wedge of finitely many copies of Sn and kn : Kn -! Qn is a map of
S-modules that satisfies (2.2).
Definition 2.8. A connective commutative S-algebra Q whose unit induces an
isomorphism on ß0 is atomic if any map of S-algebras f : Q -! Q is a weak
equivalence.
It is plausible but not obvious that the analogue of Proposition 1.5 holds.
Conjecture 2.9. A nuclear commutative S-algebra is atomic.
It might also seem plausible that a core of an S-algebra R is also a core of *
*its
underlying S-module, but we shall see that that is false. We do have the follow*
*ing
comparison, which is what remains of the third author's original program.
Proposition 2.10. For any core g : Q -! R of commutative S-algebras, there
exists a core f : X -! R of S-modules and a map , : X -! Q of S-modules such
that f = g O ,. In particular, , induces a monomorphism on homotopy groups.
Proof.We construct a commutative diagram
jn
Jn__________________//HXnH
||HHH~nH | HHH,nH
| HH | HHH
'| H##H ~kn | H$$
| PKn ________|_______//_Qn
| | | |,,
| | | |,
| | | | ,,
fflffl| P'| fflffl| | ,,
CJn ________|_______//_Xn+1 | ,,
HHH | HHH | ,,
HH | H,n+1H| gn,
nHHH | HHH | ,,
H##fflffl| H##fflffl,,|
PCKn _______________//_WWWWWQ,,n+1gE
WWWWWWWW EEE ,, n+1
~WWWWWWWW EEE,,
hn WWWWWWW,~~++""E
R.
The front part of the diagram displays underlying S-modules in our construction
of the S-algebra core g : Q -! R. The back square of the diagram is a pushout
in M that will display the inductive step of a construction of an S-module core
f : X -! R such that fn = gn O ,n for a map ,n : Xn -! Qn of S-modules that
induces a monomorphism on ßq for q n. We have X0 = S0 and Q0 = S, and we
let ,0 : X0 -! Q0 be a weak equivalence of S-modules (a cofibrant approximation*
*).
CORES OF SPACES, SPECTRA, AND E1 RING SPECTRA 9
We let f0 = ,0 O g0, where g0 is the unit of R. Assume inductively that we have
constructed ,n : Xn -! Qn and let fn = gn O,n. Let Jn be a wedge of copies of S*
*n,
one for each element in a chosen minimal set of generators for the kernel of (f*
*n)* :
ßn(Xn) -! ßn(R), and let jn : Jn -! Xn represent these generators. Recall that
Kn is a wedge of copies of Sn, one for each element in a chosen minimal set of
generators for the kernel of (gn)* : ßn(Qn) -! ßn(R), and that kn : Kn -! Qn
represents these generators. The cofiber Ckn is the pushout in M of the diagram
CKn oo'__Kn _kn_//_Qn,
and the universal property of pushouts in M gives a canonical map
æ : Ckn -! Qn+1,
which is a (2n - 1)-equivalence by inspection or use of (2.6). Since gn O ,n = *
*fn,
(,n)* : ßn(Xn) -! ßn(Qn) restricts to a homomorphism Ker(fn)* -! Ker(gn)*.
Choosing preimages in ßn(Kn) of the images under (,n)* O (jn)* of the generators
of ßn(Jn), we obtain a homomorphism ßn(Jn) -! ßn(Kn). We can realize this
homomorphism by a map fln : Jn -! Kn such that the left square commutes up
to homotopy in the diagram
jn
Jn _____//Xn____//Xn+1_____// Jn
fln|| |,n| ,0n+1|| ||fln
fflffl| fflffl| fflffl| fflffl|
Kn _kn__//Qn____//_Ckn____//_ Kn
A standard comparison of cofibration sequences argument in M gives a map ,0n+1
such that the middle square commutes and the right square commutes up to homo-
topy. Moreover, ,0n+1induces an monomorphism on ßq for q n, since in degree n
it induces the inclusion Im(fn)* -! Im(gn)* (up to isomorphism). Define
,n+1 = æ O ,0n+1: Xn+1 -! Qn+1
and define ~n and n to be the evident composites
Jn _fln//_Kn___//PKnand CJn _Cfln//_CKn__//_PCKn.
Then the cube in our main diagram is a commutative diagram in M . The composite
~hnO n in the diagram coincides with the composite hn O Cfln, which is a null
homotopy of fn Ojn. The map fn+1 = gn+1O,n+1 is induced by this null homotopy,
in agreement with our inductive prescription of a core of the S-module R. Passi*
*ng
to colimits, we obtain the maps , and f of the conclusion.
Thus an S-algebra core has larger homotopy groups than the corresponding S-
module core. In particular, with R = MU, for any S-algebra core g : Q -! MU,
we have a factorization of an S-module core f : BP -! MU as g O , for a map
, : BP -! Q of S-modules that induces a monomorphism on homotopy groups. It
is easy to see that the lowest positive degree homotopy group of Q must be 2p -*
* 2,
so it seems reasonable to hope that , is an equivalence. However, that is false*
*: ,
cannot be an equivalence, since that would contradict the following observation*
* of
the first two authors.
Proposition 2.11. There is no map g : BP -! MU of commutative S-algebras.
10 P. HU, I. KRIZ, AND J.P. MAY
Proof.If there were such a map g, it would commute with units and so induce
an isomorphism on ß0. Therefore, since BP is atomic, the composite of g and a
splitting map MU -! BP would be a self-equivalence of BP , so that g would
be the inclusion of a retract. The map g* : H*(BP ) -! H*(MU) on mod p
homology would be a monomorphism that commutes with Dyer-Lashof operations.
The Thom isomorphism ` : H*(MU) -! H*(BU) commutes with Dyer-Lashof
operations by a result of Lewis [18, IX.7.4]. Kochman [17] and Priddy [22] have
computed the Dyer-Lashof operations in H*(BU) and thus on H*(MU). Write
H*(MU) = P [ai|degai = 2i], where ai is the standard generator coming from
H*(BU(1)). If p = 2, then Q5(a1) a5 mod decomposables, and, if p > 2, then
Qp(ap-1) a(p+1)(p-1)mod decomposables, by [22] or [10, II.8.1]. Here ap-1 is *
*in
the image of H*(BP ), but H*(BP ) has no indecomposable elements in degree 10
if p = 2 or in degree 2(p + 1)(p - 1) if p > 2.
So, if BP is not a core of MU, what is? The first two authors succeeded in
answering this question when p = 2.
Theorem 2.12. If p = 2, then M(Sp=U) is a core of MU, regarded as a commu-
tative S-algebra.
Proof.We have an infinite loop map Sp=U -! BU, and work of Lewis [18, IX]
gives an E1 -ring Thom spectrum M(Sp=U), a map ff : M(Sp=U) -! MU of E1 -
ring spectra, and compatible Thom isomorphisms that commute with Dyer-lashof
operations. It is standard that Sp=U has no 2-torsion and that H*(Sp=U), with
coefficients in Z(2)or Z=2, is a polynomial algebra on generators of degrees 4r*
* + 2,
r 0, that map to generators of these degrees in H*(BU). By [22] or [10, II.8.*
*1],
Q4r(a1) a2r+1 mod decomposables for r 1 in H*(MU). It follows that Q4r
maps the generator of degree 2 in H*(M(Sp=U)) to a generator in degree 4r + 2.
Thus, intuitively, the image of H*(M(Sp=U)) is the smallest possible subalgebra*
* of
H*(MU) that contains a1 and is closed under the Dyer-Lashof operations.
The composite of ff with the canonical map MU -! BP induces an epimorphism
on homology, hence a monomorphism on cohomology. A theorem of Milnor and
Moore [21, 4.4] shows that H*(M(Sp=U)) is a free A=(fi)-module, and a theorem
of Brown and Peterson [4] shows that M(Sp=U) is a wedge of suspensions of copies
of BP . Therefore ff induces a monomorphism on homotopy groups since it induces
a monomorphism on homology groups. By Lemma 2.5, the composite of ff and a
core g : Q -! M(Sp=U) of the commutative S-algebra M(Sp=U) is a core of MU.
It is clear by construction that b1, 2-locally, must be in the image of g*, and*
* it
follows by consideration of Dyer-Lashof operations that g induces an epimorphism
on mod 2 homology and therefore on 2-local homology. Moreover, Lemmas 1.13
and 1.14 imply that BP is the core of M(Sp=U) as an S-module. By Proposition
2.10, we can factor a core BP -! M(Sp=U) through g, giving a composite map,
necessarily an equivalence
BP ____//_Qg__//M(Sp=U)____//_MU____//BP.
Now BP is complex oriented, and the image of its orientation gives Q a complex
orientation. Since Q is a 2-local commutative and associative ring spectrum, the
complex orientation can be modified if necessary to give a 2-typical formal gro*
*up
law, and then there is a map of ring spectra BP -! Q that is compatible with
the orientation. In particular, this gives Q a structure of BP -module spectru*
*m.
CORES OF SPACES, SPECTRA, AND E1 RING SPECTRA 11
Enumerate the wedge summands niBP of M(Sp=U) so that ni nj if i < j,
where i 1. Via the unit of BP , each summand is determined by an element
~i 2 ßni(M(Sp=U)). We claim that the generators lift to elements i 2 ßni(Q).
Using the BP -module structure, the idetermine maps niBP -! Q that together
give a map : M(Sp=U) -! Q such that g O ' id. This implies that g induces
an epimorphism and therefore an isomorphism on homotopy groups.
Thus assume inductively that ~i lifts to i for i < j. Let N be the wedge of *
*the
niBP , i < j, choose a splitting map ß : M(Sp=U) -! N, and let L be its fiber.
We may regard ~i as an element of ßni(L). Let K be the fiber of ß O g : Q -! N.
Comparing fiber sequences, we obtain a map f : K -! L. Inductively, g induces an
isomorphism of homotopy groups in degrees less than n. Therefore K, like L, must
be (ni-1)-connected. Now, since g induces an epimorphism on 2-local homology, so
does f. Since the 2-local Hurewicz homomorphism for K and L is an isomorphism
in degree nj, we may lift ~j to ßnj(K) and therefore to ßnj(Q).
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Department of Mathematics, University of Chicago, Chicago, IL 60637
E-mail address: pohu@math.uchicago.edu
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109
E-mail address: ikriz@math.lsa.umich.edu
Department of Mathematics, University of Chicago, Chicago, IL 60637
E-mail address: may@math.uchicago.edu