LOCALISATION AND COMPLETION with an addendum on the use of 1 1 Brown-Peterson homology 0 in stable homotopy 2 n a J by 7 J. F. Adams 1 University of Cambridge ] T A . h t a m [ 3 v 0 2 0 5 . 2 1 0 1 : v i X r a Lecture notes by Z. Fiedorowicz on a course given at The University of Chicago in Spring 1973, revised and supplemented by Z. Fiedorowicz, 2011 FOREWORD In spring 1973 Frank Adams gave a course at the University of Chicago on localisation and completion. This was in the very early days of the sub- ject, which arose from disparate constructions of Quillen, Sullivan, Mimura, Nishida, Toda and others during the period 1969-1971. In those days one usually assumed the spaces onewas localising or completing were simply con- nected. There were various proposals for extending the domain of definition of these constructions to more general spaces (e.g. [8]), but there was no clear consensus on how to proceed. In his lectures Adams gave a lucid and compelling analysis of the properties one would want of such constructions. He set up an elegant axiomatic treat- ment of localisation and completion in the framework of category theory and proposed a vast generalisation of the existing constructions. Unfortunately Adams’ program for constructing these localisation functors with respect to arbitrary generalised homology theories ran into a serious difficulty during the course of these lectures. His proposal involved the use of the Brown Representability Theorem to construct his localisation functors, but he was unabletoshowthattherelevantrepresentablefunctorswereset-valuedrather than class-valued. Subsequent work by Bousfield established the existence of these generalised localisation functors, using more technical simplicial meth- ods. These functors are now an essential tool in homotopy theory. At that time I was a graduate student at Chicago and was charged with the responsibility of taking notes for Adams’ lectures. The resulting notes were briefly available in mimeographed form from the University of Chicago Mathematics Department. However these notes were never published in a more formal venue due to this apparent flaw in the proof.1 The notes also contain an addendum devoted to establishing that a certain element in the gamma family of the stable homotopy groups of spheres is nonzero, using Brown-Peterson (co)homology. Atthattimethiswasamatterofcontroversy, as Oka and Toda claimed to have proved the contrary result. I thought I had lost my only copy of these notes a long time ago, but I re- cently rediscovered them, and I want to make them publicly available again. Besides being of historical interest, these notes give a very readable intro- 1An announcement of this work was published in [3]. duction to localisation and completion, with minimal prerequisites. I was long aware that the gap in Adams’ proof was easily mendable, so I have sup- plemented these notes to include an epilogue explaining this and have made a few other minor editorial changes. Thus it can now be seen in retrospect that Adams amazingly succeeded in his project of “constructing localizations and completions without doing a shred of work” (cf. [12]). Iwould like to takethis opportunity to thank Peter Landweber forhis careful proofreading of an earlier draft of this manuscript and Carles Casacuberta for providing me with the recent preprint [4]. Zig Fiedorowicz Columbus, Ohio January, 2011 fi[email protected] CONTENTS 1. Introduction to Localisation.......................................... 1 2. Idempotent Functors ................................................ 10 3. Axiomatic Characterisation of Classes S............................. 23 4. A Further Axiom.................................................... 36 5. Behaviour of Idempotent Functors with Respect to Fiberings; Construction of Localisation Using Postnikov Decomposition ........ 47 6. Profinite Completion................................................ 60 7. Use of Brown-Peterson Homology in Stable Homotopy............... 69 8. Epilogue ........................................................... 104 References......................................................... 109 1 Introduction to Localisation “Il ya l`a la possibilit´ed’une´etude locale(ausens arithm´etique!) des groupes d’homotopie ...” J-P. Serre [22] In homotopy theory we have known for a long time that it is sufficient to attackproblemsoneprimeatatime. Thisinsight goesbacktothepioneering work of J-P. Serre [22]. Morerecently wehavegainedaparticularlyconvenient languageandsome particularly convenient machinery for exploiting this insight. This language and machinery was introduced following an analogy from commutative alge- bra. In commutative algebra we attack our problems one prime at a time by using the method of localisation. Thus we seek a comparable method in homotopy theory. The earliest reference I have which develops such a method is Sullivan [25]. This was certainly very influential. At this point perhaps we should alsomentionMimura-Nishida-Toda [15], Mimura-O’Neill-Toda[16], Mimura- Toda[17], andZabrodsky [29]. Another reference we might suggest is Quillen [21]. In the first part of these lectures, I want to present a simple and uniform method of constructing all functors in homotopy theory which have formal properties similar to those of Sullivan’s localisation functor. This opens the way to a study of such functors along axiomatic lines. I may also say some- thing about Sullivan’s completion functor. However it is clear fromSullivan’s work that the completion functor enters it for a visibly good and sufficient reason which is particular to that piece of work. The localisation functor, however, isofverygeneraluse, andeverygraduatestudent oftopologyshould learn about it. I must beginby sketching some background, and Istart with commutative algebra. Let Rbea commutative ring with1 andS ⊂ R beamultiplicatively closed subset (i.e. a subset closed under finite products, such that 1 ∈ S). For example, if R = Z we may take S = {1,2,4,8,...,2n,...} 1 or S = {1,3,5,...2m+1,...}. Let M be an R-module. We say M is S-local if the map M → M given by multiplication by s, i.e., m 7→ ms, is an isomorphism for any s ∈ S. To every R-module M we can find a map f : M → M′ so that (i) M′ is S-local (ii) f is universal with respect to (i). That is, if g : M → M′′ is another map such that M′′ is S-local, then there is a unique map h : M′ → M′′ which makes the following diagram commute M′ M oooooofooooooo77 (cid:31)(cid:31)(cid:31)(cid:31)h OOOOOOgOOOOOOO'' (cid:15)(cid:15)(cid:31)(cid:31)(cid:31) M′′ Such a map f is called a localisation map; we say f localises M at S. The usual construction of M′ is as a module of fractions. We first take pairs (m,s), m ∈ M, s ∈ S. We then define an equivalence relation on pairs: (m,s) ∼ (m′,s′) ⇐⇒ ∃s′′ ∈ S ∋ ms′s′′ = m′ss′′. We define S−1M to be the set of all equivalence classes. The fraction m is s the equivalence class containing (m,s). We make S−1M into an R-module in the obvious way. We define the map f : M → S−1M by f(m) = m . We 1 see that S−1M is S-local and the map f is universal. Since the ring R is an R-module, S−1R is defined; we can make it into a ring so that the canonical map R → S−1R is a map of rings: r r′ rr′ = . s s′ ss′ (cid:18) (cid:19) (cid:16) (cid:17) Similarly S−1M becomes a module over S−1R: m r mr = . s′ s s′s (cid:16) (cid:17)(cid:16) (cid:17) 2 Moreover we obtain a commutative diagram M ⊗ S−1R R TTTTTTTT∼=TTTTTTTT)) ∼= S−1M (cid:15)(cid:15) jjjjjjjj∼=jjjjjjjj55 S−1M ⊗ S−1R R We often use this fact just as a matter of notation when we have a conve- nient name for S−1R. For example, suppose R = Z and S = {1,2,4,8,...,2n,...} so that S−1R = Z 1 . If M is a Z-module, we would usually write M⊗Z 1 2 2 for S−1M. (cid:2) (cid:3) (cid:2) (cid:3) The most common example of a multiplicatively closed subset S is the complement of a prime ideal P. If S = CP we write M for S−1M. For P example, if R = Z, P = (2), CP = {±1,±3,±5,...,±(2m+1),...}, then Z is the set of fractions (2) a ⊂ Q. 2b+1 (cid:26) (cid:27) The construction of localisation in commutative algebra has many good properties. Most of them need not delay us now; however, it is essential to know that localisation preserves exactness. If i i L −→ M −→ N is exact, then so is S−1L −→i S−1M −→i S−1N. Example (a) Take the two localisation functors on Z-modules −⊗Z 1 and −⊗Z . 2 (2) (cid:2) (cid:3) 3 These two functors commute up to isomorphism and for any Z-module M we have a commutative diagram M //M ⊗Z 1 2 (cid:2) (cid:3) (cid:15)(cid:15) M ⊗Z 1 ⊗Z 2 (2) (cid:0) (cid:2) ∼=(cid:3)(cid:1) (cid:15)(cid:15) (cid:15)(cid:15) M ⊗Z(2) //(M ⊗Z2)⊗Z 21 ∼= M ⊗Q (cid:2) (cid:3) This diagram is both a pullback and pushout. Conversely, if we are given : M′ a Z 1 -module, M′′ a Z -module , M′′′ 2 (2) a Q-module, and localising maps f : M′ → M′′′, g : M′′ → M′′′, then in the (cid:2) (cid:3) pullback diagram M _ _ _h1_ _ _//M′ (cid:31) (cid:31) h2 f (cid:31) (cid:15)(cid:15) g (cid:15)(cid:15) M′′ //M′′′ h and h are also localising maps. 1 2 Now I want to recall the basic theorem about Sullivan’s localisation func- tor and give one example of its use to show what it is meant for. Before we do this, however, we must consider the category on which it is to be defined. If we stick to simply-connected CW-complexes, everyone will feel happy and secure. Moreover, it might be a matter of debate exactly how far we might wish to enlarge the domain of definition of our functor; and if we take the domain too large, there might be more than one functor extending the functor we all agree about for simply-connected spaces, and it might be a matter of debate which extension is best. So let us take C to be the category whose objects are 1-connected CW- complexes with basepoint and whose morphisms are homotopy classes of maps, with both maps and homotopies preserving basepoints. Let S ⊂ Zbea multiplicatively closed subset. Then Sullivanshowed there is a functor C → C which at the level of homotopy and homology performs localisation at S. 4 Theorem 1.1 (Sullivan). The following conditions on a map f : X → Y in C are equivalent: (i) f : π (X) → π (Y) localises at S for each n ≥ 1. # n n (ii) f : H (X) → H (Y) localises at S for each n ≥ 1. ∗ n n Moreoverthere is a functorE : C → C anda natural transformationη : 1 → E so that for each X, η : X → EX satisfies both (i) and (ii). X Before we go on, let me comment. Theorems like this usually tell us that there is something with a stated property when it is not obvious that there is; but it is obvious that if there is anything with that property, then the property characterises it. On the face of it this is not a theorem of that form; the properties do characterise EX and η , but it is not obvious they do. Of X course, these are comments on the way I’ve stated the theorem not on the way Sullivan states it. Still we may make a note to look for other forms of the statement. Following the analogy from algebra we use the notation X or X⊗Z for S S EX. We defer discussing the properties of E until we are forced to do so by examples. As an application, I recall that at one time there was a conjecture of the following sort. Conjecture. Any finite CW-complex which is an H-space is homotopy- equivalent to a product of spaces from the classical list: S7, RP7, compact Lie groups. This conjecture must always have looked optimistic, and it is now known to be false. The first counterexample was due to Hilton and Roitberg. Let us see how we get one by Zabrodsky’s method of mixing homotopy groups, expressed in the language of localisation. The classical list contains two entries S3×S7 and Sp(2). These are differ- ent attheprime2andalsoattheprime3(e.g. because π (S3×S7) = Z/12Z, 6 π (Sp(2)) = 0.) However, if we apply the localisation functor − ⊗ Q they 6 become the same: (S3 ×S7)⊗Q = K(Q,3)×K(Q,7) = Sp(2)⊗Q. 5 So consider for example Sp(2)⊗Z 1 2 (cid:2) (cid:3) (cid:15)(cid:15) Sp(2)⊗Z 1 ⊗Z 2 (2) (cid:2)∼= (cid:3) (cid:15)(cid:15) (S3 ×S7)⊗Z //(S3 ×S7)⊗Z ⊗Z 1 (2) (2) 2 (cid:2) (cid:3) There should be a space X such that X ⊗Z 1 ≃ SP(2)⊗Z 1 2 2 X ⊗Z ≃ Sp(2)⊗Z . (cid:2)(2(cid:3)) ((cid:2)2) (cid:3) It is easy to construct a candidate for X. Suppose we are given a diagram X′ p g (cid:15)(cid:15) X′′ // X′′′ then the weak pullback is the space of triples (x′,x′′,ω), where ω : I → X′′′ is a path from p(x′) to g(x′′). Strictly I shall take a weakly equivalent CW- complex; anyway I get a diagram X // X′ p (cid:15)(cid:15) g (cid:15)(cid:15) X′′ // X′′′ and an exact homotopy sequence ··· −→ π (X′′′) −→ π (X) −→ π (X′)⊕π (X′′) −→ π (X′′′) −→ ··· n+1 n n n n If we apply this to our case we get a diagram X //Sp(2)⊗Z 1 2 (cid:2) (cid:3) (cid:15)(cid:15) (cid:15)(cid:15) (S3 ×S7)⊗Z // K(Q,3)×K(Q,7) (2) 6