The homotopy groups of the algebraic K-theory of the sphere spectrum
aa r X i v : . [ m a t h . K T ] M a r THE HOMOTOPY GROUPS OF THE ALGEBRAIC K -THEORYOF THE SPHERE SPECTRUM ANDREW J. BLUMBERG AND MICHAEL A. MANDELL
Abstract.
We calculate the homotopy groups of K ( S ) in terms of the homo-topy groups of K ( Z ), the homotopy groups of C P ∞− , and the homotopy groupsof S . This completes the program begun by Waldhausen, who computed therational homotopy groups (building on work of Quillen and Borel), and con-tinued by Rognes, who calculated the groups at regular primes in terms of thehomotopy groups of C P ∞− , and the homotopy groups of S . Introduction
The algebraic K -theory of the sphere spectrum K ( S ) is Waldhausen’s A ( ∗ ), thealgebraic K -theory of the one-point space. The underlying infinite loop space of K ( S ) splits as a copy of the underlying infinite loop space of S and the smoothWhitehead space of a point W h
Diff ( ∗ ). For a high-dimensional highly-connectedcompact manifold M , the second loop space of W h
Diff approximates the stableconcordance space of M , and the loop space of W h
Diff parametrizes stable h -cobordisms in low dimensions. As a consequence, computation of the algebraic K -theory of the sphere spectrum is a fundamental problem in algebraic and differ-ential topology.Early efforts in this direction were carried out by Waldhausen in the 1980’s usingthe “linearization” map K ( S ) → K ( Z ) from the K -theory of the sphere spectrum tothe K -theory of the integers. This is induced by the map of (highly structured) ringspectra S → Z . Waldhausen showed that because the map S → Z is an isomorphismon homotopy groups in degree zero (and below) and a rational equivalence onhigher homotopy groups, the map K ( S ) → K ( Z ) is a rational equivalence. Borel’scomputation of the rational homotopy groups of K ( Z ) then applies to calculatethe rational homotopy groups of K ( S ): π q K ( Z ) ⊗ Q is dimension 1 when q = 0 or q = 4 k + 1 for k > K -theory with the introduction and study of topo-logical cyclic homology T C , an analogue of negative cyclic homology that can becomputed using the methods of equivariant stable homotopy theory.
T C is thetarget of the cyclotomic trace , a natural transformation K → T C . For a map of
Date : March 14, 2016.2010
Mathematics Subject Classification.
Primary 19D10.
Key words and phrases.
Algebraic K -theory of spaces, stable pseudo-isotopy theory, White-head space, cyclotomic trace.The first author was supported in part by NSF grant DMS-1151577.The second author was supported in part by NSF grants DMS-1105255, DMS-1505579. highly structured ring spectra such as S → Z , naturality gives a diagram K ( S ) / / (cid:15) (cid:15) T C ( S ) (cid:15) (cid:15) K ( Z ) / / T C ( Z ) , the linearization/cyclotomic trace square . A foundational theorem of Dundas [7](building on work of McCarthy [18] and Goodwillie [10]) states that the squareabove becomes homotopy cartesian after p -completion, which means that the mapsof homotopy fibers become weak equivalences after p -completion.In the 2000’s, Rognes [25] used the linearization/cyclotomic trace square to com-pute the homotopy groups of K ( S ) at odd regular primes in terms of the homotopygroups of S and the homotopy groups of C P ∞− (assuming the now affirmed Quillen-Lichtenbaum conjecture). The answer is easiest to express in terms of the torsionsubgroups: Because π n K ( S ) is finitely generated, it is the direct sum of a free partand a torsion part, the free part being Z when n = 0 or n ≡ n >
1, and0 otherwise. The main theorem of Rognes [25] is that for p an odd regular primethe p -torsion of π ∗ K ( S ) istor p ( π ∗ K ( S )) ∼ = tor p ( π ∗ S ⊕ π ∗− c ⊕ π ∗− C P ∞− )(which can be made canonical, as discussed below). Here c denotes the additive p -complete cokernel of J spectrum (the connected cover of the homotopy fiber of themap S ∧ p → L K (1) S ); its homotopy groups are all torsion and are direct summandsof π ∗ S . The spectrum C P ∞− is a wedge summand of ( C P ∞− ) ∧ p [16, (1.3)], [25,p.166], ( C P ∞− ) ∧ p ≃ C P ∞− ∨ S ∧ p ; using unpublished work of Knapp, Rognes [25, 4.7]calculates the order of these torsion groups in degrees ≤ p +1)( p − −
4. Rognes’argument identifies the homotopy type of the homotopy fiber of the cyclotomictrace, assuming the now affirmed Quillen-Lichtenbaum conjecture. The regularityhypothesis comes into the argument in two ways: First, the homotopy type of K ( Z )is completely understood at regular primes by the work of Dwyer and Mitchell [8]and the Quillen-Lichtenbaum conjecture. Second, the B¨okstedt-Hsiang-Madsengeometric Soul´e embedding splits part of T C ( S ) off of K ( S ) if (and only if) p is aregular prime.This paper computes π ∗ K ( S ) ∧ p in the case of irregular primes, thereby completingthe computation of the homotopy groups of the algebraic K -theory of the spherespectrum. We take a very different approach; as a first step, we prove the followingsplitting theorem in Section 4. Theorem 1.1.
Let p be an odd prime. The long exact sequence on homotopygroups induced by the p -completed linearization/cyclotomic trace square breaks upinto non-canonically split short exact sequences −→ π ∗ K ( S ) ∧ p −→ π ∗ T C ( S ) ∧ p ⊕ π ∗ K ( Z ) ∧ p −→ π ∗ T C ( Z ) ∧ p −→ . Choosing appropriate splittings in the previous theorem, we can identify the p -torsion groups. The identification is again in terms of π ∗ S and π ∗ Σ C P ∞− butnow involves also π ∗ K ( Z ), which is not fully understood at irregular primes. Inthe statement, e K ( Z ) denotes the wedge summand of K ( Z ) ∧ p complementary to j [8,2.1,9.7], K ( Z ) ∧ p ≃ j ∨ e K ( Z ), where j is the p -complete additive image of J spectrum,the connective cover of L K (1) S . (As discussed below, this splitting is canonical.) HE HOMOTOPY GROUPS OF K ( S ) 3 Theorem 1.2.
Let p be an odd prime. The p -torsion in π ∗ K ( S ) admits canonicalisomorphisms tor p ( π ∗ K ( S )) ∼ = tor p ( π ∗ c ⊕ π ∗− c ⊕ π ∗− C P ∞− ⊕ π ∗ K ( Z ))(a) ∼ = tor p ( π ∗ S ⊕ π ∗− c ⊕ π ∗− C P ∞− ⊕ π ∗ e K ( Z )) . (b)In Formula (a), the map tor p ( π ∗ K ( S )) → tor p ( π ∗ ( K ( Z ))) is induced by thelinearization map and the maptor p ( π ∗ K ( S )) −→ tor p ( π ∗ c ⊕ π ∗− c ⊕ π ∗− C P ∞− )is induced by the composite of the cyclotomic trace map K ( S ) → T C ( S ) and acanonical splitting (2.3) of the homotopy groups π ∗ T C ( S ) ∧ p as π ∗ T C ( S ) ∧ p ∼ = π ∗ ( j ) ⊕ π ∗ ( c ) ⊕ π ∗ (Σ j ) ⊕ π ∗ (Σ c ) ⊕ π ∗ ( C P ∞− )explained in the first part of Section 2, followed by the projection onto the non- j summands.In formula (b), the map tor p ( π ∗ K ( S )) → tor p ( π ∗ e K ( Z )) is induced by the lin-earization map and the canonical map π ∗ K ( Z ) → π ∗ e K ( Z ) which is the quotient ofthe Harris-Segal summand. The maptor p ( π ∗ K ( S )) −→ tor p ( π ∗ S ⊕ π ∗− c ⊕ π ∗− C P ∞− )is induced by the composite of the cyclotomic trace map K ( S ) → T C ( S ) and thecanonical splitting of homotopy groups π ∗ T C ( S ) ∧ p ∼ = π ∗ ( S ∧ p ) ⊕ π ∗ (Σ j ) ⊕ π ∗ (Σ c ) ⊕ π ∗ ( C P ∞− )followed by projection onto the non- j summands. (The splitting of π ∗ T C ( S ) ∧ p hereis related to the splitting above by the canonical splitting on homotopy groups π ∗ S ∧ p ∼ = π ∗ ( j ) ⊕ π ∗ ( c ).)Formula (b) generalizes the computation of Rognes [25] at odd regular primesbecause e K ( Z ) is torsion free if (and only if) p is regular (see for example [30, § VI.10]). Part of the argument for the theorems above involves making certainsplittings in prior K -theory and T C computations canonical and canonically iden-tifying certain maps (or at least their effect on homotopy groups). Although weconstruct the splittings and prove their essential uniqueness calculationally, we offerin Section 5 a theoretical explanation in terms of a conjectural extension of Adamsoperations on algebraic K -theory to an action of the p -adic units and a conjectureon the consistency of Adams operations on K -theory and Adams operations on T C .This perspective leads to a new splitting of K ( S ) ∧ p and the linearization/cyclotomictrace square into p − Theorem 1.3.
Let p be an odd prime. Let α : π ∗ T C ( S ) → π ∗ j be the inducedmap on homotopy groups given by the composite of the canonical maps T C ( S ) → T HH ( S ) ≃ S and S → j ; let β : π ∗ K ( Z ) → π ∗ j be the canonical splitting of theHarris-Segal summand; and let γ : π ∗ T C ( S ) → π ∗ Σ S → π ∗ Σ j be the map inducedby the canonical splitting of (2.2) below. Then the p -torsion subgroup of π ∗ K ( S ) maps isomorphically to the subgroup of the p -torsion subgroup of π ∗ T C ( S ) ⊕ π ∗ K ( Z ) where the appropriate projections composed with α and β agree and the appropriateprojection composed with γ is zero. ANDREW J. BLUMBERG AND MICHAEL A. MANDELL
Theorems 1.2 and 1.3 provide a good understanding of what the lineariza-tion/cyclotomic trace square does on the odd torsion part of the homotopy groups.In contrast, the maps on mod torsion homotopy groups are not fully understood.We have that π n K ( S ) and π n K ( Z ) mod torsion are rank one for n = 0 and n ≡ n >
1; the map K ( S ) → K ( Z ) is a rational equivalence and an isomor-phism in degree zero, but is not an isomorphism on mod torsion homotopy groupsin degrees congruent to 1 mod 2( p −
1) by the work of Klein-Rognes (see the proof of[13, 6.3.(i)]). The mod torsion homotopy groups of
T C ( S ) ∧ p are rank one in degreezero and odd degrees ≥ −
1; the map K ( S ) ∧ p → T C ( S ) ∧ p on mod torsion homotopygroups is an isomorphism in degree zero, by necessity zero in degrees not congruentto 1 mod 4, and for odd regular primes an isomorphism in degrees congruent to 1mod 4. For irregular primes, the map is not fully understood.In principle, we can use Theorem 1.2 to calculate π ∗ K ( S ) in low degrees. Inpractice, we are limited by a lack of understanding of π ∗ C P ∞− and π ∗ K ( Z ); weknow π ∗ S and π ∗ c in a comparatively larger range. The calculations of π ∗ C P ∞− for ∗ < | β | = (2 p + 1)(2 p − − π ∗ K ( Z ) in terms of Bernoulli numbers; at other primes we know π ∗ K ( Z ) in odddegrees, but do not know K n Z at all (except n = 0 ,
1) and only know the orderof π n +2 K ( Z ) and again only in terms of Bernoulli numbers. As the formula [25,4.7] for π ∗ C P ∞− is somewhat messy, we do not summarize the answer here, but forconvenience, we have included it in Section 6.The authors observed in [2] that as a consequence of the work of Rognes [25],the cyclotomic trace trc p : K ( S ) ∧ p −→ T C ( S ) ∧ p is injective on homotopy groups at odd regular primes. In Theorem 1.2 above,the contribution in tor p ( π ∗ K ( S )) of p -torsion from tor p ( π ∗ e K ( Z )) maps to zero in π ∗ T C ( S ) under the cyclotomic trace. This then gives the following complete answerto the question the authors posed in [2]: Corollary 1.4.
For an odd prime p , the cyclotomic trace trc p : K ( S ) ∧ p → T C ( S ) ∧ p is injective on homotopy groups if and only if p is regular. Theorem 1.1 contains the following more general injectivity result.
Corollary 1.5.
For an odd prime p , the map of ring spectra K ( S ) ∧ p −→ T C ( S ) ∧ p × K ( Z ) ∧ p is injective on homotopy groups. Conventions.
Throughout this paper p denotes an odd prime. For a ring R , wewrite R × for its group of units. Acknowledgments.
The authors especially thank Lars Hesselholt for many help-ful conversations and for comments on a previous draft that led to substantialsharpening of the results. The authors also thank Mark Behrens, Bill Dwyer,Tom Goodwillie, John Greenlees, Mike Hopkins, and Tyler Lawson for helpful con-versations, John Rognes for useful comments, and the IMA and MSRI for theirhospitality while some of this work was done.
HE HOMOTOPY GROUPS OF K ( S ) 5 The spectra in the linearization/cyclotomic trace square
We begin by reviewing the descriptions of the spectra
T C ( S ) ∧ p , K ( Z ) ∧ p , and T C ( Z ) ∧ p in the p -completed linearization/cyclotomic trace square K ( S ) ∧ p / / (cid:15) (cid:15) T C ( S ) ∧ p (cid:15) (cid:15) K ( Z ) ∧ p / / T C ( Z ) ∧ p . These three spectra have been identified in more familiar terms up to weak equiv-alence. We discuss how canonical these weak equivalences are and what choicesparametrize them. Often this will involve studying splittings of the form X ≃ Y ∨ Z (or with additional summands). We will say that the splitting is canonical whenwe have a canonical isomorphism in the stable category X ≃ X ′ ∨ X ′′ with X ′ and X ′′ (possibly non-canonically) isomorphic in the stable category to Y and Z ; wewill say that the identification of the summand Y is canonical when further theisomorphism in the stable category X ′ ≃ Y is canonical.To illustrate the above terminology, and justify its utility, consider the examplewhen X is non-canonically weakly equivalent to Y ∨ Z and [ Y, Z ] = 0 = [
Z, Y ](where [ − , − ] denotes maps in the stable category). In the terminology above, thisgives an example of a canonical splitting X ≃ Y ∨ Z without canonical identificationof the summands. As another example, if we have a canonical map Y → X and acanonical map X → Y giving a retraction, then we have a canonical splitting withcanonical identification of summands X ≃ Y ∨ F where F is the homotopy fiber ofthe retraction map X → Y . In this second example if we also have a non-canonicalweak equivalence Z → F , we then have a canonical splitting X ≃ Y ∨ Z with acanonical identification of the summand Y (but not the summand Z ). The splitting of
T C ( S ) ∧ p . Historically,
T C ( S ) ∧ p was the first of the terms in thelinearization/cyclotomic trace square to be understood. Work of B¨okstedt-Hsiang-Madsen [3, 5.15,5.17] identifies the homotopy type of T C ( S ) ∧ p as(2.1) T C ( S ) ∧ p ≃ S ∧ p ∨ Σ( C P ∞− ) ∧ p . The inclusion of the S summand is the unit of the ring spectrum structure andis split by the canonical map T C ( S ) ∧ p → T HH ( S ) ∧ p and canonical identification S ∧ p ≃ T HH ( S ) ∧ p (also induced by the ring spectrum structure). We therefore get acanonical isomorphism in the stable category between T C ( S ) ∧ p and S ∧ p ∨ F where F is the homotopy fiber of the map T C ( S ) ∧ p → T HH ( S ) ∧ p ; in the terminology at thebeginning of the section, this is a canonical splitting with canonical identificationof the S ∧ p summand. The identification of the other summand as Σ( C P ∞− ) ∧ p ispotentially somewhat non-canonical with indeterminacy parametrized by a lim term; however, all maps in this class induce the same splitting on homotopy groups π ∗ T C ( S ) ∧ p ∼ = π ∗ S ∧ p ⊕ π ∗ (Σ( C P ∞− ) ∧ p ) , which is then canonical. In more detail, the homotopy fiber of T C ( S ) ∧ p → S ∧ p maybe identified up to weak equivalence as the homotopy fiberholim Σ ∞ + BC p n −→ S ∧ p ANDREW J. BLUMBERG AND MICHAEL A. MANDELL (where C p n denotes the cyclic group of order p n ) with the homotopy limit takenover the transfer maps, and the maps Σ ∞ + BC p → S ∧ p also the transfer map. It ispossible that the system of comparison maps with BC p n can be made rigid enoughto specify a canonical weak equivalence, but without more work we have the lim indeterminacy indicated above. (Recent results of Reich and Varisco [21] on a point-set model for the Adams isomorphism may help here.) We then have a canonicalweak equivalence(ΣΣ ∞ + C P ∞ ) ∧ p ∼ = (ΣΣ ∞ + B T ) ∧ p −→ holim Σ ∞ + BC p n (where T denotes the circle group) and a canonical isomorphism in the stable cat-egory from C P ∞− to the homotopy fiber of the T -transfer ΣΣ ∞ + C P ∞ → S [16, § §
3] (see alsothe remarks preceding [16, (1.3)]),(2.2) ( C P ∞− ) ∧ p ≃ S ∧ p ∨ C P ∞− , induced by the splitting Σ ∞ + C P ∞ ≃ Σ ∞ C P ∞ ∨ S . The splitting exists after inverting2, but for notational convenience, we use it only after p -completion. As the splittingand identification of summands for Σ ∞ + C P ∞ is canonical, and the null homotopyof the composite map Σ S ∧ p −→ (Σ C P ∞− ) ∧ p −→ S ∧ p is canonical, the splitting in (2.2) and identification of the summand S ∧ p is canonical.Finally, we have a canonical isomorphism of homotopy groups π ∗ S ∧ p ∼ = π ∗ ( j ) ⊕ π ∗ ( c ) from classical work in homotopy theory on Whitehead’s J homomorphismand Bousfield’s work on localization of spectra [5, § j denotes theconnective cover of the K (1)-localization of the sphere spectrum and c denotes thehomotopy fiber of the map S ∧ p → j . The map S ∧ p → j induces an isomorphism fromthe p -Sylow subgroup of the image of J subgroup of π q S ∧ p to π q j .Putting this all together, we have a canonical isomorphism(2.3) π ∗ T C ( S ) ∧ p ∼ = π ∗ ( j ) ⊕ π ∗ ( c ) ⊕ π ∗ (Σ j ) ⊕ π ∗ (Σ c ) ⊕ π ∗ (Σ C P ∞− ) . The splitting of
T C ( Z ) ∧ p . Next up historically is
T C ( Z ), which was first identi-fied by B¨okstedt-Madsen [4] and Rognes [23]. They expressed the answer on theinfinite-loop space level and (equivalently) described the connective cover spectrum T C ( Z ) ∧ p [0 , ∞ ) as having the homotopy type of j ∨ Σ j ∨ Σ ku ∧ p ≃ j ∨ Σ j ∨ Σ ℓ ∨ Σ ℓ ∨ · · · ∨ Σ p − ℓ (non-canonical isomorphism in the stable category). Here ku denotes connectivecomplex topological K -theory (the connective cover of periodic complex topological K -theory KU ), and ℓ denotes the Adams summand of ku ∧ p (the connective coverof the Adams summand L of KU ∧ p ). A standard calculation (e.g., see [15, 2.5.7])shows that before taking the connective cover π − ( T C ( Z ) ∧ p ) is free of rank oneover Z ∧ p , and the argument of [25, 3.3] shows that the summand Σ p − ℓ abovebecomes Σ − ℓ in T C ( Z ) ∧ p .The non-canonical splitting above rigidifies into a canonical splitting and canon-ical identification(2.4) T C ( Z ) ∧ p ≃ j ∨ Σ j ′ ∨ Σ − ℓ T C (0) ∨ Σ − ℓ T C ( p ) ∨ Σ − ℓ T C (2) ∨· · ·∨ Σ − ℓ T C ( p − . Here the numbering replaces 1 with p but otherwise numbers sequentially 0 , . . . , p −
2. Each Σ − ℓ T C ( i ) is a spectrum that is non-canonically weakly equivalent to HE HOMOTOPY GROUPS OF K ( S ) 7 Σ i − ℓ ∼ = Σ i (Σ − ℓ ) but that admits a canonical description by work of Hesselholt-Madsen [11, Theorem D] and Dwyer-Mitchell [8, §
13] in terms of units of cyclotomicextensions of Q ∧ p ; we omit a detailed description of the identification as the specificsdo not come into the argument. The spectrum j ′ is the connective cover of the K (1)-localization of the ( Z ∧ p ) × -Moore spectrum M ( Z ∧ p ) × , or equivalently, the p -completionof j ∧ M ( Z ∧ p ) × . Since the p -completion of ( Z ∧ p ) × is non-canonically isomorphic to Z ∧ p ,we have that j ′ is non-canonically weakly equivalent to j but with π j ′ canonicallyisomorphic to (( Z ∧ p ) × ) ∧ p (and isomorphisms in the stable category from j to j ′ arein canonical bijection with isomorphisms Z ∧ p → (( Z ∧ p ) × ) ∧ p ). Much of the canonicalsplitting in (2.4) follows by a calculation of maps in the stable category. Temporarilywriting x (0) = j ∨ Σ − ℓ , x (1) = Σ j ∨ Σ p − ℓ , and x ( i ) = Σ i − ℓ for i = 2 , . . . , p − T C ( Z ) ∧ p ≃ x (0) ∨ · · · ∨ x ( p − i = i ′ , we have [ x ( i ) , x ( i ′ )] = 0; we provide a detailed computationas Proposition 2.13 at the end of the section. The canonical map S → j in-duces an isomorphism [ j, T C ( Z ) ∧ p ] ∼ = π ( T C ( Z ) ∧ p ) and so we have a canonical map η : j → T C ( Z ) ∧ p coming from the unit of the ring spectrum structure. Likewise, thecanonical map M ( Z ∧ p ) × → j ′ induces an isomorphism[Σ j ′ , T C ( Z ) ∧ p ] ∼ = [Σ M ( Z ∧ p ) × , T C ( Z ) ∧ p ] ∼ = Hom((( Z ∧ p ) × ) ∧ p , π T C ( Z ) ∧ p ) . The canonical isomorphism π T C ( Z ) ∧ p ∼ = (( Z ∧ p ) × ) ∧ p [11, Th. D], then gives a canon-ical map u : Σ j ′ → T C ( Z ) ∧ p . The restriction along η and u induce bijections[ T C ( Z ) ∧ p , j ] ∼ = [ j, j ] and [ T C ( Z ) ∧ p , Σ j ′ ] ∼ = [Σ j ′ , Σ j ′ ](q.v. Proposition 2.14 below), giving retractions to η and u that are unique in thestable category. This gives a canonical splitting and identification of the j and Σ j ′ summands in (2.4). The splitting of K ( Z ) ∧ p . The homotopy type of K ( Z ) ∧ p is still not fully understoodat irregular primes even in light of the confirmation of the Quillen-Lichtenbaum con-jecture. The Quillen-Lichtenbaum conjecture implies that K ( Z ) ∧ p can be understoodin terms of its K (1)-localization L K (1) K ( Z ), and work of Dwyer-Friedlander [9] orDwyer-Mitchell [8, 12.2] identifies the homotopy type of K ( Z ) ∧ p at regular primesas j ∨ Σ ko ∧ p (non-canonically). At any prime, Quillen’s Brauer induction andreduction mod r (for r prime a generator of ( Z /p ) × as above) induce a splitting(2.5) K ( Z ) ∧ p ≃ j ∨ e K ( Z )for some p -complete spectrum we denote as e K ( Z ) (see for example, [8, 2.1,5.4,9.7]).We argue in Proposition 2.16 at the end of this section (once we have reviewed moreabout e K ( Z )) that [ j, e K ( Z )] = 0 and [ e K ( Z ) , j ] = 0, and it follows that the splittingin (2.5) is canonical; the identification of the summand j is also canonical, as themap j → K ( Z ) described is then the unique one taking the canonical generator of π j to the unit element of π K ( Z ) in the ring spectrum structure.The splitting K ( Z ) ∧ p ≃ j ∨ e K ( Z ) corresponds to a splitting L K (1) K ( Z ) ≃ J ∨ L K (1) e K ( Z ) ANDREW J. BLUMBERG AND MICHAEL A. MANDELL where J = L K (1) S is the K (1)-localization of S . We have that e K ( Z ) is 4-connectedand the Quillen-Lichtenbaum conjecture (as reformulated by Waldhausen [28, § K ( Z ) → L K (1) K ( Z ) is an isomorphism on homotopy groups in degrees2 and above. Thus, e K ( Z ) is the 4-connected cover of L K (1) e K ( Z ). This makes itstraightforward to convert statements about the homotopy type of L K (1) K ( Z ) intostatements about the homotopy type of K ( Z ) ∧ p .By [8, 1.7], L K (1) e K ( Z ) is a KU ∧ p -theory Moore spectrum of type ( M, − KU ∧ p -cohomology is concentrated in odd degrees and is projective di-mension 1 over the ring ( KU ∧ p ) ( KU ∧ p ). In particular, it is a E (1) ∗ -Moore spectrumin the sense of Bousfield [6] and so canonically splits as L K (1) e K ( Z ) ≃ Y ∨ · · · ∨ Y p − where π ∗ Y i = 0 unless ∗ is congruent to 2 i or 2 i + 1 mod 2( p − L ∗ ( Y i ) = 0 for ∗ not congruent to 2 i + 1 mod 2( p −
1) (where as above L denotesthe Adams summand of KU ∧ p , or equivalently, in terms of [6], the p -completion of E (1)). Here [ Y i , Y i ′ ] = 0 unless i = i ′ . Furthermore, Y i is completely determinedby L i Y i : Y i is the fiber of a map from a finite wedge of copies of Σ i +1 L to a finitewedge of copies of Σ i +1 L , giving a L L -resolution of the projective dimension 1module L i +1 Y i after applying L i +1 .Letting y i be the 4-connected cover of Y i , we show below in Proposition 2.17 that[ y i , y i ′ ] = 0 for i = i ′ and so obtain a canonical splitting and canonical identificationof summands(2.6) K ( Z ) ∧ p ≃ j ∨ y ∨ · · · ∨ y p − . The L L -modules L i +1 Y i have a close relationship to class groups of cyclotomicfields. Writing A m for the p -Sylow group of the class group of the integers ofthe cyclotomic field Q ( ζ p m +1 ), where ζ p m +1 is a primitive p m +1 th root of unity,the relevant object is the inverse limit of A m over the norm maps. This inverselimit turns out to be a ( KU ∧ p ) ( KU ∧ p )-module by a mathematical pun explainedin [8, § §
6] and is denoted there by A ∞ (which in the case we are discussingis also isomorphic to the modules they denote as L ∞ and A ′∞ , q.v. ibid. , 12.2).Any ( KU ∧ p ) ( KU ∧ p )-module X has a canonical “eigensplitting” into a direct sum ofpieces corresponding to the powers of the Teichm¨uller character: The ω i -characterpiece ǫ i X is the submodule where the ( p -adically interpolated) Adams operation ψ ω ( α ) acts by multiplication by ω i ( α ) ∈ Z ∧ p for all α ∈ ( Z /p ) × . We then regard ǫ i X as a L L -module via the projection( KU ∧ p ) ( KU ∧ p ) −→ (Σ i L ) (Σ i L ) ∼ = L L induced by the splitting Σ i L → KU ∧ p → Σ i L . The precise relationship between L i +1 ( Y i ) and A ∞ is given concisely by the 4-term exact sequence of [8, 12.1], whichafter eigensplitting becomes0 −→ Ext L L ( ǫ − i A ∞ , L L ) −→ L i +1 Y i −→ Hom L L ( ǫ − i E ′∞ (red) , L L ) −→ Ext ( ǫ − i A ∞ , L L ) −→ . Here Hom L L ( ǫ − i E ′∞ (red) , L L ) is zero when i is odd and a free L L -module ofrank 1 when i is even.For fixed p , several of the ω j -character pieces of A ∞ are always zero. In fact ǫ j A ∞ = 0 if and only if ǫ j A = 0 (see for example [29, 13.22] and apply Nakayama’s HE HOMOTOPY GROUPS OF K ( S ) 9 lemma). In particular, for the trivial character, ǫ A = 0 (because it is canon-ically isomorphic to the p -Sylow subgroup of the class group of Z ). From theexact sequence above, L Y ∼ = L L (non-canonically) and so Y is non-canonicallyweakly equivalent to Σ L . It follows that y is non-canonically weakly equivalentto Σ p − ℓ . In terms of K ( Z ) ∧ p , we obtain a further canonical splitting (withoutcanonical identification) e K ( Z ) ≃ Σ p − ℓ ∨ e K ( Z ) for some p -complete spectrum e K ( Z ). We use the identification of y as a key step in the proof of Theorem 1.1in Section 4.Another useful vanishing result is ǫ A = 0 [29, 6.16]. As a consequence, since ω − = ω p − , we see that(2.7) Y p − ≃ ∗ . This simplifies some formulas and arguments.Although these are the only results we use, other vanishing results for ǫ j A giveother vanishing results for the summands Y i . Herbrant’s Theorem [29, 6.17] andRibet’s Converse [22], [29, 15.8] state that for 3 ≤ j ≤ p − ǫ j A = 0 if andonly if p | B p − j where B n denotes the Bernoulli number, numbered by the convention te t − = P B n t n n ! . Using ω − j = ω p − − j and i = p − − j , we see that for p − ≥ i ≥ Y i ≃ ∗ when p does not divide B i +1 . In particular, Y , Y , Y , Y , and Y are trivial, Y is trivial for p = 691, and for every odd i , Y i is only nontrivial forfinitely many primes.A prime p is regular precisely when p does not divide the class number of Q ( ζ p ),or in other words, when A = 0 (and therefore A ∞ = 0). Then for an odd regularprime, we have that Y k is non-canonically weakly equivalent to Σ k +1 L and Y k +1 is trivial for all k . It follows that L K (1) e K ( Z ) is non-canonically weakly equivalentto Σ KO ∧ p and e K ( Z ) is non-canonically weakly equivalent to Σ ko ∧ p , since e K ( Z )is the 4-connected cover of L K (1) e K ( Z ). This leads precisely to the description of K ( Z ) ∧ p as non-canonically weakly equivalent to j ∨ Σ ko ∧ p , as indicated above.A prime p satisfies the Kummer-Vandiver condition precisely when p does notdivide the class number of the ring of integers of Q ( ζ p + ζ − p ) (the fixed field of Q ( ζ p ) under complex conjugation). The p -Sylow subgroup is precisely the subgroupof A fixed by complex conjugation, which is the internal direct sum of ǫ i A for0 ≤ i < p − ǫ i A = 0 for i even, and so again Y k isnon-canonically weakly equivalent to Σ k +1 L . Now the odd summands Y k +1 maybe non-zero, but the L L -modules ǫ i A ∞ are cyclic for i odd (see for example, [29,10.16]) and Y k +1 is (non-canonically) weakly equivalent to the homotopy fiber of amap Σ k +3 L → Σ k +3 L determined by the p -adic L -function L p ( s ; ω k +2 ) [8, 12.2].As above, Y p − ≃ ∗ and in the other cases, for n > n ≡ k + 1 (mod p − π n Y k +1 ∼ = Z ∧ p /L p ( − n, ω k +2 ) = Z ∧ p / ( B n +1 / ( n + 1)) π n +1 Y k +1 = 0(non-canonical isomorphisms). The groups are of course zero for n k + 1(mod p − n < n ≡ k + 1 (mod p − L -function formula for π n Y k +1 still holds, and π n +1 Y k +1 = 0 still holds provided the value of the L -function is non-zero. If the value of the L -function is zero, then π n +1 Y k +1 ∼ = Z ∧ p ,though it is conjectured [26, 14] that this case never occurs.) For p not satisfying the Kummer-Vandiver condition, the even summands satisfy π n Y k = finite π n +1 Y k ∼ = Z ∧ p (non-canonical isomorphism) for n ≡ k (mod p −
1) (and zero otherwise) with thefinite group unknown. As always Y p − ≃ ∗ , and the Mazur-Wiles theorem [17], [29,15.14] implies that in the other odd cases 2 k + 1, for n > n ≡ k + 1 (mod p − π n Y k +1 ) = Z ∧ p / ( B n +1 / ( n + 1))) π n +1 Y k +1 = 0(and zero for n k + 1 (mod p − n < n ≡ k + 1 (mod p − π n Y k +1 ) = Z ∧ p /L p ( − n, ω k +2 )) and π n +1 Y k +1 = 0, provided L p ( − n, ω k +2 )is non-zero. If L p ( − n, ω k +2 ) = 0, then π n Y k +1 ∼ = Z ∧ p ⊕ finite and π n +1 Y k +1 ∼ = Z ∧ p , non-canonically.) For more on the homotopy groups of K ( Z ), see for exam-ple [30, § VI.10].
Supporting calculations.
In several places above, we claimed (implicitly or ex-plicitly) that certain hom sets in the stable category were zero. Here we reviewsome calculations and justify these claims. All of these computations follow fromwell-known facts about the spectrum L together with standard facts about mapsin the stable category. In particular, in several places, we make use of the fact thatfor a K (1)-local spectrum Z , the localization map X → L K (1) X induces an isomor-phism [ L K (1) X, Z ] → [ X, Z ]; also, several times we make use of the fact that if X is ( n − n − Z [ n, ∞ ) → Z inducesan isomorphism [ X, Z [ n, ∞ )] → [ X, Z ]. We begin with results on [ ℓ, Σ q ℓ ]. Proposition 2.8.
The map [ ℓ, Σ q ℓ ] → [ ℓ, Σ q L ] ∼ = [ L, Σ q L ] is an injection for q ≤ p − ℓ, Σ q ℓ ] = 0 if q p −
2) and q < p − Proof.
We have a cofiber sequenceΣ q − − (2 p − H Z ∧ p −→ Σ q ℓ −→ Σ q − (2 p − ℓ −→ Σ q − (2 p − H Z ∧ p , and a corresponding long exact sequence · · · −→ [ ℓ, Σ q − − (2 p − H Z ∧ p ] −→ [ ℓ, Σ q ℓ ] −→ [ ℓ, Σ q − (2 p − ℓ ] −→ · · · . First we note that the map [ ℓ, Σ q ℓ ] → [ ℓ, Σ q − (2 p − ℓ ] is injective for q ≤ p − q = (2 p −
2) + 1 this follows from the fact that[ ℓ, Σ q − − (2 p − H Z ∧ p ] = H q − − (2 p − ( ℓ ; Z ∧ p ) = 0for q − − (2 p − < (2 p −
2) unless q − − (2 p −
2) = 0. In the case q = (2 p −
2) + 1,the image of [ ℓ, Σ H Z ∧ p ] in [ ℓ, Σ q ℓ ] in the long exact sequence is still zero becausethe map [ ℓ, ℓ ] → [ ℓ, H Z ∧ p ] ∼ = Z ∧ p is surjective. Now when q − (2 p − < p − ℓ, Σ q − (2 p − ℓ ] ∼ = [ ℓ, Σ q − (2 p − L ] ∼ = [ L, Σ q − (2 p − L ]since then Σ q − (2 p − ℓ → Σ q − (2 p − L is a weak equivalence on connective covers.For the remaining case q = 2(2 p − ℓ, Σ q ℓ ] → [ ℓ, Σ q − (2 p − ℓ ] is an injection and the map [ ℓ, Σ q − (2 p − ℓ ] → [ ℓ, Σ q − (2 p − L ] is aninjection. (cid:3) HE HOMOTOPY GROUPS OF K ( S ) 11 Next, using the cofiber sequenceΣ − ℓ −→ Σ (2 p − − ℓ −→ j −→ ℓ and applying the previous result, we obtain the following calculation. Proposition 2.9. [ j, Σ q ℓ ] = 0 if q p −
2) and q < p − Proof.
Looking at the long exact sequence · · · −→ [ ℓ, Σ q ℓ ] −→ [ j, Σ q ℓ ] −→ [Σ (2 p − − ℓ, Σ q ℓ ] −→ [Σ − ℓ, Σ q ℓ ] −→ · · · and using the isomorphism [Σ (2 p − − ℓ, Σ q ℓ ] ∼ = [ ℓ, Σ q +1 − (2 p − ℓ ] we have that both[ ℓ, Σ q ℓ ] and [Σ (2 p − − ℓ, Σ q ℓ ] are 0 when q , − p −
2) and q ≤ p − q ≡ − p − − ℓ, Σ q ℓ ] ∼ =[ ℓ, Σ q +1 ℓ ], we have a commutative diagram[ ℓ, Σ q +1 − (2 p − ℓ ] / / (cid:15) (cid:15) (cid:15) (cid:15) [ ℓ, Σ q +1 ℓ ] (cid:15) (cid:15) (cid:15) (cid:15) [ L, Σ q +1 − (2 p − L ] / / / / [ L, Σ q +1 L ]where the feathered arrows are known to be injections. The statement now followsin this case as well. (cid:3) For maps the other way, we have the following result. The proof is similar tothe proof of the previous proposition.
Proposition 2.10. [Σ q ℓ, j ] = 0 if q
6≡ − p −
2) and q ≥ − (2 p − q = − Proposition 2.11. [Σ − ℓ, j ] = 0 Proof.
Let j − = J [ − , ∞ ) where J = L K (1) S ≃ L K (1) j . Then we have a cofibersequence Σ − Hπ − J → j → j − → Σ − Hπ − J and a long exact sequence · · · −→ [Σ − ℓ, Σ − Hπ − J ] −→ [Σ − ℓ, j ] −→ [Σ − ℓ, j − ] −→ [Σ − ℓ, Σ − Hπ − J ] −→ · · · . Since Σ − ℓ is ( − j − in J induces a bijection[Σ − ℓ, j − ] −→ [Σ − ℓ, J ] ∼ = [Σ − L, J ] . It follows that a map Σ − ℓ → j − is determined by the map on π − , and thereforethat the image of [Σ − ℓ, j ] in [Σ − ℓ, j − ] is zero. But H − (Σ − ℓ ; π − J ) = 0, so[Σ − ℓ, j ] = 0. (cid:3) In the case of maps between suspensions of j , we only need to consider two cases: Proposition 2.12. [ j, Σ j ] = 0 and [Σ j, j ] = 0. Proof.
As in the previous proof, we let j − = J [ − , ∞ ), and we use the cofibersequence Σ − Hπ − J → Σ j → Σ j − → Hπ − J and the induced long exact sequence · · · −→ [ j, Σ − Hπ − J ] −→ [ j, Σ j ] −→ [ j, Σ j − ] −→ [ j, Hπ − J ] −→ · · · . Since the connective cover of Σ J is Σ j − , we have that the map [ j, Σ j − ] → [ j, Σ J ]is a bijection, and the maps[ J, Σ J ] −→ [ j, Σ J ] −→ [ S , Σ J ] ∼ = π − J are isomorphisms. It follows that the map [ j, Σ j − ] → [ j, Hπ − J ] is an isomor-phism. Since [ j, Σ − Hπ − J ] = 0, this proves [ j, Σ j ] = 0. For the other calculation,the map j → J induces a weak equivalence of 1-connected covers, and the inducedmap [Σ j, j ] −→ [Σ j, J ] ∼ = [Σ J, J ] ∼ = [Σ S , J ] = π J = 0is a bijection. (cid:3) The following propositions are now clear.
Proposition 2.13.
In the notation above, the summands x ( i ) of T C ( Z ) ∧ p satisfy[ x ( i ) , x ( i ′ )] = 0 for i = i ′ . Proposition 2.14.
Let k = 0 ,
1. In the notation above, [ x ( i ) , Σ k j ] = 0 for i = k and the inclusion of Σ k j in x ( k ) induces a bijection [ x ( k ) , Σ k j ] → [Σ k j, Σ k j ].Eliminating the summands where maps out of j or Σ j are trivial, and looking atthe connective and 0-connected covers of K (1)-localizations, we get the followingproposition. Proposition 2.15.
The map S → j induces isomorphisms [ j, T C ( Z ) ∧ p ] ∼ = π T C ( Z ) ∧ p and [Σ j, T C ( Z ) ∧ p ] ∼ = π T C ( Z ) ∧ p .For the summands of K ( Z ) ∧ p , we first consider the splitting of j . Proposition 2.16. [ j, e K ( Z )] = 0 and [ e K ( Z ) , j ] = 0 Proof.
As indicated above e K ( Z ) ≃ y ∨ · · · ∨ y p − . We have that y is (non-canonically) weakly equivalent to Σ p − ℓ , and applying Propositions 2.9, 2.10, wesee that [ j, y ] = 0 and [ y , j ] = 0. In addition, y ≃ ∗ and y p − ≃ ∗ . For1 < i < p − y i is the cofiber of a map from a finite wedge of copies of Σ i ℓ to afinite wedge of copies of Σ i ℓ . Looking at the long exact sequences · · · −→ M [ j, Σ i − ℓ ] −→ [ j, y i ] −→ M [ j, Σ i ℓ ] −→ · · ·· · · −→ Y [Σ i ℓ, j ] −→ [ y i , j ] −→ Y [Σ i − ℓ, j ] −→ · · · , we again see from Propositions 2.9 and 2.10 that [ j, y i ] = 0 and [ y i , j ] = 0. (cid:3) Finally, Bousfield’s work shows that (by construction) the spectra Y i satisfy[ Y i , Y i ′ ] = 0 for i = i ′ ; we now verify that the same holds for the covers y i . Proposition 2.17.
In the notation above, the summands y i of e K ( Z ) satisfy [ y i , y i ′ ] =0 for i = i ′ . Proof.
Each y i is the cofiber of a map from a finite wedge of copies of Σ i ℓ to afinite wedge of copies of Σ i ℓ except that y ≃ Σ p − ℓ (non-canonically), y ≃ ∗ ,and y p − ≃ ∗ . First, for i = 0, looking at the long exact sequence · · · −→ Y [Σ i ℓ, Σ q ℓ ] −→ [ y i , Σ q ℓ ] −→ Y [Σ i − ℓ, Σ q ℓ ] −→ · · · , we see from Proposition 2.8 that [ y i , Σ q ℓ ] = 0 when q i, i − p −
2) and q ≤ p −
2) + 2 i −
1. In particular, [ y i , y ] = 0 for i = 0. For i ′ = 0, looking atthe long exact sequence · · · −→ M [ y i , Σ i ′ − ℓ ] −→ [ y i , y i ′ ] −→ M [ y i , Σ i ′ ℓ ] −→ · · · , we see that [ y i , y i ′ ] = 0 for i = i ′ in the remaining cases. (cid:3) HE HOMOTOPY GROUPS OF K ( S ) 13 The maps in the linearization/cyclotomic trace square
The previous section discussed the corners of the linearization/cyclotomic tracesquare; in this section, we discuss the edges. The main observation is that withrespect to the canonical splittings of the previous section, the cyclotomic trace isdiagonal and the linearization map is diagonal on the p -torsion part of the homotopygroups. Theorem 3.1.
In terms of the splittings (2.4) and (2.6) of the previous section, thecyclotomic trace K ( Z ) ∧ p → T C ( Z ) ∧ p splits as the wedge of the identity map j → j and maps y −→ Σ − ℓ T C ( p ) y −→ Σ − ℓ T C (2) ... ... y p − −→ Σ − ℓ T C ( p − y p − −→ Σ − ℓ T C (0) . Proof.
We have that each y i fits in to a cofiber sequence of the form _ Σ i ℓ −→ _ Σ i ℓ −→ y i , except in the case i = 0 where the suspension is Σ p − ℓ rather than Σ ℓ (and thecase i = 1, where y i = ∗ anyway). Choosing a non-canonical weak equivalenceΣ q − ℓ ≃ Σ − ℓ T C ( q ), and looking at the long exact sequences of maps into Σ q − ℓ ,Proposition 2.8 implies [ y i , Σ − ℓ T C ( q )] = 0 unless q − i ≡ p − j, Σ − ℓ T C ( q )] = 0 for all q by Proposition 2.9. (cid:3) Next, we turn to the linearization map.
Theorem 3.2.
In terms of the splittings (2.1) , (2.2) , and (2.4) of the previoussection, the linearization map T C ( S ) ∧ p → T C ( Z ) ∧ p admits factorizations as follows: (i) The map S ∧ p → T C ( Z ) ∧ p factors through the canonical map S ∧ p → j and isa canonically split surjection on homotopy groups in all degrees. (ii) The map Σ S ∧ p → T C ( Z ) ∧ p factors through a map Σ S ∧ p → Σ j ′ that is anisomorphism on π and is a split surjection on homotopy groups in alldegrees. (iii) The map Σ C P ∞− → T C ( Z ) ∧ p factors through j ∨ W ℓ T C ( i ) ; on p -torsion,the map tor p ( π ∗ (Σ C P ∞− )) → tor p ( π ∗ ( T C ( Z ) ∧ p )) is zero.Proof. The statement (i) is clear from the construction of the map j → T C ( Z ) ∧ p since the linearization map is a map of ring spectra. For (ii), the composite mapΣ S ∧ p → T C ( Z ) ∧ p is determined by where the generator goes in π T C ( Z ) ∧ p , but theinclusion of Σ j ′ in T C ( Z ) ∧ p induces an isomorphism on π , and so Σ S ∧ p factorsthrough Σ j ′ . The map T C ( S ) ∧ p → T C ( Z ) ∧ p is a (2 p − S ∧ p → H Z ∧ p is (2 p − p − T HH ( S ∧ p ) ∧ p ≃ T HH ( S ) ∧ p −→ T HH ( Z ) ∧ p ≃ T HH ( Z ∧ p ) ∧ p . The map Σ S → Σ j ′ is therefore an isomorphism on π . Using the image of thegenerator of π Σ S as a generator for π Σ j ′ gives a weak equivalence Σ j → Σ j ′ such that the map Σ S → Σ j ′ → Σ j obtained by composing with the inverse is thesuspension of the canonical map S → j . This completes the proof of (ii).To show the factorization of Σ C P ∞− for (iii), it suffices to check that the com-posite map Σ C P ∞− −→ T C ( Z ) ∧ p −→ Σ j ′ is trivial. For this we show that [Σ C P ∞− , Σ j ′ ] = 0. Looking at the cofibrationsequence Σ − S ∧ p −→ Σ C P ∞− −→ Σ(Σ ∞ C P ∞ ) ∧ p −→ S ∧ p since π − Σ j ′ = 0, the map [Σ(Σ ∞ C P ∞ ) ∧ p , Σ j ′ ] → [Σ C P ∞− , Σ j ′ ] is surjective andso it suffices to show that[Σ(Σ ∞ C P ∞ ) ∧ p , Σ j ′ ] ∼ = [(Σ ∞ C P ∞ ) ∧ p , j ′ ] ∼ = [(Σ ∞ C P ∞ ) ∧ p , L K (1) j ′ ] ∼ = [ L K (1) Σ ∞ C P ∞ , L K (1) j ′ ]is zero. Ravenel [19, 9.2] identifies the K (1)-localization of Σ ∞ C P ∞ as an infinitewedge of copies of KU ∧ p . Since there are no essential maps KU ∧ p → J , there are noessential maps Σ(Σ ∞ C P ∞ ) ∧ p → Σ j ′ and hence no essential maps from Σ C P ∞− toΣ j ′ .Finally, to see that the map Σ C P ∞− → T C ( Z ) ∧ p is zero on the torsion subgroupof π ∗ Σ C P ∞− , it suffices to note that the composite map Σ C P ∞− → T C ( Z ) ∧ p → j is zero on the torsion subgroup of π ∗ Σ C P ∞− , or equivalently that the compositemap to J is zero on the torsion subgroup of π ∗ Σ C P ∞− . The map Σ C P ∞− → J factors through L K (1) Σ C P ∞− , and again using the result on the K (1)-localizationof Σ ∞ C P ∞ , we have a cofibration sequenceΣ − J −→ L K (1) Σ C P ∞− −→ _ Σ KU ∧ p −→ J. The image of tor p ( π ∗ Σ C P ∞− ) in π ∗ L K (1) C P ∞− can therefore only possibly be non-zero in even degrees, and hence maps to zero in π ∗ J . This completes the proof of(iii). (cid:3) It would be reasonable to expect that the augmentations
T C ( S ) ∧ p → S ∧ p and T C ( Z ) ∧ p → j are compatible, although we see no K -theoretic, T HH -theoretic, orcalculational reasons why this should hold. Such a compatibility would imply thatthe map C P ∞− → T C ( Z ) ∧ p factors through W ℓ T C ( i ) and would then (combinedwith the observations in Section 5) say that the linearization map is fully diagonalwith respect to the splittings of the previous section.4. Proof of main results
We now apply the work of the previous two sections to prove the theoremsstated in the introduction. We begin with Theorem 1.1, which is an immediateconsequence of the following theorem.
Theorem 4.1.
The map π n T C ( S ) ∧ p ⊕ π n K ( Z ) ∧ p −→ π n T C ( Z ) ∧ p is (non-canonically) split surjective. We apply the splittings of
T C ( S ) ∧ p , T C ( Z ) ∧ p , and K ( Z ) ∧ p and the maps on homo-topy groups to prove the previous theorem by breaking it into pieces and showingthat different pieces in the splitting induce surjections on homotopy groups. Indeed HE HOMOTOPY GROUPS OF K ( S ) 15 Theorem 3.2.(i) and (ii) give the first piece. For the next piece, we look at the map
T C ( S ) ∧ p → T C ( Z ) ∧ p . The following lemma is essentially due to Klein-Rognes [13] Lemma 4.2.
Under the splittings (2.1) , (2.2) , and (2.4) , the composite map Σ C P ∞− −→ T C ( S ) ∧ p −→ T C ( Z ) ∧ p −→ Σ − ℓ T C (0) ∨ · · · Σ − ℓ T C ( p − induces a split surjection on π q +1 for q p − Proof.
Klein and Rognes [13, 5.8,(17)] (and independently Madsen and Schlichtkrull [16,1.1]) construct a space-level map SU ∧ p −→ Ω ∞ (Σ C P ∞− ) ∧ p . They study the composite map(4.3) SU ∧ p −→ Ω ∞ (Σ C P ∞− ) ∧ p −→ Ω ∞ ( T C ( Z ) ∧ p ) −→ SU ∧ p induced by the linearization map T C ( S ) ∧ p → T C ( Z ) ∧ p , the projection map T C ( Z ) ∧ p [0 , ∞ ) −→ Σ ku ∧ p , and the Bott periodicity isomorphism Ω ∞ Σ ku ≃ SU . In [13, 6.3.(i)], Klein andRognes show that their map (4.3) induces an isomorphism of homotopy groups inall degrees except those congruent to 1 mod 2( p − −
1, where it follows from the fact that the linearization map
T C ( S ) ∧ p → T C ( Z ) ∧ p is a (2 p − (cid:3) For the final piece, we need a split surjection onto π ∗ T C ( Z ) ∧ p in degrees congruentto 1 mod 2( p − Lemma 4.4.
Under the splittings of (2.4) and (2.6) , the composite map y −→ K ( Z ) ∧ p −→ T C ( Z ) ∧ p −→ Σ − ℓ T C ( p ) is a weak equivalence.Proof. Since y and Σ − ℓ T C ( p ) are both (non-canonically) weakly equivalent toΣ p − ℓ , it suffices to show that the map becomes a weak equivalence after K (1)-localization. Indeed, it suffices to show that the map on K (1)-localizations is anisomorphism on any odd dimensional homotopy group. We have a canonical iden-tification of π L K (1) K ( Z ) with the p -completion of the unit group of Z [1 /p ], whichis isomorphic to Z ∧ p by the homomorphism sending the generator 1 of Z ∧ p to p in Z [1 /p ] × . We likewise have a canonical identification of π L K (1) T C ( Z ) with the p -completion of the unit group of Q ∧ p and the map π L K (1) K ( Z ) → π L K (1) T C ( Z )is the inclusion of ( Z [1 /p ] × ) ∧ p in (( Q ∧ p ) × ) ∧ p . By construction, the inclusion of Σ j ′ in T C ( Z ) ∧ p corresponds to the inclusion of (( Z ∧ p ) × ) ∧ p in (( Q ∧ p ) × ) ∧ p ; the quotient groupis isomorphic to Z ∧ p by the homomorphism sending the generator to p ∈ (( Q ∧ p ) × ) ∧ p .The summand Y = L K (1) y of L K (1) K ( Z ) is the only one that contributes to π L K (1) K ( Z ), and the summands L K (1) Σ j ′ ∨ L K (1) ℓ T C ( p ) of L K (1) T C ( Z ) are theonly ones that contribute to π L K (1) T C ( Z ), so we can identify the composite map Y = L K (1) y −→ L K (1) T C ( Z ) −→ L K (1) ℓ T C ( p )on π as the composite map Z [1 /p ] × → ( Q ∧ p ) × → ( Q ∧ p ) × / ( Z ∧ p ) × , which is an iso-morphism. (cid:3) We now have everything we need for the proof of Theorem 4.1.
Proof of Theorem 4.1.
Combining previous results, we have two families of (non-canonical) splitting π ∗ ( S ∨ Σ S ∨ Σ C P ∞− ∨ j ∨ y ∨ · · · ∨ y p − ) −→ π ∗ ( j ∨ Σ j ′ ∨ Σ − ℓ T C (0) ∨ · · · ∨ ℓ T C ( p − . For both splittings we use Lemma 4.4 to split the ℓ T C ( p ) summand in the codomain(canonically) using the y summand of the domain, we use Theorem 3.2.(ii) to splitthe π ∗ Σ j ′ summand in the codomain (canonically) using the Σ S summand in thedomain, and we use Lemma 4.2 to split the π ∗ ( ℓ T C (0) ∨ ℓ T C (2) ∨ · · · ℓ T C ( p − π ∗ Σ C P ∞− summand in thedomain. We then have a choice on the remaining summand of the codomain, π ∗ j .We can use Theorem 3.1 to split this (canonically) using the π ∗ j summand in thedomain or use Theorem 3.2 to split this (canonically) using the π ∗ S summand inthe domain. (cid:3) This completes the proof of Theorem 1.1. We now prove the remaining theoremsfrom the introduction.
Proof of Theorems 1.2 and 1.3.
Since the long exact sequence of the homotopycartesian linearization/cyclotomic trace square breaks into split short exact se-quences, we get split short exact sequences on p -torsion subgroups0 → tor p ( π n K ( S )) −→ tor p ( π n T C ( S ) ∧ p ⊕ π n K ( Z )) −→ tor p ( π n T C ( Z ) ∧ p ) −→ . Using the splittings of (2.1), (2.4), and (2.5), leaving out the non-torsion summands,we can identify tor p ( π n K ( S )) as the kernel of a maptor p ( π n S ⊕ π n − S ⊕ π n − C P ∞− ⊕ π n j ⊕ π n e K ( Z )) −→ tor p ( π n j ⊕ π n − j )which by Theorems 3.1 and 3.2 is mostly diagonal: It is the direct sum of thecanonical maps tor p ( π n S ) ⊕ tor p ( π n j ) −→ tor p ( π n j )tor p ( π n − S ) −→ tor p ( π n − j )and the zero maps on tor p ( π n − C P ∞− ) and tor p ( π n e K ( Z )). Here we have used theimage of the generator of π T C ( S ) ∧ p to produce the weak equivalence of Σ j ′ with Σ j (as in the proof of Theorem 3.2). The isomorphism (a) uses the canonical splitting π n j → π n S on the π n S summand, while the isomorphism (b) uses the identity of π n j on the π n j summand. (cid:3) Conjecture on Adams Operations
In Section 2 we produced canonical splittings on K ( Z ) ∧ p and T C ( Z ) ∧ p and inSection 3, we showed that the cyclotomic trace is diagonal with respect to thesesplittings. In both cases we argued calculationally based on the paucity of maps inthe stable category between the summands; however, there is a conceptual reasonto expect much of this behavior. Specifically, the diagonalization would be a con-sequence of the existence of p -adically interpolated Adams operations on algebraic K -theory analogous to the p -adically interpolated Adams operations on topological K -theory. We begin with the following conjecture, which appears to be new. HE HOMOTOPY GROUPS OF K ( S ) 17 Conjecture 5.1.
Let R be an E ∞ ring spectrum. There exists a homomorphismfrom ( Z ∧ p ) × to the composition monoid [ K ( R ) ∧ p , K ( R ) ∧ p ] , which is natural in theobvious sense and satisfies the following properties. (i) When R is a ring, the restriction to Z ∩ ( Z ∧ p ) × give Quillen’s Adams oper-ations on the zeroth space. (ii) The induced map ( Z ∧ p ) × → Hom( π ∗ K ( R ) ∧ p , π ∗ K ( R ) ∧ p ) is continuous wherethe target is given the p -adic topology. The preceding conjecture on p -adic interpolation of the Adams operations in p -completed algebraic K -theory is weaker than the (known) results in the case oftopological K -theory in that we are only asking for continuity on homotopy groupsrather than continuity for (some topology on) endomorphisms. Nevertheless, itimplies a natural action of Z ∧ p [( Z /p ) × ] (in the stable category) on p -completed al-gebraic K -theory spectra (of connective ring spectra). This then implies an eigen-splitting of K ( R ) ∧ p (and a fortori, L K (1) K ( R )) into the p − p or a strict Henselian ring A with p invertible, the Adamsoperation ψ k acts on π s L K (1) K ( A ) by multiplication by k s . As a consequence, forany scheme satisfying the hypotheses for Thomason’s spectral sequence [27, 4.1],the eigensplitting on homotopy groups is compatible with the filtration from E ∞ in the sense that the subquotient of H s ( R ; Z /p n ( i )) comes from the ω i summand.For R = Z [1 /p ], we see that the ω i summand of L K (1) K ( R ) has homotopy groupsonly in degrees congruent to 2 i − i − p −
1) except when i = 0where the unit of π L K (1) ( R ) is also in the trivial character summand. As a con-sequence, we see that this splitting agrees with the splitting described above for L K (1) K ( Z [1 /p ]) ∼ = L K (1) K ( Z ). Specifically, the summand corresponding to trivialcharacter is J and the summand corresponding to ω i is Y i − for i = 1 , . . . , p − K -theory conjecture also leads to a splitting of T C ( Z ) ∧ p . Hesselholt-Madsen [11, Th. D,Add. 6.2] shows that the completion map and cyclotomic trace T C ( Z ) ∧ p −→ T C ( Z ∧ p ) ∧ p ←− K ( Z ∧ p ) ∧ p are weak equivalences after taking the connective cover and so in particular in-duce weak equivalences after K (1)-localization. The Quillen localization sequenceidentifies the homotopy fiber of the map K ( Z ∧ p ) → K ( Q ∧ p ) as K ( F p ). Since the p -completion of K ( F p ) is weakly equivalent to H Z ∧ p , its K (1)-localization is trivial.Combining these maps, we obtain a canonical isomorphism in the stable categoryfrom L K (1) T C ( Z ) to L K (1) K ( Q ∧ p ). The Hesselholt-Madsen proof of the Quillen-Lichtenbaum conjecture for certain local fields [12, Th. A] (or in this case, inspectionfrom the calculation of T C ( Z ) ∧ p ), shows that the map T C ( Z ) ∧ p −→ L K (1) T C ( Z ) ≃ L K (1) K ( Q ∧ p )becomes a weak equivalence after taking 1-connected covers. Again looking atThomason’s spectral sequence, we see that the conjectural Adams operations wouldthen split L K (1) K ( Q ∧ p ) into summands as follows. The summand correspondingto the trivial character is J ∨ Σ − L T C (0), the summand corresponding to ω isΣ J ∨ Σ − L T C (1), and the summand corresponding to ω i is Σ − L T C ( i ) for i =2 , . . . , p −
2. This perspective does not help split the trivial character summandand ω summand into their constituent smaller pieces, but does further imply by naturality of the Adams operations, that the map L K (1) K ( Z ) → L K (1) K ( Q ∧ p ) isdiagonal in the eigensplitting.To bring T C ( S ) into the picture, we can relate T C ( S ) ∧ p to K ( S ∧ p ) ∧ p , but in thiscase it is easier to work with T C ( S ) ∧ p directly. As discussed in [1, 10.7], the Adamsoperation on T C constructed there lead to a eigensplitting of
T C ( S ) ∧ p in terms ofa familiar wedge decomposition of C P ∞− previously used by Rognes [25, § C P ∞− ≃ C P ∞− [ − ∨ C P ∞− [0] ∨ · · · ∨ C P ∞− [ p − . Here we are following the numbering of Rognes [25, p. 169], which has its rationale inthat the [ i ] piece has its ordinary cohomology concentrated in degrees 2 i mod 2 p − i ; in terms of characters, Σ C P ∞− [ i ] corresponds to ω i +1 .For T C ( S ) ∧ p , the summand corresponding to the trivial character is then S ∧ p ∨ Σ C P ∞− [ −
1] and for i = 1 , . . . , p −
2, the summand corresponding to ω i is thenΣ C P ∞− [ i − T C ( S ) ∧ p → T C ( Z ) ∧ p is diagonal on the eigensplitting, sending the ω i summand of T C ( S ) ∧ p to the ω i summand of T C ( Z ) ∧ p , as there are no maps into theother summands. This proves the following result (independently of the conjectureabove). Theorem 5.3.
The spectrum K ( S ) ∧ p splits into p − summands, K ( S ) ∧ p ≃ K ∨· · · ∨ K p − , and the linearization/cyclotomic trace square splits into the wedge sumof p − homotopy cartesian squares K / / (cid:15) (cid:15) j (cid:15) (cid:15) K / / (cid:15) (cid:15) y (cid:15) (cid:15) S ∧ p ∨ Σ C P ∞− [ − / / j ∨ Σ − ℓ T C (0) Σ C P ∞− [0] / / Σ j ∨ Σ − ℓ T C ( p ) for i = 0 , , and K i / / (cid:15) (cid:15) y i (cid:15) (cid:15) Σ C P ∞− [ i − / / Σ − ℓ T C ( i ) for i = 2 , . . . p − . The splitting of the preceding theorem fails to be canonical because of the lim problem in the identification of T C ( S ) ∧ p as S ∧ p ∨ Σ C P ∞− . We are using the notationabove and in Section 2, and in the ω diagram, we have used that y p − = ∗ , as notedin Section 2. In using the above squares for studying the homotopy type of K i , wecan simplify the i = 0 square to a cofiber sequence K −→ S ∧ p ∨ Σ C P ∞− [ − −→ Σ − ℓ T C (0) −→ Σ K since Theorem 3.1 indicates that the map j → j ∨ Σ − ℓ T C (0) factors through theidentity map j → j . For the i = 1 summand, the splitting of C P ∞− fits into a fibersequence with the splitting of(Σ ∞ C P ∞ ) ∧ p ≃ Σ ∞ K ( Z ∧ p , ≃ C P ∞ [1] ∨ · · · ∨ C P ∞ [ p − , which identifies C P ∞− [0] as S ∧ p ∨ C P ∞ [ p − y → Σ j ∨ Σ − ℓ T C ( p ) factors through a weak equivalence y → ℓ T C ( p ), HE HOMOTOPY GROUPS OF K ( S ) 19 and we get a weak equivalence K ≃ Σ c ∨ C P ∞ [ p − , where by definition c is the homotopy fiber of the canonical map S → j .We have deduced the previous theorem calculationally, but other than the iden-tification of the Σ C P ∞− factors, it would follow from directly from Conjecture 5.1,at least after passing to connective covers. For the identification of the Σ C P ∞− fac-tors, we would also have to relate the Adams operations on T C to the (conjectural)Adams operations on K -theory. To this end, we propose the following second-levelconjecture. Conjecture 5.4.
For R a connective E ∞ ring spectrum, the cyclotomic trace K ( R ) ∧ p → T C ( R ) ∧ p commutes with the (conjectural) p -adically interpolated Adamsoperations. Low degree computations
Theorem 1.2 describes the p -torsion in K ( S ) in terms of the p -torsion in variouspieces. For convenience, we review in Proposition 6.1 below what is known aboutthe homotopy groups of these pieces at least up to the range in which [25] describesthe homotopy groups of C P ∞− . As a consequence of Theorem 1.2, irregular primespotentially contribute in degrees divisible by 4 but otherwise make no contributionto the torsion of K ( S ) until degree 22. Thus, π ∗ K ( S ) in degrees ≤
21 not divisible by4 is fully computed (up to some 2-torsion extensions) by the work of Rognes [24, 25].For convenience, we assemble the computation of π ∗ K ( S ) for ∗ ≤
22 in Table 1 onpage 20.
Proposition 6.1.
The p -torsion groups tor p ( π ∗ S ), tor p ( π ∗ c ), tor p ( π ∗ e K ( Z )), andtor p ( π ∗ Σ C P ∞− ) are known in at least the following ranges, as follows.(i) π ∗ S , π ∗ j , see for example [20, 1.1.13]. π ∗ S splits as π ∗ S = π ∗ j ⊕ π ∗ c. tor p ( π k j ) is zero unless 2( p −
1) divides k + 1, in which case it is cyclicof order p s +1 where k + 1 = 2( p − p s m for m relatively prime to p . Seebelow for π ∗ c .(ii) π ∗ c , see for example [20, 1.1.14]. In degrees ≤ p ( p − −
6, tor p ( π ∗ c ) is Z /p in the following degrees and zero in all others. (In the table, α ∈ π p − j .)Generator Degree β p ( p − − α β p + 1)( p − − β p ( p − − α β p + 1)( p − − β p + 1)( p − − α β p + 1)( p − − β p ( p − − Table 1.
The homotopy groups of K ( S ) in low degrees n π n K ( S )0 Z Z / Z / Z / × Z / ⊕ Z /
24 05 Z Z / Z / × Z / × Z / ⊕ Z /
28 ( Z / ⊕ K ( Z )9 Z ⊕ ( Z / ⊕ Z / Z / × Z / ⊕ Z / × ( Z / Z / × Z / × Z / ⊕ Z / ⊕ Z / Z / ⊕ Z / ⊕ K ( Z )13 Z ⊕ Z /
314 ( Z / ⊕ Z / ⊕ Z / ⊕ Z / Z / × Z / × Z / × Z / ⊕ ( Z /
16 ( Z / ⊕ Z / × Z / ⊕ Z / ⊕ K ( Z )17 Z ⊕ ( Z / ⊕ ( Z / Z / × Z / ⊕ Z / × ( Z / ⊕ Z / × Z / Z / × Z / × Z / × Z / ⊕ [64]20 Z / × Z / ⊕ [128] ⊕ Z / ⊕ K ( Z )21 Z ⊕ ( Z / ⊕ [16] ⊕ Z /
322 ( Z / ⊕ [2 ? ] ⊕ Z / ⊕ Z / (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) The table compiles the results reviewed in Proposition 6.1 (q.v. for sources) and[24, 5.8] into the computation of π n K ( S ) for n ≤
22. The description of π n K ( S ) isdivided into columns:1. The non-torsion part2. The contribution from the torsion of S
3. The remaining 2-torsion (from [24])4. The contribution from Σ c for odd primes5. The torsion contribution from Σ C P ∞− for odd primes6. The torsion contribution from e K ( Z ) for odd primesPresently K n ( Z ) is unknown for n >
1, conjectured to be 0 (the Kummer-Vandiverconjecture) and if non-zero is a finite group with order a product of irregular primes,each of which is > .Summands denoted as [ m ] are finite groups of order m whose isomorphism class isnot known. HE HOMOTOPY GROUPS OF K ( S ) 21 (iii) π ∗ e K ( Z ), see for example [30, § VI.10] or Section 2. tor p ( π ∗ e K ( Z )) is zeroin odd degrees. If p is regular, then tor p ( π k e K ( Z )) = 0. If p satisfies theKummer-Vandiver condition, thentor p ( π k e K ( Z )) = 0 and tor p ( π k +2 e K ( Z )) = Z ∧ p / ( B k +2 k +2 ) , where B n denotes the Bernoulli number, numbered by the convention te t − = P B n t n n ! . If p does not satisfy the Kummer-Vandiver conditionthen tor p ( π k e K ( Z )) = 0 for k = 1 and is an unknown finite group for k > p ( π k +2 e K ( Z )) is an unknown group of order Z ∧ p / ( B k +2 / (2 k +2)) for all k .(iv) π ∗ Σ C P ∞− , see [25, 4.7]. • In odd degrees ≤ | β | − p + 1)( p − − p ( π n +1 Σ C P ∞− ) = Z /p in degrees n = p − p − m or n = 2 p − p − m for 1 ≤ m ≤ p − • In even degrees ≤ p ( p − p ( π n Σ C P ∞− ) = Z /p for m ( p − < n < mp for 2 ≤ m ≤ p − p ( π p ( p − − Σ C P ∞− ) = 0 . Beyond this range, [25, 4.7] only describes the size of the group.In all even degrees ≤ | β | − p + 1)( p − −
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