MINIMAL ATOMIC COMPLEXES A.J. BAKER AND J.P. MAY Abstract.Hu, Kriz and May recently reexamined ideas implicit in Priddy's elegant homotopy theoretic construction of the Brown-Peterson spectrum a* *t a prime p. They discussed May's notions of nuclear complexes and of cores * *of spaces, spectra, and commutative S-algebras. Their most striking conclus* *ions, due to Hu and Kriz, were negative: cores are not unique up to equivalenc* *e, and BP is not a core of MU considered as a commutative S-algebra, althou* *gh it is a core of MU considered as a p-local spectrum. We investigate the* *se ideas further, obtaining much more positive conclusions. We show that nu- clear complexes have several non-obviously equivalent characterizations.* * Up to equivalence, they are precisely the irreducible complexes, the minimal a* *tomic complexes, and the Hurewicz complexes with trivial mod p Hurewicz homo- morphism above the Hurewicz dimension, which we call complexes with no mod p detectable homotopy. Unlike the notion of a nuclear complex, these other notions are all invariant under equivalence. This simple and conce* *ptual criterion for a complex to be minimal atomic allows us to prove that many familiar spectra, such as ko, eo2, and BoP at the prime 2, all BP at * *any prime p, and the indecomposable wedge summands of 1 CP1 and 1 HP1 at any prime p are minimal atomic. Introduction Atomic spaces and spectra have long been studied. They are so tightly bound together that a self-map which induces a isomorphism on homotopy in the Hurewicz dimension must be an equivalence. Atomic spaces and spectra can often be shrunk to ones with smaller homotopy groups. Minimal ones can be shrunk no further. Clearly, these are very natural objects of study. They seem to have been first introduced in [10]. Spheres, 2-cell complexes that are not wedges, and K(ß, n)* *'s for cyclic groups ß are obviously minimal atomic, but there are many much more interesting examples. Nuclear complexes are atomic complexes (spaces or spectra) that are built up in an especially economical way. They are minimal atomic, and we shall see that every minimal atomic complex is equivalent to a nuclear complex. We regard the invariant notion of a minimal atomic complex as the more fundamental one. The combinatorial notion of a nuclear complex provides us with a tool for proving t* *hings about minimal atomic complexes. We think of atomic complexes as analogues of ä tomic modules", namely mod- ules for which a non-trivial self-map is an isomorphism. We think of minimal atomic complexes as analogues of irreducible modules. We give a definition of an irreducible complex that makes this analogy more transparent, and we prove that ____________ 1991 Mathematics Subject Classification. Primary 55P15 55P42 55P43; Secondar* *y 55S12. Key words and phrases. atomic space, atomic spectrum, nuclear space, nuclear* * spectrum. 1 2 A.J. BAKER AND J.P. MAY the irreducible complexes are precisely the minimal atomic complexes. In one di- rection, the implication gives a homotopical analogue of Schur's lemma. Just as* * in algebra, we suggest that the irreducible, or minimal atomic, complexes are more basic mathematical objects than the atomic complexes. Carrying the analogy fur- ther, we see that the spectra we consider admit (dual) composition series of in* *finite length, constructed in terms of irreducible complexes. There is a different and much more elementary notion of minimality, implicitly due to Cooke [5], such that any complex is equivalent to a minimal complex. This notion is also combinatorial and noninvariant. We prove that a Hurewicz complex that is minimal in this sense is nuclear if and only if it has no mod p detecta* *ble homotopy. It is well known that the latter condition implies wedge indecompos- ability (e.g. [14, 5.4]). The fact that it implies irreducibility is much stron* *ger. We also prove a converse, leading to the conclusion that a complex is irreducible,* * or equivalently minimal atomic, if and only if it has no mod p detectable homotopy. This allows us to show that a variety of familiar spectra are in fact minimal a* *tomic. 1. Definitions and statements of results Here we give the precise definitions needed to make sense of the introduction* * and state our main theorems. We write things so that the stable reader can view the* *se as statements about spectra, and the unstable reader can view them as statements about (based) spaces. We adopt the following conventions throughout. They allow us to treat spaces and spectra uniformly and to avoid repeated mention of the f* *act that we are working p-locally under connectivity and finite type hypotheses. We agree once and for all that all spaces and spectra X are to be localized a* *t a fixed prime p. Thus Sn, for example, means a p-local sphere. We also agree that all spaces and spectra are to be p-local CW spaces or spectra, so that the doma* *ins of their attaching maps are p-local spheres. Spaces are to be simply connected,* * and their attaching maps are to be based. Spectra are to be bounded below. In either case, we say that X has Hurewicz dimension n0 if X is (n0 - 1)-connected, but not n0-connected. Thus n0 > 2 in the case of spaces, and there is no real loss * *of generality if we take n0 = 0 in the case of spectra. We may assume without loss* * of generality that there are no cells (except the base vertex) of dimension less t* *han n0. We assume further that there are only finitely many cells in each dimension. We agree to use the ambiguous term öc mplex" to mean such a p-local CW space or spectrum. We say that X is a Hurewicz complex if it has a single cell in dimens* *ion n0. We write Xn for the n-skeleton of a complex X. Thus Xn+1 is the cofiber of a map jn : Jn -! Xn, where Jn is a finite wedge of (p-local) n-spheres Sn. If X i* *s a Hurewicz complex, Xn0 = Sn0. We shall use these notations generically. By H*(X), we always understand (reduced) homology with p-local coefficients. Any (n0 - 1)-connected space or spectrum such that each Hn(X) is a finitely gen- erated Z(p)-module is weakly equivalent to a complex in the sense that we have just specified. If, further, Hn0(X; Fp) = Fp or, equivalently, ßn0(X) is a cyc* *lic Z(p)-module, then X is weakly equivalent to a Hurewicz complex. We begin with definitions of concepts that are invariant under equivalence and the statement of our main characterization theorem relating them. Definition 1.1. Consider complexes X and Y of Hurewicz dimension n0. Think of Y as fixed but X as variable. MINIMAL ATOMIC COMPLEXES 3 (i)A map f : X -! Y is a monomorphism if f*: ßn0(X) Fp -! ßn0(Y ) Fp and all f*: ßn(X) -! ßn(Y ) are monomorphisms (ii)Y is irreducible if any monomorphism f : X -! Y is an equivalence. (iii)Y is atomic if it is a Hurewicz complex and a self-map f : Y - ! Y that induces an isomorphism on ßn0 is an equivalence. (iv)Y is minimal atomic if it is atomic and any monomorphism f : X -! Y from an atomic complex X to Y is an equivalence. (v) Y has no mod p detectable homotopy if Y is a Hurewicz complex and the mod p Hurewicz homomorphism h : ßn(Y ) -! Hn(Y ; Fp) is zero for all n > n0. (vi)Y is H*-monogenic if H*(Y ; Fp) is a cyclic algebra (in the case of spac* *es) or module (in the case of spectra) over the mod p Steenrod algebra A. Remarks 1.2. We offer several comments on these notions. (i)The structure theory for finitely generated modules over a PID implies t* *hat if f :X -! Y is a monomorphism, then f*(ßn0(X)) is a direct summand of ßn0(Y ). If X is a Hurewicz complex, this summand is cyclic. If X and Y are both Hurewicz complexes, then f induces an isomorphism on ßn0. (ii)In [10, 1.1], following [21] and other early sources, Y was defined to * *be irreducible if it has no non-trivial retracts. On the space level, that * *concept has its uses, but we think that "irreducible" is the wrong name for it. * *We suggest "irretractible". On the spectrum level, irretractibility is equi* *valent to wedge indecomposability. However, just as in algebra, irreducibility should be stronger rather than weaker than atomic. That is, there should be implications irreducible =) atomic =) indecomposable. One could avoid the conflict with the earlier literature by using the word "simple" instead of "irreducible", the two being synonymous in algebra, but that risks confusion with the standard use of the term "simple" in topology. (iii)A complex can well have more than one cell in its Hurewicz dimension and still have the property that a self-map that induces an isomorphism on ß* *n0 is an equivalence. A particularly interesting example is given in [2, x4* *]. It might be sensible to delete the requirement that Y be a Hurewicz complex from the definition of atomic. By Theorem 1.9 below, the notion of minim* *al atomic would not change. (iv)Since our methods are cellular, we definitely mean to consider p-local r* *ather than p-complete spaces and spectra. However, Definition 1.1 makes just as much sense in the p-complete case as the p-local case, and it is well wo* *rth studying there. Since a finite type p-complete space or spectrum is the * *p- completion of a finite type p-local space or spectrum, one can easily de* *duce conclusions in the p-complete case from the results here. We leave the details to the interested reader. (v) A Hurewicz complex Y has no mod p detectable homotopy if and only if there are no permanent cycles in dimension greater than n0 on the zeroth row of the classical (unstable or stable) mod p Adams spectral sequence for Y . It is a much more computable condition than the others. Theorem 1.3 (The characterization theorem). The following conditions on a com- plex Y are equivalent. (i)Y is irreducible. 4 A.J. BAKER AND J.P. MAY (ii)Y is minimal atomic. (iii)Y has no mod p detectable homotopy. The fact that irreducible complexes are atomic should be viewed as a homotopi* *cal analogue of Schur's lemma since its intuitive content is that a non-trivial sel* *f-map of an irreducible complex must be an equivalence. Of course, it is consistent w* *ith the analogy that not all atomic complexes are irreducible. When [10] was writte* *n, examples of minimal atomic spectra seemed hard to come by. The following result now gives us many interesting examples. Corollary 1.4. If Y is H*-monogenic, then Y has no mod p detectable homotopy and is therefore minimal atomic. Proof.For spectra, the long exact sequence on Ext arising from the epimorphism n0A -! H*(Y ; Fp) implies that the zeroth row Hom 0,*A(H*(Y ; Fp), Fp) of the Adams spectral sequence is Fp concentrated in degree n0. Similarly for spaces. Remark 1.5. The converse of Corollary 1.4 fails. For q > 2, Moore spaces and spectra M(Z=pq, n) and Eilenberg-Mac Lane spaces and spectra K(Z=pq, n) are elementary examples of complexes that are minimal atomic but not H*-monogenic. The proof of Theorem 1.3 proceeds by first showing in Theorem 1.12 that (i) and (ii) are each equivalent to a statement about one noninvariant cellular con* *struc- tion and then showing in Theorem 1.18 that (ii) and (iii) are each equivalent t* *o a statement about another noninvariant cellular construction. The first noninvari* *ant construction, codified in Theorems 1.9 and 1.11, is based on the notions of nuc* *lear complexes and cores introduced in [10]. Definition 1.6. A nuclear complex is a Hurewicz complex X such that (1.7) Ker(jn* : ßn(Jn) -! ßn(Xn)) p . ßn(Jn) for each n. Observe that X is nuclear if and only if each Xn for n > n0 is nucl* *ear. A core of a complex Y is a nuclear complex X together with a monomorphism f : X -! Y . This notion of a core is more general than that of [10, 1.7], where it was as* *sumed that ßn0(Y ) is cyclic; that is, the definition there was restricted to Hurewic* *z com- plexes Y . Since cores are not unique even when Y is a Hurewicz complex, the mo* *re general notion seems preferable. Of course, with our perspective that cores are analogues of irreducible sub-modules, the non-uniqueness is only to be expected. With the present language, the following results are proven in [10, 1.5, 1.6]. Proposition 1.8. A nuclear complex is atomic. Theorem 1.9. If Y has Hurewicz dimension n0 and C is a cyclic direct summand of ßn0(Y ), then there is a core f : X -! Y such that f*(ßn0(X)) = C. This allows us to construct homotopical analogues of composition series. That is, for spectra, we can shrink homotopy groups inductively by successively taki* *ng cofibers of cores, as discussed in [10, 1.8]. This works less well for spaces, * *where we would have to take fibers and so gradually decrease the Hurewicz dimension. To make the analogy with algebra precise, recall that a (countably infinite) compo* *sition series of a module Y is a sequence of monomorphisms Y = Y0 oi0oY1_oo___. .o.o_Ynooin_Yn+1oo___ . . . MINIMAL ATOMIC COMPLEXES 5 such that the cokernels of the in are irreducible and limYn = 0. A dual composi* *tion series of Y is a sequence of epimorphisms Y = Y0 _p0_//_Y1__//_._._.//_Ynpn//_Yn+1__//. . . such that the kernels of the pn are irreducible and colimYn = 0. For a spectrum* * Y , we construct an analogous sequence by letting pn : Yn -! Yn+1 be the cofiber of* * a core fn : Xn -! Yn. Each pn induces an epimorphism on all homotopy groups, we kill ßn0(Y ) in finitely many steps, then kill ßn0+1(Y ) in finitely many steps* *, and so on. If Y has non-zero homotopy groups in only finitely many dimensions, then this sequence has only finitely many terms. Theorem 1.9 has the following immediate consequence. Corollary 1.10. A core of a minimal atomic complex is an equivalence, hence a minimal atomic complex is equivalent to a nuclear complex. The converse, which completes Proposition 1.8, was conjectured in [10, 1.12]. We will prove it in Section 3. Theorem 1.11. A nuclear complex is a minimal atomic complex. Theorem 1.12. Conditions (i) and (ii) of Theorem 1.3 are each equivalent to the condition that any core f : X -! Y of Y is an equivalence. Proof.Corollary 1.10 and Theorem 1.11 show that Y is minimal atomic if and only if any core of Y is an equivalence. If Y is irreducible and f : X -! Y is a cor* *e, then f is an equivalence by the definition of irreducibility. Conversely, suppo* *se that Y is minimal atomic and let f : X -! Y be a monomorphism. Let g : W -! X be a core of X. Then the composite f O g : W -! Y is a core of Y and therefore * *an equivalence. This implies that f induces an epimorphism and hence an isomorphism on homotopy groups. Thus f is an equivalence. Remark 1.13. With these implications in place, it is perhaps better to redefine the notion of core invariantly, taking X to be minimal atomic but not necessari* *ly nuclear. There is no substantive difference. To tie in the Hurewicz homomorphism condition (iii) of Theorem 1.3, we use an- other, very different, noninvariant notion of minimality for a complex X. Of co* *urse, our complexes have p-local chain complexes specified by Cn(X) = Hn(Xn=Xn-1). Definition 1.14. A complex X is minimal if the differential on its mod p chain complex C*(X; Fp) is zero. It is minimal Hurewicz if it is minimal and Hurewicz. Observe that X is minimal if and only if each Xn is minimal. A simple inductive argument gives a homological reformulation of this notion. Lemma 1.15. A complex X is minimal if and only if the inclusion of skeleta Xn -! Xn+1 induces an isomorphism Hn(Xn; Fp) -! Hn(Xn+1; Fp) = Hn(X; Fp) for each n. The following two results codify our second noninvariant construction. The fi* *rst is implicit in Cooke's paper [5, Theorem A], which gives an integral space level version. Cooke described the result as ä well-known, basic fact". For a rece* *nt reappearance, see [8, 4.C.1]. The proof is very easy, but we shall run through * *it in Section 5 in view of the importance of the result to our work. 6 A.J. BAKER AND J.P. MAY Theorem 1.16. For any complex Y , there is a minimal complex X and an equiv- alence f : X -! Y . We prove the following result in Section 4. Theorem 1.17. If X is a nuclear complex, then X has no mod p detectable ho- motopy. If X is a minimal Hurewicz complex, then X is nuclear if and only if it has no mod p detectable homotopy. Theorem 1.18. Conditions (ii) and (iii) of Theorem 1.3 are each equivalent to the condition that any equivalence X -! Y from a minimal complex X to Y is a core of Y ; that is, a minimal complex equivalent to Y is nuclear. Proof.Since a minimal atomic complex is equivalent to a nuclear complex, the second statement of Theorem 1.17 implies that (iii) is equivalent to the condit* *ion specified in the statement and that (iii) implies (ii). Similarly, the first st* *atement of Theorem 1.17 implies that (ii) implies (iii). Corollary 1.4 and, more generally, the implication (iii) implies (ii) provide* * a powerful tool for detecting minimal atomic complexes. We give some general resu* *lts that illustrate the use of this criterion in the next section. Restricting to spectra, we see in Section 6 that ko and eo2 at the prime 2, B* *P at any prime p, and the indecomposable wedge summands of 1 CP1 and 1 HP1 at any prime p are minimal atomic. We give a few other examples and remarks, but we regard this section as just a beginning. Our results imply that minimal atomic complexes exist in abundance, and something closer to a classification of them would be desirable. In Section 7, we describe Pengelley's 2-local spectrum BoP as a nuclear compl* *ex and thereby give it a new construction that is independent of [14]. This is in * *the same spirit as Priddy's construction of BP [16] which motivated the definition * *of nuclear complexes and is recalled in Section 6. The key step in the proof is de* *ferred to Section 8. A brief Appendix corrects minor errors in one of the proofs in [1* *0]. 2. Constructions on minimal atomic complexes We indicate briefly how the collection of minimal atomic complexes behaves wi* *th respect to some basic topological constructions. The proofs are direct conseque* *nces of the ön mod p detectable homotopy" characterization of minimal atomic. The following triviality may help the reader see the various implications. Lemma 2.1. Consider a commutative diagram f A _____//A0 h|| |h0| fflffl|gfflffl| B ____//_B0 of Abelian groups. If f is an epimorphism and h = 0, then h0 = 0. If g is a monomorphism and h0= 0, then h = 0. We begin by recording an immediate consequence of Theorems 1.17 and 1.18. Proposition 2.2. If Y is minimal atomic, then Y is equivalent to a complex X whose skeleta Xn for n > n0 are minimal atomic. MINIMAL ATOMIC COMPLEXES 7 There is no reason to believe that the skeleta of Y itself are minimal atomic* *. We have a more invariant analogue for Postnikov sections, which we denote by Y [n]. Proposition 2.3. A complex Y is minimal atomic if and only if Y [n] is minimal atomic for each n > n0. Proof.We have the following commutative diagram. ßq(Y )_______//_ßq(Y [n]) h|| h|| fflffl| fflffl| Hq(Y ; Fp)____//Hq(Y [n]; Fp) Since ßq(Y [n]) is zero for q > n and the horizontal arrows are isomorphisms for q 6 n, the conclusion is immediate from Lemma 2.1. In the following result, which is only of interest for spaces, we consider th* *e loop and suspension functors. Proposition 2.4. If either Y or Y is minimal atomic, then so is Y . Proof.This is immediate from the following commutative diagrams. ~= ßq( Y )________//_ßq+1(Y ) ßq(Y )__________//ßq+1( Y ) h || h|| h || |h| fflffl| fflffl| fflffl| fflffl| Hq( Y ; Fp)_oe_//Hq+1(Y ; Fp) Hq(Y ; Fp)~=_//_Hq+1( Y ; Fp) This result has an analogue that relates minimal atomicity for spaces and spe* *ctra. Here, exceptionally, we must distinguish the two contexts notationally. Proposition 2.5. If E is a spectrum of Hurewicz dimension n0 > 2 whose 0th space 1 E is minimal atomic, then E is minimal atomic. If Y is a simply con- nected space whose suspension spectrum 1 Y is minimal atomic, then Y is minimal atomic. Proof.This is immediate from the following commutative diagrams. ~= ßq( 1 E) ________//_ßq(E) ßq(Y )__________//ßq( 1 Y ) h|| h|| h || |h| fflffl| fflffl| fflffl| fflffl| Hq( 1 E; Fp)__oe_//Hq(E; Fp) Hq(Y ; Fp)~=_//_Hq( 1 Y ; Fp) 3.The proof of Theorem 1.11 The proof in [10, 1.5] that a nuclear complex X is atomic starts with a self map f : X -! X which is an isomorphism on ßn0(X) and deduces that f is an equivalence. The cited proof readily adapts to give the following analogue. Proposition 3.1. Let X and Y be nuclear complexes of Hurewicz dimension n0 and let f :X -! Y be a core of Y . Then f is an equivalence. 8 A.J. BAKER AND J.P. MAY Proof.Take f :X - ! Y to be cellular. Since f is a monomorphism between Hurewicz complexes, f : Xn0 -! Yn0 is an equivalence. Assume that f : Xn -! Yn is an equivalence. We must show that f : Xn+1 -! Yn+1 is an equivalence. The attaching maps of X and Y give rise to the following map of cofibre sequences. jn Jn _____//Xn____//_Xn+1 f|| f|| |f| fflffl| fflffl| fflffl| Kn __kn_//Yn____//_Yn+1 Passing to homology, this gives rise to a commutative diagram with exact rows. jn* 0_____//Hn+1(Xn+1)_____//Hn(Jn)____//Hn(Xn)____//_Hn(Xn+1)____//0 f*|| |f*| ~=f*|| |f*| fflffl| fflffl| fflffl| fflffl| 0_____//Hn+1(Yn+1)____//Hn(Kn)_kn*_//Hn(Yn)_____//_Hn(Yn+1)___//0 It suffices to prove that the left and right vertical arrows are isomorphisms. * *By the five lemma and the Hurewicz theorem, this holds if f*: ßn(Jn) -! ßn(Kn) is an isomorphism. To see that this is so, consider the following diagram. jn* ßn(Jn) ____//_ßn(Xn)____//ßn(Xn+1)____//0 f*|| |f*| |f*| fflffl| fflffl| fflffl| ßn(Kn) _kn*_//ßn(Yn)____//_ßn(Yn+1)___//0 The rows are exact, and a chase of the diagram shows that the right arrow f* is* * an epimorphism. Now consider the following diagram. ~= ßn(Xn+1) ____//_ßn(X) f*|| f*|| fflffl| fflffl| ßn(Yn+1)__~=_//ßn(Y ) Since its right arrow f* is a monomorphism, its left arrow f* is a monomorphism and therefore an isomorphism. This implies that the right vertical arrow is an isomorphism in the following diagram. ~= 0_____//Kerjn*__i__//_ßn(Jn+1)___//_Imjn*___//0 | | ~| | |f* =| fflffl| fflffl| fflffl| 0_____//Kerkn*__i_//_ßn(Kn+1)_~=//_Imkn*____//0 In view of (1.7), both maps i become 0 after tensoring with Fp. This implies th* *at f* Fp is an isomorphism, and therefore so is f*. Proof of Theorem 1.11.Let Y be a nuclear complex of Hurewicz dimension n0 and let f :X -! Y be a monomorphism, where X is atomic. The same argument as in the last part of the proof of Theorem 1.12 shows that f is an equivalence. MINIMAL ATOMIC COMPLEXES 9 4.The proof of Theorem 1.17 We start with the following observation, which is implicit in Priddy [16]. It* * gives a homological recasting of the definition of a nuclear complex. Lemma 4.1. A Hurewicz complex of dimension n0 is nuclear if and only if the mod p Hurewicz homomorphism h : ßn(Xn) -! Hn(Xn; Fp) is zero for n > n0. Proof.Recall the defining property (1.7) of a nuclear complex. In the case of spaces, our assumption that X is simply connected allows us to quote the relati* *ve and absolute Hurewicz theorem to deduce that ßn+1(Xn+1, Xn) ~=ßn+1( Jn) ~=ßn(Jn) from the trivial analogue in p-local homology. In either the space or the spect* *rum context, we obtain the following commutative diagram with exact rows. ßn+1(Xn) _______//ßn+1(Xn+1)_______//_ßn(Jn)_j*___//_ßn(Xn) |h| |h| h|| fflffl| |fflffl fflffl| 0 ________//_Hn+1(Xn+1; Fp)__//_Hn(Jn; Fp)j*//_Hn(Xn; Fp) An easy diagram chase gives that (1.7) holds for n if and only if the left arro* *w h is zero. The conclusion follows. We prove the following reformulation of Theorem 1.17 by relating this skeletal criterion to the Hurewicz homomorphism of X itself. Proposition 4.2. Let X be a Hurewicz complex of dimension n0 and consider the following conditions. (i)X has no mod p detectable homotopy. (ii)X is a minimal complex. If (i) and (ii) hold, then X is nuclear. Conversely, if X is nuclear, then (i) * *holds. Proof.We have the following commutative diagram, where n > n0. ßn(Xn) ________//_ßn(X) h|| |h| fflffl| fflffl| Hn(Xn; Fp) ____//_Hn(X; Fp) The conclusion is immediate from Lemmas 1.15, 2.1, and 4.1. 5.The proof of Theorem 1.16 We are given a complex Y . Recall that our complexes are simply connected in the case of spaces and bounded below in the case of spectra. More fundamentally, everything is p-local. We have assumed that H*(Y ) is of finite type, so that e* *ach Hn(Y ) is a direct sum of finitely many cyclic Z(p)-modules An,i. We must const* *ruct a minimal complex X and an equivalence f : X -! Y , and it suffices for the lat* *ter to ensure that f induces an isomorphism on H*. The complex X will have an n-cell jn,ifor each free cyclic summand An,iand an n-cell jn,iand an (n + 1)-cell kn,i with differential qijn,ifor each summand An,iof order qi. Since each qi must be a power of p, it will be immediate that the differential on C*(X; Fp) is zero. * *The 10 A.J. BAKER AND J.P. MAY cells jn,iwill map to cycles that represent the generators of the An,i, and the* * cells kn,iwill map to chains with boundary qif*(jn,i). Assume inductively that we have constructed the n-skeleton Xn together with a (based) map fn : Xn -! Y that induces an isomorphism on homology in dimen- sions less than n and an epimorphism on Hn. More precisely, assume that Hn(Xn) is Z(p)-free on basis elements given by cells jn,ithat map to chosen generators* * of the An,i. Let Cfn be the cofiber of fn. Then Hm (Cf) = 0 for m 6 n. The kernel * *of f* : Hn(Xn) -! Hn(Y ) is free on the basis qijn,ifor those i such that An,ihas * *finite order. These elements are the images of elements k00n,iin Hn+1(Cfn), and k00n,i= h(k0n,i) for unique elements k0n,iin ßn+1(Cfn). Similarly, the chosen generator* *s of the An+1,i Hn+1(Y ) map to elements j00n+1,i2 Hn+1(Cf) with j00n+1,i= h(j0n+1,* *i). For spectra, we have the connecting homomorphism ßn+1(Cfn) -! ßn(Xn). For spaces, the relative Hurewicz theorem gives ßn+1(Mfn, Xn) ~=ßn+1(Cf), and we have the connecting homomorphism ßn+1(Mfn, Xn) -! ßn(Xn). Thus in either case the elements k0n,iand j0n+1,idetermine elements of ßn(Xn). Choose maps Sn -! Xn that represent these elements and use them as attaching maps for the construction of Xn+1 from Xn by attaching cells kn,iand jn+1,i. Since the sequence ßn+1(Cfn) -! ßn(Xn) -! ßn(Y ) is exact, these attaching maps become null homotopic in Y , and there is an extension fn+1 : Xn+1 -! Y of fn. Thus we can construct the following map of cofiber sequences. Xn _____//Xn+1____//_Xn+1=Xn____//_ Xn || | | || || fn+1| | || || fflffl| fflffl| || Xn __fn__//_Y________//Cfn______// Xn This gives rise to the following commutative diagram with exact rows. 0_____//Hn+1(Xn+1)___//_Hn+1(Xn+1=Xn)___//_Hn(Xn)___//_Hn(Xn+1)___//0 (fn+1)*|| || |||| (fn+1)*|| fflffl| fflffl| || fflffl| 0______//_Hn+1(Y_)_____//_Hn+1(Cfn)_____//Hn(Xn)______//_Hn(Y_)___//_0 Of course, the differential on Cn+1(Xn+1) is the composite Hn+1(Xn+1=Xn) -! Hn(Xn) -! Hn(Xn=Xn-1), where the second arrow is a monomorphism. By construction, the first arrow sends the basis elements kn,ito qijn,iand the basis elements jn+1,ito zero, so that Hn+1(Xn+1) is Z(p)-free on the basis elements jn+1,i. By construction and a cha* *se of the diagram, the map fn+1 induces an isomorphism on Hn and sends the basis elements jn+1,ito generators of the groups An+1,i. This completes the inductive step in the construction of f : X -! Y . 6.Spectrum level examples As a first example, we revisit Priddy's construction [16] of a nuclear spectr* *um equivalent to BP . Although it was the motivating example for [10], it was not explicitly discussed there. We work with p-local spectra in this section. Unl* *ess otherwise stated, p is unrestricted. MINIMAL ATOMIC COMPLEXES 11 Example 6.1. BP is a minimal atomic spectrum, hence the canonical monomor- phism BP -! MU is a core of MU. Proof.H*(BP ; Fp) is a cyclic A-module, hence Corollary 1.4 applies. Proposition 6.2. Let X be the nuclear complex of [16] defined by starting with S0 and inductively killing the homotopy groups in odd degrees. Then there is an equivalence X -! BP . Proof.A minimal complex equivalent to BP has cells only in even degrees and is nuclear. By construction, X also has cells only in even degrees and is nuclear,* * and its non-zero homotopy groups only occur in even degrees. Obstruction theory giv* *es maps f :X -! BP and g :BP -! X that extend the identity on the bottom cell. The composites g O f :X -! X and f O g :BP -! BP are equivalences since X and BP are atomic. Recall that, for an odd prime p, there is a splitting of ku with BP <1>as a wedge summand. In [10, 1.18] it is conjectured that the core of ku is BP <1>. H* *ere ku = BP <1>if p = 2. Since H*(BP <1>; Fp) is a cyclic A-module, this is now immediate. Example 6.3. The spectrum BP <1>is minimal atomic, hence the canonical mono- morphism BP <1>-! ku is a core. More generally, H*(BP ) is a cyclic A-module for all n > -1, the extreme cases being BP <-1>= HFp and BP <0>= HZ(p). Example 6.4. For n > -1, BP is a minimal atomic spectrum. The following example of the non-uniqueness of cores generalizes [10, 1.17]. Example 6.5. For n > 0, the canonical maps BP ____//_BP ^ BP oo_BP induced by the units of BP and BP are both cores of BP . Proof.The left map is a monomorphism since it factors the (p-local) Hurewicz homomorphism of BP . The right map is a monomorphism since it is split by the BP -action BP ^ BP -! BP . Since H*(ko; F2) = A==A(1) and H*(eo2; F2) = A==A(2) are cyclic A-modules, we have the following complement to Proposition 6.3. Proposition 6.6. At p = 2, ko and eo2 are minimal atomic spectra. Some well-known Thom complexes give further examples. Proposition 6.7. Let X be RP1-1, CP1-1, or HP1-1, that is, the Thom spectrum of the negative of the canonical real, complex, or quaternionic line bundle. At p * *= 2, X is minimal atomic. Proof.Let d = 1, 2, and 4 and P = RP1 , CP1 , and HP1 in the respective cases. Then H*(P ; F2) = F2[x], where x 2 Hd(P ; F2) is the dth Stiefel-Whitney class of the canonical line bundle. Since X is a Thom spectrum, H*(X; F2) is the free H*(P ; F2)-module generated by the Thom class ~ in degree -d. A standard calculation shows that Sqnd~ = xn~ for n > 1, so H*(X; F2) is cyclic over A. 12 A.J. BAKER AND J.P. MAY To give some examples where we must check the ön mod p homotopy" condition directly, we consider a few suspension spectra and another Thom spectrum. Let ,3 -! HP1 be the bundle associated to the adjoint representation of S3 and let M,3 be its Thom complex (also known as a quaternionic quasi-projective space). It has one cell in each positive dimension congruent to 3 (mod 4). By [9, 6, 3], for each odd prime p, there is a splitting of p-local spaces CP1 ' W1 _ W2 _ . ._.Wp-1, where Wr has cells in all dimensions of the form 2(p - 1)k + 2r + 1 with k > 0. Proposition 6.8. At the prime 2, 1 CP1 , 1 HP1 and 1 M,3 are minimal atomic spectra. At an odd prime p, each 1 Wr is minimal atomic. Proof.Let a(n) = 1 if n is even and a(n) = 2 if n is odd. By [17], the Hurewicz homomorphisms h: ß2n( 1 CP1 ) -! H2n(CP1 ) ~=Z and h: ß4n( 1 HP1 ) -! H4n(HP1 ) ~=Z have images of index n! and (2n)!=a(n), respectively. Thus, for n > 1, the cor- responding mod 2 Hurewicz homomorphisms are trivial. By [19], the Hurewicz homomorphism h: ß4n+3( 1 M,3) -! H4n+3(M,3) ~=Z has image of index a(n)(2n - 1)! , so for each n > 1 the associated mod 2 Hurew* *icz homomorphism is also trivial. The odd primary results follow similarly from the calculation of h for 1 CP1 . Remark 6.9. We raise a few questions here. (i)There are many basic results in the literature in which interesting spac* *es are split p-locally into products of indecomposable factors and interest* *ing spectra are split p-locally into wedges of indecomposable summands. (The notion of wedge indecomposability is less interesting in the case of spa* *ces). It is a very interesting set of problems to revisit these splittings and* * deter- mine which of the summands are atomic rather than just indecomposable, and which are minimal atomic rather than just atomic. The results above just give particularly elementary examples. (ii)The suspension spectrum of RP1 presents an interesting challenge. It is a standard observation that H*(RP1 ; F2) is an atomic, but not cyclic, A- module, in the sense that any A-endomorphism which is the identity on H1(RP1 ; F2) is an isomorphism. This implies that 1 RP1 is atomic. However, since the top cell of RP3 splits off stably, the stable Hurewic* *z ho- momorphism ß3( 1 RP1 ) -! H3(RP1 ; F2) is non-trivial, hence 1 RP1 cannot be minimal atomic. It would be interesting to identify a core of 1 RP1 . For an odd prime p, similar remarks apply to the (p - 1) wedge summands of 1 BZ=p, one of which is (B p)(p). (iii)Fred Cohen observes that it is an open question whether or not K(Z=2, n) or the p - 1 summands of K(Z=p, n) are stably atomic for n > 2. 7.A construction of the spectrum BoP In this section, all spectra are understood to be localized at 2, and S = S0. Recall the spectrum BoP of Pengelley [14]. It has no mod 2 detectable homotopy [14, 5.5] and it is a retract of MSU , so we have a monomorphism j : BoP -! MSU* * . MINIMAL ATOMIC COMPLEXES 13 Example 7.1. The monomorphism j :BoP -! MSU is a core of MSU. We recall a further property of BoP , proven in Pengelley [14, 6.15, 6.16]. Proposition 7.2. There is a map p : BoP -! ko that induces an epimorphism on homotopy groups in all degrees and an isomorphism in odd degrees. Corollary 7.3. The odd degree homotopy groups of the fiber F p are zero. We now give a description of BoP as a nuclear spectrum, thus providing a simp* *le construction of it that is independent of [14]. Guided by Proposition 7.2, we c* *on- struct a nuclear spectrum X and a map q : X -! ko that induces a monomorphism on homotopy groups in odd degrees, and we prove that it induces an epimorphism on homotopy groups. That turns out to imply that X is equivalent to BoP . We begin with X0 = S, and we inductively attach even dimensional cells, letti* *ng X2n = X2n+1 for all n > 0. Suppose that we have factored the unit ' : S -! ko through a map qn :X2n-1 -! ko. We enlarge X2n-1 to X2n by attaching 2n-cells minimally, so that (1.7) is satisfied. We do this so as to kill the kernel of qn*: ß2n-1(X2n-1) -! ß2n-1(ko). Thus, in the resulting cofiber sequence J2n-1 -! X2n-1 -! X2n, Im(ß2n-1(J2n-1) -! ß2n-1(X2n-1)) = Ker(ß2n-1(X2n-1) -! ß2n-1(ko)). Clearly qn extends to a map qn+1: X2n = X2n+1 -! ko. In the limit we obtain a nuclear complex X and a map q :X -! ko that induces an isomorphism on ß0 and a monomorphism on ß* in odd degrees. Proposition 7.4. q :X -! ko induces an epimorphism on homotopy groups. Corollary 7.5. The odd degree homotopy groups of the fiber F q are zero. Let 2 ß3(S) and oe 2 ß7(S) be the Hopf maps. If x 2 X has even degree, then x and oex are odd degree elements of the kernel of q*, hence they are zer* *o. The proposition is therefore a direct consequence of the following result, whic* *h is presumably known. Since we do not know of a reference for it, we will give a pr* *oof in the next section. Proposition 7.6. Let X be a Hurewicz complex of dimension 0 with inclusion of the bottom cell i: S -! X and let q :X -! ko be a map such that the composite S -i!X -q!ko is the unit ' : S -! ko. If x = 0 and oex = 0 in ß*(X) for every even degree element x 2 ß*(X), then q*: ß*(X) -! ß*(ko) is an epimorphism. Theorem 7.7. There are equivalences f :X -! BoP and g :BoP -! X such that the following diagram is homotopy commutative. f g X _____//EEBoP___//X EE p| yyyy qEEE""Efflffl||q__yyyy ko 14 A.J. BAKER AND J.P. MAY Proof.We construct maps f and g such that the diagram is homotopy commuta- tive. The maps f and g, hence also the composites g O f and f O g, then induce isomorphisms on ß0. Since X and BoP are atomic, these composites are equiv- alences and therefore f and g are equivalences. We may take BoP and ko to be Hurewicz complexes and take p to be the identity map on the bottom cell. Taking f0: X0 = X1 = S -! BoP to be the identity map on the bottom cell and h0 to be the constant homotopy at the identity map, we assume inductively that we have a map fn :X2n-1 -! BoP and a homotopy hn :qn ' p O fn. Consider the following diagram, where we implicitly precompose maps already specified with the map of cells CJ2n-1 -! X2n+1 that constructs X2n+1 from X2n-1. J2n-1_______i0_____//J2n-1^ I+oo_____i1_______J2n-1 | hn rrrr | fntttt | | rrr | ttt | | yyrrr | p zztt | | ko oo_________|__________BoP | | w;; eeLL | ddJ | | qn+1www L | J J | | www hn+1L | f J | fflffl|ww L fflffl| n+1 J fflffl| CJ2n-1 ______i0_____//CJ2n-1^ I+oo____i1______CJ2n-1 Since J2n-1 is a wedge of (2n - 1)-spheres and ß2n-1(F p) = 0, [J2n-1, F p] = 0* *. A standard result, given in just this form in [13, Lemma 1], shows that there are* * maps fn+1 and hn+1 that make the diagram commute. Passing to colimits, we obtain f and a homotopy h: q ' p O fn. Since the homology groups of BoP are concentrated in even degrees [14], we can replace it by a minimal complex, with cells only i* *n even degrees. This allows us to reverse the roles of X and BoP to construct g. A similar argument proves the following result. Proposition 7.8. There is a map r :MSU -! X such that the following diagram is homotopy commutative. MSUFF_____r______//_X" FFF """" t FF""F~~"q"" ko It is not clear that BoP is the only core of MSU up to equivalence, but we conjecture that it is. The following consequence of Lemma 7.6 may shed some lig* *ht on this question. Proposition 7.9. If Y - ! MSU is a core, the composite Y - ! MSU -! ko induces an epimorphism on homotopy groups. Remark 7.10. It might be of interest to revisit the results of [11, 14] from our present perspective. However, it is not clear how to construct a map X -! MSU that induces the identity on ß0 and how the distinguished map of [11] fits in. * *It might be of more interest to revisit the results of [11, 14] from the perspecti* *ve of S-modules [7]. Pengelley constructs BoP by first constructing another spectrum, which he denotes by X, and then taking a fiber to kill BP summands in it. His X is obtained from MSU by using the Baas-Sullivan theory of manifolds with singularities to kill a regular sequence of elements in ß*(MSU). We can instead use the results of [7, Ch. V] to construct X as an MSU -module together with a * *map MINIMAL ATOMIC COMPLEXES 15 of MSU -modules MSU -! X. It seems plausible that the methods of [7, 18] can be used to construct BoP as a commutative MSU -ring spectrum. 8.The proof of Proposition 7.6 We continue to work with spectra localized at 2. Recall that (8.1) ß*(ko) = Z(2)[j, ff, fi]=(2j, j3, jff, ff2 - 4fi), where degj = 1, degff = 4, and degfi = 8. We will describe elements of ß*(X) that map to each of the additive generators of ß*(ko). Note that, since we do n* *ot know that X is a ring spectrum, we cannot exploit the algebra structure of ß*(k* *o). The essential point is to describe additive generators in terms of Toda bracket* *s in ß*(ko) that admit analogues in ß*(X). We are interested in Toda brackets of the form where a and b are ele* *ments of ß*(S) and c is an element of ß*(Y ) for a spectrum Y . We require ab = 0 and bc = 0, and then is a coset of elements in ß|a|+|b|+|c|+1(Y ) with res* *pect to the indeterminacy subgroup indeteraß|b|+|c|+1(Y ) + (ß|a|+|b|+1(S))c. Such Toda brackets are natural with respect to maps Y -! Z. Remark 8.2. We remark parenthetically that the theory of Toda brackets simplifi* *es greatly if one defines them in terms of the associative smash product in one of* * the modern categories of spectra, such as the category of S-modules of [7]. A syste* *matic exposition would be of value. In brief, the conclusion must be that all of the * *results that are catalogued in [12] for matric Massey products in the homology of DGA's carry over verbatim to S-modules. Now take X as in Proposition 7.6. Recall that 8 = 0 and 16oe = 0 in ß*(S) and that, by hypothesis, and oe annihilate all even degree elements of ß*(X). Let b0 denote i : S - ! X regarded as an element of ß0(X) and choose coset representatives in iterated Toda products as follows: a1 2 <8, , b0>, bk 2 <16, oe, bk-1>, and ak+1 2 <16, oe,,ak> where k > 1. The indeterminacies are benign for our purposes since they are indetera1= (ß4(S))b0 + 8ß4X = 8ß4(X), indeterbk= (ß8(S))bk-1 + 16ß8k(X) 16ß8k(X) mod Ker(q*) indeterak= (ß8(S))ak-1 + 16ß8k-4(X) 16ß8k-4(X) mod Ker(q*). Here the congruences hold since ß8(S) is 2-torsion and there are no torsion ele* *ments in the relevant degrees of ß*(ko). For k > 0, we also have the elements ~8k+1b0 2 ß8k+1(X) and ~8k+2b0 2 ß8k+2(X), where ~8k+1and ~8k+2are the usual elements in ß*(S). Now q*: ß*(X) -! ß*(ko) maps these elements to elements of the same form in ß*(ko), where b0 2 ß0(ko) is the unit of ko. In the familiar periodic pattern Z2, Z2, 0, Z, 0, 0, 0, Z, the * *additive positive degree generators of ß*(ko) are jfik = ~8k+1b0, j2fik = ~8k+2b0, fffik, and fik+1, where k > 0. The following known result gives that fffik = ak+1 and fik+1 = bk+1 in ß*(ko), and this completes the proof that q* is an epimorphism. 16 A.J. BAKER AND J.P. MAY Lemma 8.3. In ß*(ko), k-1ff k k-1ff ff 2 <8, , b0>, fik 2 16, oe, fi, and fffi 2 16, oe, fffi for k > 1, where the indeterminancy is 0 mod2 in each case. An unstable version of the lemma is stated without proof in [20, p.64], where* * it is attributed to Barratt. One quick way to see the result is to use the convergenc* *e of Massey products to Massey products in the May spectral sequence and of Massey products to Toda brackets in the Adams spectral sequence, but the details would take us too far afield. Appendix: Errata to [10] We take this opportunity to correct some minor errors in the proof of [10, 2.* *11]. In brief, the last two sentences of the cited proof should be replaced with the following two sentences. "If p = 2, then Q8(a1) a5 mod decomposables, and, if p > 2, then Q2p(ap-1) a(2p+1)(p-1)mod decomposables, by [15] or [4, 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(2p + 1)(p - 1) if p > 2." References [1]J. F. Adams. Stable Homotopy and Generalised Homology. University of Chicag* *o Press (1974). [2]J.F. Adams and N.J. Kuhn. Atomic spaces and spectra. Proc. Edinburgh Math. * *Society 32 (1989), 473-481. [3]D. Carlisle, P. Eccles, S. Hilditch, N. 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Math. 32 (1981), 467-489. [20]G. W. Whitehead. Recent advances in homotopy theory. Amer. Math. Soc. Conf.* * Board of the Mathematical Sciences Regional Conference Series in Mathematics 5 (1970). MINIMAL ATOMIC COMPLEXES 17 [21]C. Wilkerson. Genus and cancellation. Topology 14 (1975), 29-36. Mathematics Department, University of Glasgow, Glasgow G12 8QW, Scotland. E-mail address: a.baker@maths.gla.ac.uk Department of Mathematics, University of Chicago, Chicago, IL 60637, USA. E-mail address: may@math.uchicago.edu