TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES John Rognes Abstract. We compute the mod 2 cohomology of Waldhausen's algebraic K-theo* *ry spectrum A(*) of the category of finite pointed spaces, as a module over t* *he Steen- rod algebra. This also computes the mod 2 cohomology of the smooth Whitehe* *ad spectrum of a point, denoted Wh Diff(*). Using an Adams spectral sequence* * we compute the 2-primary homotopy groups of these spectra in dimensions * 18, and up to extensions in dimensions 19 * 21. As applications we show that the linearization map L:A(*) ! K(Z) induces the zero homomorphism in mod 2 spectrum cohomology in positive dimensions, the space level Hatcher-Waldha* *usen map hw: G=O ! Wh Diff(*) does not admit a four-fold delooping, and there i* *s a 2-complete spectrum map M :WhDiff(*) ! g=o which is precisely 9-connected. Here g=o is a spectrum whose underlying space has the 2-complete homotopy* * type of G=O. Introduction Let A(X) be Waldhausen's algebraic K-theory of spaces functor evaluated on the space X, see [Wa1]. When X is a manifold, A(X) provides the fundamental link between algebraic K-theory and the geometric topology of X _ in particular with the concordance space, the h-cobordism space and the automorphism space of X, see [Wa3]. We are therefore interested in evaluating its homotopy type. It is the aim of this paper to compute the 2-primary homotopy type of A(X) in the case when X = * is the one-point space. We achieve this by computing the mod 2 spectrum cohomology of A(*) as a module over the mod 2 Steenrod algebra. The result is a complete calculation valid in all dimensions; we also compute t* *he homotopy groups of A(*) modulo odd torsion in dimensions * 18, and up to extensions in dimensions 19 * 21. We begin by discussing some definitions and interpretations of A(X), in order to explain why this is an important homotopy type. One way to define A(X) is as the algebraic K-theory of a category with cofi- brations and weak equivalences Rf(X), whose objects are retractive spaces over X subject to a relative finiteness condition, see [Wa5]. When X = * this category Rf(*) is the category of finite pointed CW-complexes and pointed cellular maps, and is the category of pointed spaces alluded to in the title. The cofibrations* * are the cellular embeddings, and the weak equivalences are the homotopy equivalence* *s. Let hRf(X) be the subcategory of Rf(X) obtained by restricting the morphisms to be homotopy equivalences, and let |hRf(X)| denote its geometric realization.* * As a space, A(X) is defined as the loop space |hSoRf(X)|, where So is Waldhausen's simplicial construction of the same name. This construction can be iterated, and Typeset by AM S-TEX 1 2 JOHN ROGNES in fact A(X) is an infinite loop space with nth delooping |hS(n)oRf(X)| for each n 1. There is a canonical map e: |hRf(X)| -! A(X) from the geometric realization of the category of finite pointed spaces and hom* *otopy equivalences to the infinite loop space A(X). There is a natural isomorphism ss0A(X) ~=Z, and for every object Y 2 hRf(X) the image under ss0(e) of the corresponding point in |hRf(X)| is the relative E* *uler characteristic O(Y; X) = O(Y ) - O(X) of Y . From this point of view the map e is a lift of the usual Euler characteristic that takes values in the integers* *, to a map that takes values in the infinite loop space A(X). Furthermore, a diagram of spaces and homotopy equivalences given as a functor F :C ! hRf(X) gives rise to a map e O |F |: |C| ! A(*), which will detect more information than just the Euler characteristics of the individual spaces in the diagram. For example* * a pointed G-space Y gives rise to a map BG ! A(*) whose homotopy class is a refined invariant of Y . We think of e as a homotopy theoretic improvement on the Euler characteristic, able also to detect information about diagrams of spa* *ces and homotopy equivalences, rather than just individual spaces, and A(X) is the receptacle for this improved Euler characteristic. In fact A(*) is a kind of universal receptacle for homotopy invariants of fi* *nite pointed spaces that take values in infinite loop spaces and are subject to the * *fol- lowing additivity condition: for each cofiber sequence Y 0! Y ! Y 00we have [Y 0] + [Y 00] = [Y ] where [Y ] 2 ss0A(*) denotes the path component in A(*) o* *f the invariant applied to Y . Of course, the corresponding universal invariant taki* *ng values in an abelian group is just the Euler characteristic. We shall not make * *the universality claim more precise in this introduction, but note that a similar d* *is- cussion applies for A(X) and suitably additive homotopy invariants of retractive spaces over X. Hereafter it will be more convenient to work with spectra than infinite loop spaces. The infinite loop space A(X) determines a unique connective spectrum, and from now on A(X) will refer to this spectrum. The body of this paper is also written in terms of spectra rather than infinite loop spaces, partly because a * *few non-connective spectra will appear. Suspension of retractive spaces over X induces an equivalence on the level of algebraic K-theory, and so A(X) can also be considered as the algebraic K-theory of a category of spectra over X. It is simplest to make this precise for X = *,* * when A(*) is equivalent to the algebraic K-theory of the category of finite CW-spect* *ra, with respect to suitable notions of cofibrations and stable equivalences, see [* *Wa4]. Let S be the sphere spectrum in some good closed symmetric monoidal category of spectra and spectrum maps, for example the S-modules of [EKMM] or the - spaces of [Se] and [Ly]. In either case the ring spectrum S is a monoid object * *with respect to the internal smash product, and a spectrum is a module over S, so we* * can sensibly refer to spectra as S-modules. Then A(*) can be described as the algeb* *raic K-theory of a category of S-modules subject to suitable finiteness conditions, * *and briefly A(*) is the algebraic K-theory of the ring spectrum S. See [BHM] for a discussion in terms of FSPs. More generally, for a unital and associative ring spectrum A we may consider a category of finitely generated free A-modules, and form its algebraic K-theor* *y, TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 3 see [D2]. These ring spectra are unital and associative monoids in one of the categories of spectra considered above, and may conveniently be called S-algebr* *as. For each ring R in the algebraic sense, the Eilenberg-Mac Lane spectrum HR is an S-algebra whose algebraic K-theory agrees with Quillen's K(R), see [Q2]. For* * a simplicial monoid G the unreduced suspension spectrum 1 (G+ ) is an S-algebra whose algebraic K-theory agrees with Waldhausen's A(X) for X = BG. Thus S- algebras encompass the previous examples of inputs for algebraic K-theory, see * *also [Wa7]. Now S is a commutative S-algebra, so its algebraic K-theory K(S) = A(*) is itself a ring spectrum, and furthermore the algebraic K-theory K(A) of any S- algebra is a module spectrum over A(*). Hence every algebraic K-theory spectrum considered so far is a module spectrum over A(*), which further emphasizes the special role played by A(*). The relationship of A(X) to geometric topology is through the splitting of s* *pectra A(X) ' 1 (X+ ) _ Wh Diff(X) for the smooth category, and the cofiber sequence of spectra A(*) ^ X+ -ff!A(X) -! Wh PL(X) for the piecewise linear category, see [Wa3] and [Wa6]. Here ff is the assembly* * map, one construction of which uses that A(X) is a homotopy functor in X, see [WW2]. The spectra Wh Diff(X) and Wh PL(X) are the smooth and PL Whitehead spec- tra, respectively. The topological Whitehead spectrum Wh Top(X) is equivalent to the PL one by [KS] and [BuLa]. Thus knowledge of A(*) determines Wh Diff(*) and is the ingredient needed to pass from A(X) to Wh PL(X) ' Wh Top(X). The underlying infinite loop spaces of these Whitehead spectra are called Whitehead spaces, and it is perhaps more common to work in terms of these. When X is a smooth manifold, 1 Wh Diff(X) gives the homotopy functor that best approximates the space CDiff(X) of smooth concordances (= pseudoisotopies) of X. By Igusa's stability theorem [Ig] there is a stabilization map DiffX:CDiff(X) -! 21 Wh Diff(X) which is at least roughly n=3-connected where n is the dimension of X. Similar results relate Wh PL(X) and Wh Top(X) to the PL- and topological concordance spaces CPL(X) and CTop(X) when X is a PL- or topological manifold, respectively. Furthermore there is a geometrically significant involution on A(X), related through the Whitehead spectra to the involution on concordance spaces arising from `turning a concordance upside-down', see [H] and [Vog]. By [WW1] there is a map DiffX:gDiff(X)=Diff(X) -! 1 (EC2+ ^C2 Wh Diff(X)) which is at least as connected as the stabilization map considered by Igusa. The C2-action on Wh Diff(X) on the right is given by the involution, and the homoto* *py orbit construction is formed on the spectrum level. This is a space level inter* *pre- tation of the output of the Hatcher spectral sequence [H], which works on the l* *evel of homotopy groups. The space Dgiff(X)=Diff(X) measures the difference between the topological group Diff(X) of diffeomorphisms of the smooth manifold X and the simplicial group gDiff(X) of `block diffeomorphisms', which is computable in terms of surg* *ery theory, see [H]. Thus knowledge of the homotopy orbits for the involution acting 4 JOHN ROGNES on the spectrum Wh Diff(X), or equivalently on the spectrum A(X), can be viewed as giving knowledge of the homotopy type of the space of diffeomorphisms Diff(X) in dimensions up to roughly n=3, where n is the dimension of X. Similar results apply for the spaces of PL homeomorphisms of PL manifolds and homeomorphisms of topological manifolds. See [WW3] for a more detailed survey. In this paper we shall determine the homotopy type of the 2-primary completi* *on of the spectrum Wh Diff(*). Since the Whitehead spectrum is a homotopy functor and preserves connectivity of maps, for any smooth n-manifold X which is roughly n=3-connected the map DiffXcomposed with the natural map 1 (EC2+ ^C2 Wh Diff(X)) -! 1 (EC2+ ^C2 Wh Diff(*)) is roughly n=3-connected. Thus when our 2-primary calculation is extended to a calculation of the C2-homotopy orbits of Wh Diff(*), we will have complete info* *rma- tion about the 2-primary homotopy type of the space of diffeomorphisms Diff(X) of roughly n=3-connected manifolds up to dimension roughly n=3. We leave these calculations for a future paper. We now turn to a description of the contents of the present paper. We are able to access the homotopy type of A(*) by means of a comparison of algebraic K-theory with the topological cyclic homology theory of B"okstedt, Hs* *iang and Madsen [BHM], relying on a theorem of Dundas [D1]. In Chapter 1 we review these notions, and are led in Theorem 1.11 to the homotopy cartesian square A(*) ___L__//_K(Z) |trc*| |trcZ| fflffl|L fflffl| T C(*)_____//T C(Z) : Here T C denotes the topological cyclic homology functor, and the natural trans- formation trcis the cyclotomic trace map of [BHM]. We are able to access A(*) after 2-adic completion because the 2-primary homotopy types of the three other spectra in this diagram are known, together with sufficient information about t* *he maps in the diagram. More specifically, the homotopy type of T C(*) was deter- mined in [BHM], for odd primes p the p-adic completion of T C(Z) was computed in [BM], and the 2-adic completion was determined in [R5]. The 2-adic completion of K(Z) was found in [RW], by arguments based on Voevodsky's proof of the Milnor conjecture [Voe] and the Bloch-Lichtenbaum spectral sequence [BlLi]. The 2-adic map trcZ:K(Z) ! T C(Z) was also studied in [R5], in sufficient detail that we c* *an describe A(*) as an extension of T C(*) by the common homotopy fiber of the maps labelled trc*and trcZin the diagram above. At odd primes p, the missing information needed to determine the p-primary homotopy type of A(*) is the identification of the p-adic completion of K(Z), i* *.e., a proof of the p-primary Lichtenbaum-Quillen conjecture for the integers, and t* *he determination of how A(*) is an extension of T C(*) by the homotopy fiber of tr* *cZ, after p-adic completion. Since A(*) has finite type, and is rationally equivale* *nt to K(Z), this would suffice to determine the integral homotopy type of A(*). TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 5 Also in Chapter 1 we make precise a part of the calculation of T C(*) from [* *BHM], relating its p-adic completion to the Thom spectrum CP-11= T h(-fl1) of minus t* *he canonical complex line bundle over CP 1. See Theorem 1.16 and Corollary 1.21, which when combined yield a homotopy equivalence T C(*) ' 1 S0_CP-11after p-adic completion. In Chapter 2 we analyze the 2-primary homotopy type of CP-11by classical methods. We obtain its homotopy groups in dimensions * 20 in Theorem 2.13, by use of the Atiyah-Hirzebruch spectral sequence for stable homotopy associated to the skeleton filtration of CP-11by the subspectra CP-s1for s -1. The E1-term in this spectral sequence is given in terms of the stable homotopy groups of sp* *heres, ssS*, and the differentials depend on the attaching maps for the cells in CP-11* *. This involves primary and secondary operations in homotopy, somewhat along the lines of Toda's book [To], and we build on previous work for CP 1 by Mosher [Mo] and Mukai [Mu1], [Mu2] and [Mu3]. It is much easier to describe CP-11cohomologically, and in Proposition 2.15 * *we find that the mod 2 spectrum cohomology of CP-11is cyclic as an A-module, where A is the mod 2 Steenrod algebra, and we describe the annihilator ideal C of the generator in Definition 2.14. The squaring operations Sqi with i odd together w* *ith the admissible monomials SqI of length 2 form a basis for C as an F2-vector space. Thus H*spec(CP-11; F2) ~=-2A=C as left graded A-modules. This allows us to describe the E2-term of the Adams spectral sequence for the 2-adically compl* *eted homotopy of CP-11in a range in Tables 2.18(a) and (b). Combined with the results from the Atiyah-Hirzebruch spectral sequence, we are also able to determine the differentials that land in homotopical degree t - s 20 in this spectral sequen* *ce. The details of this computation will be applied in Chapter 5, where Adams filtr* *ation and sparseness in the Adams spectral sequence will make it easier for us to stu* *dy the homotopy type of A(*) (and Wh Diff(*)) in terms of its spectrum cohomology and the differentials in its Adams spectral sequence, rather than by means of t* *he long exact sequences in homotopy arising from Dundas' homotopy cartesian square. In Chapter 3 we familiarize ourselves with the spectrum hofib(trc) defined a* *s the homotopy fiber of the (implicitly 2-completed) map trcZ:K(Z) ! T C(Z) : By Dundas' theorem this is also the homotopy fiber of the map trc*:A(*) ! T C(** *). The principal result is Theorem 3.13, which expresses this common homotopy fiber as the homotopy fiber of the spectrum map ffi :-2ku -! 4ko given as a suitably connected cover of the explicit composite map 4r O fi-2 O ( 3 - 1) O fi-1 :-2KU -! 4KO : From this description it is easy to extract other homotopical information about hofib(trc), such as its homotopy groups (Corollary 3.16), its spectrum cohomolo* *gy (Theorem 4.4), or its Adams spectral sequence (Tables 3.18(a) and (b)). The calculations in Chapter 3 are based on the spectrum level description of K(Z[1_2]) given in Theorem 3.4, and of K(Q2) given in Theorem 3.6, which were obtained in [RW] and [R5, 8.1] respectively. The calculation of K(Z[1_2]) relie* *d on the proven Lichtenbaum-Quillen conjecture in this case [RW], using essential in- puts from algebraic geometry [Voe] and [BlLi], while the identification of K(Q2* *) in 6 JOHN ROGNES [R5] amounted to the calculation of T C(Z) completed at 2, which used topologic* *al cyclic homology and calculational spectral sequence techniques from stable homo- topy theory. The results in Chapter 3 also rely on knowing how the natural map j0:K(Z[1_2]) ! K(Q2) acts on the level of homotopy groups, which was determined in [R5, 7.7 and 9.1]. Those results depended on knowing the structure of the K- theory spectra involved, not just their homotopy groups, and were feasible beca* *use the prime 2 is so small, or perhaps because it is regular. These inputs allow us to obtain a spectrum level description of the homotopy fiber of j0in Propositions 3.10 and 3.11, with a more convenient reformulation * *given in Proposition 3.12. The arguments rely on knowing the endomorphism algebras of the 2-completed connective topological K-theory spectra ko and ku, as well as a* *ll the maps between them, which stems from [MST]. Using Quillen's localization se- quence in algebraic K-theory, and Hesselholt and Madsen's link between K(Z2) and T C(Z) from [HM, Thm. D], we rework the description of hofib(j0) into a spectrum level description of hofib(trc) in Theorem 3.13, as desired. In Chapter 4 we use the cofiber sequence (3.14) CP-11-i!hofib(trc) -j!Wh Diff(*) ; and the splitting A(*) ' 1 S0 _ Wh Diff(*), to reduce the identification of A(*) to that of CP-11, which was studied in Chapter 2, to that of hofib(trc), which * *was settled in Chapter 3, and the map i between the two. At the prime 2 we are in the fortunate situation that the mod 2 spectrum cohomology of CP-11is cyclic as an A-module on a generator in degree -2, so because Wh Diff(*) is 2-connected it follows that i induces a surjection on cohomology in all degrees. Thus we can o* *mit any discussion of the linearization map L: T C(*) ! T C(Z) in Dundas' homotopy cartesian square, and still obtain a complete cohomological description of Wh D* *iff(*). This is achieved in the main Theorem 4.5. We have an isomorphism of left graded A-modules H*spec(A(*); F2) ~=H*spec(1 S0; F2) H*spec(Wh Diff(*); F2) where H*spec(1 S0; F2) = F2 is the trivial A-module in dimension zero, and there is a unique nontrivial extension of left graded A-modules -2C=A(Sq1; Sq3) -! H*spec(Wh Diff(*); F2) -! 3A=A(Sq1; Sq2) characterizing H*spec(Wh Diff(*); F2). Here C A is the annihilator ideal of t* *he generator for H*spec(CP-11; F2), introduced in Definition 2.14. The assertion o* *f the theorem is that abstractly there are precisely two such extensions of left grad* *ed A-modules, and H*spec(Wh Diff(*); F2) is the one which does not split. In Chapter 5 we turn to a homotopical analysis of the smooth Whitehead spec- trum Wh Diff(*), and thus also of A(*). Our approach is to study the Adams spec* *tral sequence (5.5) Es;t2= Exts;tA(H*spec(Wh Diff(*); F2); F2) =) sst-s(Wh Diff(*))^2: Here we can in principle compute the E2-term in a large range of bidegrees, but there will be many families of differentials and a complete determination of the homotopy groups of Wh Diff(*) is out of reach. TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 7 The cofiber sequence (3.14) displayed above has the special property that its connecting map induces the zero map in mod 2 spectrum cohomology, so its as- sociated long exact sequence breaks up into short exact sequences, which in turn induce long exact sequences of ExtA -groups. Thus the E2-terms of the Adams spectral sequences for CP-11, hofib(trc) and Wh Diff(*) are linked in a long ex* *act sequence (5.6). The spectral sequence for hofib(trc) was completely described * *in Chapter 3, and in Chapter 5 we use the long exact sequence of E2-terms to trans* *late the information from Chapter 2 about differentials in the Adams spectral sequen* *ce for CP-11to information about differentials in the Adams spectral sequence (5.5) for Wh Diff(*). This is a convenient approach, because the Adams spectral seque* *nce of hofib(trc) is concentrated above the line t - s = 2s + 3, while the differen* *tials in the spectral sequence for CP-11mostly originate below this line. The only subtle point concerns whether certain h1-divisible classes in bidegrees (s; t) = (4k; * *12k+3) of (5.5) are hit by differentials, but a comparison with [R5, 9.1] reveals that* * they indeed survive to the E1 -term. Thus the complexity of determining the homotopy groups of Wh Diff(*) is in practice equivalent to that of determining the homot* *opy groups of CP-11, which is a well-explored but not exhaustively analyzed subject. The Adams E2-term for Wh Diff(*) is displayed in part in Tables 5.7(a) and (* *b), and the nonzero differentials landing in homotopical dimension t - s 21 are li* *sted in Proposition 5.9. This leads to a calculational conclusion in Theorem 5.10, w* *here the 2-completed homotopy groups of Wh Diff(*) are listed in dimensions * 18, a* *nd up to group extensions in dimensions 19 * 21. Previously only the homotopy groups in dimensions 3 were known, see [BW]. We do not give names to the classes identified in ss*(Wh Diff(*)), but in Theorem 7.5 we show that the (spa* *ce level) Hatcher-Waldhausen map hw :G=O ! Wh Diff(*) constructed in [Wa3, x3] induces an isomorphism on 2-primary homotopy groups in dimensions * 8, and an injection on 2-primary homotopy groups in dimensions * 13. Thus the better known homotopy groups of G=O ' BSO x CokJ account for much of the low-dimensional homotopy of Wh Diff(*). In Chapter 6 we use the known spectrum level description of K(Z) completed a* *t 2 to compute its mod 2 spectrum cohomology in Theorem 6.4, and to show in Corol- lary 6.8 that the linearization map L: A(*) ! K(Z) induces the zero map in mod 2 spectrum cohomology in positive dimensions. Thus the linearization map does not itself provide a good cohomological approximation to A(*). In Remark 6.9 we ex- plain why the Hatcher-Waldhausen map hw does not admit a four-fold delooping, using that multiplication by the Hopf map oe 2 ssS7is nonzero on ss4(Wh Diff(*)* *), but is zero on ss4(G=O). We also explain how this relates to the results of [R* *1], where an infinite loop map from G=O to a different infinite loop space structur* *e on Wh Diff(*) is obtained. Following Miller and Priddy [MP], we describe in (6.3) a spectrum g=o as the homotopy fiber of the 2-completed unit map 1 S0 ! K(Z). Its underlying space G=O has the same 2-adic homotopy type as the usual G=O. Although there is no spectrum map g=o ! Wh Diff(*) inducing a ss3-isomorphism, we construct in Chapter 7 a 2-complete spectrum map M :Wh Diff(*) ! g=o which induces an isomorphism on mod 2 spectrum cohomology in all dimensions * 9. This is a best possible approximation, since the cohomology groups differ in dimension 10. The comparison of Wh Diff(*) with g=o finally allows us to evaluate the Hatche* *r- Waldhausen map on 2-completed homotopy groups in dimensions * 13, leading 8 JOHN ROGNES to the previously cited Theorem 7.5. Acknowledgement. The main part of this work was done in December 1997 during visits to Aarhus and Bielefeld. The author thanks M. B"okstedt, I. Madse* *n, J. Tornehave and F. Waldhausen for helpful discussions and hospitality. 1. Algebraic K-theory and topological cyclic homology We commence by discussing the cyclotomic trace map from algebraic K-theory to topological cyclic homology, and a special case of Dundas' theorem comparing relative algebraic K-theory to relative topological cyclic homology. 1.1. -spaces and S-algebras. Let S* be the category of pointed simplicial sets, and let op be the category of finite pointed sets k+ = {0; 1; : :;:k} based at * *0, and base-point preserving functions. This is the opposite of Segal's category from* * [Se]. Let S* be the category of -spaces, i.e., functors F :op ! S* with F (0+ ) = *. Each -space F naturally extends to a functor F :S* ! S*, which when evaluated on spheres determines a (pre-)spectrum {n 7! F (Sn)}. We write ss*(F ) for the homotopy groups of this spectrum. The natural inclusion op ! S* is a -space denoted S, whose associated spectrum is the sphere spectrum. The groups ss*(S) are the stable homotopy groups of spheres. There is a smash product ^ of -spaces defined by Lydakis in [Ly], making (S*; ^; S) a symmetric monoidal category. A monoid A in this symmetric monoidal category will be called an S-algebra. Its associated spectrum is an associative* * ring spectrum, conveniently thought of as an algebra over the sphere spectrum. 1.2. Examples of S-algebras. When G is a simplicial group the functor 1 (G+ ) given by 1 (G+ )(k+ ) = G+ ^ k+ is a -space. The group multiplication and unit define the structure maps : 1 (G+ ) ^ 1 (G+ ) -! 1 (G+ ) and j :S ! 1 (G+ ) making 1 (G+ ) an S-algebra. Its associated ring spectrum is the unreduced suspension spectrum on G, with product map induced by the multiplication on G. When R is a (discrete) ring the functor HR given by HR(k+ ) = R{1; : :;:k} (the free R-module on the non-basepoint elements in k+ ) is a -space. The ring multiplication and unit define the structure maps : HR ^ HR -! HR and j :S ! HR making HR an S-algebra. Its associated ring spectrum is the Eilenberg-Mac Lane spectrum representing ordinary cohomology with coefficients in R. Let G be a simplicial group, with group of path components ss0(G), and let R = Z[ss0(G)] be the its integral group ring. The linearization map is the map * *of S-algebras L: 1 (G+ ) ! HR taking g ^ i 2 G+ ^ k+ to [g] . i 2 R{1; : :k:}, whe* *re g 2 G, i 2 {1; : :k:} and [g] denotes the path component of g viewed as an elem* *ent of ss0(G) R. TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 9 1.3. Algebraic K-theory, topological Hochschild homology and topolog- ical cyclic homology. Let A be an S-algebra. The extended functor A: S* ! S* comes equipped with a product and unit map making it an FSP (functor with smash product) in the sense of [B2]. In [BHM] B"okstedt, Hsiang and Madsen functorially define the algebraic K-theory spectrum K(A), topological Hochschild homology spectrum T HH(A) and topological cyclic homology spectrum T C(A; p) of an FSP A. Here p is any prime. An integral functor A 7! T C(A) has been defined by Goodwillie (unpublished), together with a natural p-adic equivalence T C(A) ! T C(A; p) for each prime p. When G is a simplicial group and X = BG its classifying space we write A(X) = K(1 (G+ )), T HH(X) = T HH(1 (G+ )) and T C(X; p) = T C(1 (G+ ); p). Here A(X) is naturally homotopy equivalent to Waldhausen's algebraic K-theory spec- trum A(X) of the space X [Wa1], i.e., the algebraic K-theory of the category of finite retractive spaces over X. When R is a ring we write K(R) = K(HR), T HH(R) = T HH(HR) and T C(R; p) = T C(HR; p). Here K(R) is naturally homotopy equivalent to Quillen's algebraic K-theory spectrum K(R) of the ring R [Q2], i.e., the algebraic K-theo* *ry of the category of finitely generated projective R-modules. We recall from [BHM, 3.7] that there are C-equivariant homotopy equivalences (1.4) T HH(X) 'C 1C(X+ ) for each finite subgroup C S1. Here 1C denotes the C-equivariant suspension spectrum, and C S1 acts on the free loop space X by rotating the loops. 1.5. Trace maps. A trace map trX:A(X) ! T HH(X) was defined by Wald- hausen in [Wa2], and B"okstedt defined a trace map trA:K(A) ! T HH(A) in [B2], as a natural transformation of functors from FSPs to spectra. The cyclotomic tr* *ace map trcAof [BHM] gives a factorization K(A) trcA---!T C(A; p) fiA--!T HH(A) of trA, although the map to T C(A; p) was initially only defined up to homotopy. The map fiA is a projection map from the homotopy limit defining T C(A; p). When A = 1 (G+ ) with X = BG or A = HR we substitute X or R, respectively, for A in the notations trcA, fiA and trA. Thus trcX:A(X) ! T C(X; p), etc. In the case A = 1 (G+ ) with X = BG the six authors of [6A] gave a model for the cyclotomic trace map trcXas a natural transformation in X. When A = HR, Dundas and McCarthy [DuMc] gave models for K(R) and T C(R) such that trcR is a natural transformation. Finally Dundas [D2] has provided a construction of functors K, T HH and T C from S-algebras to spectra, and natural transformations trc:K ! T C, fi :T C ! T HH and tr:K ! T HH with tr= fi O trc, which agree up to natural homotopy equivalence with the preceding definitions. 1.6. Dundas' theorem. The following theorem of Dundas [D1] generalizes to maps of S-algebras a theorem of McCarthy [Mc] valid for maps of simplicial ring* *s. Both results are analogous to an older theorem about rational algebraic K-theory due to Goodwillie [Go]. 10 JOHN ROGNES Theorem 1.7 (Dundas). Let OE: A ! B be a map of S-algebras, such that the ring homomorphism ss0(OE): ss0(A) ! ss0(B) is a surjection with nilpotent kerne* *l. Then the commutative square of spectra K(A) ___OE_//K(B) |trcA| trcB|| fflffl|OE fflffl| T C(A) ____//_T C(B) is homotopy cartesian. Corollary 1.8 (Dundas). Let G be a simplicial group, and write X = BG and R = Z[ss1(X)] = Z[ss0(G)]. The linearization map L: 1 (G+ ) ! HR induces a homotopy cartesian square A(X) ___L__//_K(R) trcX|| |trcR| fflffl|L fflffl| T C(X) ____//_T C(R) : In particular, the vertical homotopy fiber hofib(trcX) only depends on the fund* *a- mental group ss1(X), for a pointed connected space X. For the last claim we used that any pointed connected space X is homotopy equivalent to BG for a simplicial group G, e.g. the Kan loop group of X. 1.9. Whitehead spectra. There are natural cofiber sequences of spectra 1 (X+ ) -i!A(X) -! Wh Diff(X) and A(*) ^ X+ -ff!A(X) -! Wh PL(X) where Wh Diff(X) is the smooth Whitehead spectrum of X, and Wh PL(X) is the piecewise linear Whitehead spectrum of X. The sequences are constructed geo- metrically in [Wa3], where Wh Diff(X) is interpreted in terms of stabilized smo* *oth concordance spaces and stabilized spaces of smooth h-cobordisms, and similarly * *in the piecewise linear case. The identification of the upper left hand homology t* *heory in X with 1 (X+ ) uses the `vanishing of the mystery homology theory' establish* *ed in [Wa6]. The composite 1 (X+ ) -i!A(X) trX--!T HH(X) ' 1 (X+ ) ev-!1 (X+ ) is homotopic to the identity. Here ev :X ! X is the map evaluating a free loop S1 ! X at the identity 1 2 S1. Hence ev O trXprovides a natural splitting for t* *he cofiber sequence above, as in A(X) ' 1 (X+ ) _ Wh Diff(X) : We can therefore identify Wh Diff(X) with the homotopy fiber of the splitting m* *ap ev O trX. TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 11 1.10. The smooth Whitehead spectrum of a point. Suppose G = 1, so X = *. Then ev :X ! X is the identity map, T HH(*) ' 1 S0, and the splitting above identifies Wh Diff(*) with the homotopy fiber of tr*. We obtai* *n a map of horizontal cofiber sequences of spectra: tr* Wh Diff(*)____//_A(*)____//T HH(*) |ftrc| trc*|| |||| fflffl| fflffl|fi || gT C(*)_____//_T C(*)___/*/T HH(*) Here gT C(*) is defined as the homotopy fiber of fi*, and ftrcis the induced map of homotopy fibers over trc*and the identity map on T HH(*). The unit map 1 S0 ! A(*) ! T C(*) and fi* yield a splitting T C(*) ' 1 S0 _ gT C(*) : Theorem 1.11. The two squares Wh Diff(*)____//A(*)__L__//K(Z) ftrc|| |trc*| trcZ|| fflffl| fflffl| fflffl| gT C(*)_____//_T C(*)L_//_T C(Z) are homotopy cartesian, and induce homotopy equivalences of vertical homotopy fibers hofib(ftrc) '-!hofib(trc*) '-!hofib(trcZ) : We denote either of these by hofib(trc). 1.12. The topological cyclic homology of a point. The topological cyclic homology T C(X; p) of a pointed connected space X was computed by B"okstedt, Hsiang and Madsen in [BHM]. We recall their result, making precise a point that was omitted in the published argument. See [Ma, x4.4] for more details about the following review. Fix a prime p. From 1.4 there is an equivalence T HH(X)Cpn ' 1Cpn(X+ )Cpn for each n 0. The Segal-tom Dieck splitting Yn 1Cpn(X+ )Cpn ' 1 (ECpkxCpk XCpn-k)+ k=0 and the power map homeomorphisms n-kp:X ~=XCpn-k combine to give an equivalence Yn (1.13) T HH(X)Cpn ' 1 (ECpkxCpk X)+ : k=0 12 JOHN ROGNES The pth power map p: 1 X+ ! 1 X+ is induced by taking a free loop S1 ! X to its precomposition by the usual degree p map S1 ! S1. Let tp: 1 (ECpn xCpn X)+ ! 1 (ECpn-1xCpn-1X)+ be the Becker-Gottlieb transfer for the principal Cp-bundle ECpn-1xCpn-1X ! ECpn xCpn X. There are restriction and Frobenius maps R; F :T HH(X)Cpn ! T HH(X)Cpn-1. Up to homotopy these are given by the formulas: R(x0; x1; : :;:xn)= (x0; x1; : :;:xn-1) F (x0; x1; : :;:xn)= (p(x0) + tp(x1); tp(x2); : :;:tp(xn)) : Here xk refers to the factor in 1 (ECpkxCpk X)+ in the equivalence 1.13, and the formulas must be interpreted as giving maps defined in terms of this splitt* *ing. Writing 1Y (1.14) T R(X; p) = holimn;RT HH(X)Cpn ' 1 (ECpn xCpn X)+ n=0 we have R(x0; x1; x2; : :):= (x0; x1; x2; : :):and F (x0; x1; x2; : :):= (p(x0)* * + tp(x1); tp(x2); tp(x3); : :):up to homotopy. The topological cyclic homology sp* *ec- trum T C(X; p) is defined as the homotopy equalizer __R__// T C(x; p)_ss_//T R(X; p)___//T R(X; p) ; F and is homotopy equivalent to the homotopy fiber of 1-F :T R(X; p) ! T R(X; p). Let T; D :T R(X; p) ! T R(X; p) be given up to homotopy by the formulas: T (x0; x1; x2; :=:):(tp(x1); tp(x2); tp(x3); : :): D(x0; x1; x2; :=:):(p(x0); 0; 0; : :):: The following observation allows us to calculate T C(X; p). Lemma 1.15. The composite (1-T )O(1-D): T R(X; p) ! T R(X; p) is homotopic to (1 - F ). Proof. In terms of the splitting 1.14, it is clear that (1 - D)(x0; x1; x2; : :* *):= (x0 - p(x0); x1; x2; : :):is mapped by (1 - T ) to (x0 - p(x0) - tp(x1); x1 - tp(x2); x2 - tp(x3); : :):, which is homotopic to (1 - F )(x0; x1; x2; : :):. Given such a choice of commuting homotopy for the right hand square below, there is an induced map of horizontal fiber sequences T C(X; p)__ss//_T R(X;_p)1-F//_T R(X; p) |ffX| |1-D| |||| fflffl| fflffl|1-T || C(X; p)_____//_T R(X;_p)__//T R(X; p) : TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 13 Here we have written C(X; p) for the homotopy limit holimn;tp1 (ECpn xCpn X)+ , which is homotopy equivalent to the homotopy fiber of 1-T in view of (1.1* *4). When ffX is determined by the commuting homotopy, the left hand square is stric* *tly commutative and homotopy cartesian. Let pr :T R(X; p) ! T HH(X) ' 1 X+ denote projection to the zeroth term in the homotopy limit defining T R(X; p). Then there is clearly a commuting and homotopy cartesian square T R(X; p)_pr__//_1 X+ |1-D| |1-p| fflffl|pr fflffl| T R(X; p)____//_1 X+ : We can combine these two homotopy cartesian squares horizontally. Then the upper composite fiX = pr O ss :T C(X; p) ! T R(X; p) ! T HH(X) ' 1 X+ agrees with the natural transformation fi of 1.5. The lower composite is the projecti* *on pr0: C(X; p) ! 1 X+ from the homotopy limit system over the Becker-Gottlieb transfer maps to its zeroth term. Theorem 1.16. [BHM, x5] Let X be a pointed connected space and write C(X; p) = holimn;tp1 (ECpn xCpn X)+ . The diagram T C(X; p)ffX_//_C(X; p) fiX|| |pr0| fflffl|1-p fflffl| 1 X+ _____//_1 X+ homotopy commutes, and there exists a commuting homotopy making the diagram homotopy cartesian. This is now clear. (The proofs in [BHM] and [Ma] only show that the horizont* *al homotopy fibers in this diagram are homotopy equivalent, not necessarily by the map induced by fiX and pr0.) Specializing to X = * we have the following coroll* *ary, which is what we will use in the rest of this paper. Corollary 1.17. There is a cofiber sequence of spectra gT C(*; p) -! holim1 (BCpn+) pr0--!1 S0 : n;tp For each n 0 there is a dimension-shifting S1-transfer map trfnS1:1 ((CP+1)) -! 1 (BCpn+) associated to the S1-bundle BCpn ! BS1 ' CP 1. See [K], [LMS] or [Mu1]. These induce a map 1 ((CP+1)) -! holimn;t1 (BCpn+) p which is a homotopy equivalence after p-adic completion. Hence we can identify * *the map pr0 above with the S1-transfer map trf0S1, briefly denoted trfS1, after p-a* *dic completion. Combined with the p-adic equivalence T C(*) ! T C(*; p) we obtain: 14 JOHN ROGNES Corollary 1.18. [BHM, 5.15] There is a homotopy equivalence TgC(*) ' hofib(trfS1:1 ((CP+1)) ! 1 S0) after p-adic completion, for each prime p. 1.19. A Thom spectrum. Let CPk1 denote the truncated complex projective space with one cell in each even dimension greater than or equal to 2k, interpr* *eted as a spectrum when k < 0. There is a homotopy equivalence CPk1 ~=T h(kfl1) where the right hand side is the Thom spectrum of k times the canonical complex line bundle over CP 1, see [At]. We shall be concerned with the case k = -1, i.* *e., with the spectrum CP-11, which can be thought of as the Thom spectrum of minus the canonical line bundle on CP 1. Theorem 1.20 (Knapp). There is a homotopy equivalence CP-11' hofib(trfS1:1 ((CP+1)) ! 1 S0) : See [K, 2.14] for a proof. Bringing these results together we have shown: Corollary 1.21. There is a homotopy equivalence (CP-11)^p' gT C(*)^p of p-adically completed spectra. 2. Two-primary homotopy of CP-11 In this chapter we study the 2-primary homotopy type of the Thom spectrum CP-11of minus the canonical complex line bundle over CP 1. We first use a rein- dexed Atiyah-Hirzebruch spectral sequence for stable homotopy to compute the 2-completed homotopy groups ss*(CP-11)^2in dimensions * 20, and next com- pare with the Adams spectral sequence with the same abutment to determine the differentials in the latter spectral sequence in the same range of dimensions. The reindexed Atiyah-Hirzebruch spectral sequence in question is derived from the stable homotopy exact couple associated to the filtration of CP-11by the su* *b- spectra CP-s1, for s -1. Its E1-term is (2.1) E1s;t= sss+t(CP-s1=CP-s-11) ~=ssSt-s for s -1, and zero elsewhere. Here ssSk= ssk(1 S0) is the kth stable stem. To determine the differentials in the reindexed Atiyah-Hirzebruch spectral s* *e- quence, we compare with the computation by Mosher [Mo] of the differentials in * *the corresponding spectral sequence for the stable homotopy of CP 1. The E1-term of the latter spectral sequence is obtained from (2.1) by restricting to filtra* *tions s 1, i.e., by omitting the columns s = -1 and s = 0, and the collapse map j :CP-11! CP 1 induces a map of spectral sequences. From here on we often use the same notation for a based space and its suspension spectrum, such as writing S0 for 1 S0. TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 15 The differentials in (2.1) landing in filtration s = 0 are always zero, due * *to the splitting CP01 = CP+1 ' CP 1 _ S0. The differentials in (2.1) landing in filtra* *tion s = -1 arise from the connecting map in the cofiber sequence S-2 ! CP-11! CP+1. This is the wedge sum of the (desuspended) S1-transfer map CP 1 ! S-1, and the (desuspended) multiplication by j map S0 ! S-1. The image of the S1- transfer map was computed in dimensions * 20 by Mukai in [Mu1], [Mu2] and [Mu3], and we use these results to determine the differentials in (2.1) landing* * in filtration s = -1 in the same range of dimensions. For ease of reference we use similar notation for classes in our spectral se* *quence (2.1) as in [Mo]. Thus we write fis 2 E1s;s+tfor the class corresponding to fi * *2 ssSt, and write Z=n(fi) for a cyclic group of order n with generator fi. In Tables 2* *.5 and 2.12 we briefly write n(fi) for Z=n(fi) and (fi) for Z(fi), to save some sp* *ace. Hereafter we concentrate on the 2-primary components, and all spectra and groups are implicitly 2-completed. Differentials are mostly given only up to multiplic* *ation by a 2-adic unit. In dimensions * 22, we will use the following presentation for the stable s* *tems ssS*, following the tables in [To, XIV] and [Ra, A3.3]. ssS0 = Z(), ssS1 = Z=2(j), ssS2 = Z=2(j2), ssS3 = Z=8(), ssS4 = 0, ssS5 = 0, ssS6= Z=2(2), ssS7= Z=16(oe), ssS8= Z=2( ) Z=2(ffl), ssS9= Z=2(3) Z=2(jffl) Z=2(), ssS10= Z=2(j), ssS11= Z=8(i), ssS12= 0, ssS13= 0, ssS14= Z=2(oe2) Z=2(), ssS15= Z=32(ae) Z=2(j), ssS16= Z=2(j*) Z=2(jae), ssS17= Z=2(jj*) Z=2() Z=2(j2ae)Z=2( ), ssS18= Z=8(*)Z=2(j ), ssS19= Z=2(oe)Z=8(i), ssS20= Z=8( ), ssS21= Z=2(*) Z=2(j ) and ssS22= Z=2(oe) Z=2(j2 ). For a fixed r, the dr-differentials in the spectral sequence for ssS*(CP 1) * *are periodic in the filtration degree s, see [Mo, 4.4], and this periodicity propag* *ates to the spectral sequence (2.1). Hence Mosher's description of the d1-, d2- and * *d3- differentials for CP 1 in [Mo, 5.1, 5.2, and 5.4] extends to give the formulas * *2.2, 2.3 and 2.4 for the corresponding differentials in (2.1). Let fi 2 ssS*. Proposition 2.2. d1(fis) = 0 for s odd and d1(fis) = jfis-1 for s even. Proposition 2.3. d2(fis) = fis-2 for s 0; 1; 4; 5 mod 8, d2(fis) = 2fis-2 for s 3; 6 mod 8 and d2(fis) = 0 for s 2; 7 mod 8. Proposition 2.4. d3(fis) = 0 for s odd. If s is even then d3(fis) = fls-3, where fl 2 for s 0 mod 8, fl 2 <; j; fi> for s 2 mod 8, fl 2 <2; j; fi> +* * for s 4 mod 8 and fl 2 <; j; fi> + for s 6 mod 8. The d1-differentials in (2.1) are given by the following multiplicative rela* *tions in ssS*, see [Ra] and [To]. j . = j, j . j = j2, j . j2 = 4, j . = 0, j . 2 = 0, j . oe = + ffl, j . * * = 3, j . ffl = jffl, j . 3 = 0, j . jffl = 0, j . = j, j . j = 4i, j . i = 0, j . o* *e2 = 0, j . = j, j . ae = jae, j . j = 0, j . j* = jj*, j . jae = j2ae, j . jj* = 4*, j . = 0, * *j . j2ae = 0, j . = j , j . * = 0, j . j = 4i, j . oe= 0, j . i= 0, j . = j , j . * = 0 a* *nd j . j = j2 . For example, oe= <; joe; oe>, so j . oe= -oe = 0 with zero indeter* *minacy. The d2-differentials in (2.1) are given by the following multiplicative rela* *tions in ssS*, see [Ra] and [To]. . = , . = 2, . 2 = 3, . oe = 0, . = 0, . 3 = 0, . jffl = 0, . = 0, . i = 0, . oe2 = 0, . = , . ae = 0, . j = 0, . j* = 0, . = 4 , . j2ae = 0, . = 0, . * = *, . j = 0, . oe= oeand . i= 0. 16 JOHN ROGNES The d3-differentials are given by the following secondary compositions, from [MT], [Mo, 10.1] and [To]. <; j; > = , = = {ffl; }, <; j; i> {0; jae}, = * *{0; jae}, <; j; oe2> = oe, <; j; 2ae> = {0; 4 } and <; j; j> = 2 by [MT]. The resulting E4-term is shown in Table 2.5, accounting for all differentials landing in total degree s + t 20. In lemmas 2.6 to 2.11, we only consider differentials landing in total degree s + t 20. Lemma 2.6. The nonzero d4-differentials in (2:1) are d4(23) = 2oe-1, d4(45) = 8oe1, d4(46) = 8oe2, d4(7) = 2oe3, d4(88) = 8oe4, d4(49) = 4oe5, d4(210) = 2oe6* * and d4(oe3) = oe2-1. Proof. The d4-differentials landing in filtration s 1 and total degree s + t * *19 are determined by those in the spectral sequence for ssS*(CP 1), and are given * *in [Mo, 5.6 and 6.4]. In total degree 20, d4(i5) = 0 by the computation of ssS20(CP 5) following [* *Mu3, 4.2], and d4(oe7) = 0 by the proof of [Mu3, 4.3] (the formula fl6oe = 2i"oe0oe). The differentials landing in filtration s = 0 are always zero, as noted abov* *e. The differentials landing in filtration s = -1 are determined by the computation of the S1-transfer in [Mu1] and [Mu2]. Thus d4(23) = 2oe-1 by [Mu1, 13.1(iii)], d4(oe3) = oe2-1by the proof of [Mu2, 5.3] (the formula g4"oe0= oe2), d4(3 ) = 0* * by the proof of [Mu2, 5.3] (the formula g8i" = j), d4(3) = 0 by the proof of [Mu2, 5.4] (the formula g4" = 0), and d4(i3) = 0 by the proof of [Mu2, 5.5] (the form* *ula g4ssS17(CP 3) = 0). Lemma 2.7. The nonzero d5-differentials in (2:1) are d5(86) = 1, d5(168) = 3 and d5(1610) = 5. Proof. The d5-differentials landing in filtration s 1 and total degree s + t * *19 are determined by those in the spectral sequence for ssS*(CP 1), and are given * *in [Mo, 6.5]. In total degree 20, d5(jffl6) = 0 by the calculation of ssS20(CP 6) followin* *g [Mu3, 4.2]. The differentials landing in filtration s = -1 are d5(84) = 0 by [Mu1, 13.1(* *iv)], 0 5 3 d5(2oe4) = 0 by the proof of [Mu2, 5.4] (the formula g5f2oe 0 mod oe), d (4) = 0 and d5(jffl4) = 0 by the proof of [Mu2, 5.5] (the formulas g5"3 0 mod {4*; j } and g5 = 0, where was chosen as a coextension of j2oe before [Mu2, 4.7]), and d5(i4) = 0 by [Mu3, 5.1]. Lemma 2.8. The nonzero d6-differentials in (2:1) are d6(85) = i-1, d6(87) = 2i1, d6(328) = 2i2, d6(169) = i3, d6(3210) = 4i4, d6(25) = -1, d6(oe5) = *-1 and d6(oe7) = 2*1. Proof. The differentials landing in filtration s 1 and total degree s + t 19 come from [Mo, 6.6], and d6(oe7) = 2*1by [Mu3, 4.3] and its proof (namely, fl6o* *e = 2i"oe0oe = 2i*). Also d6(85) = i-1 by [Mu1, 13.1(v)], d6(25) = -1 by the proof of [Mu2, 5.3] 000 2 6 2 (the formula g8(if2 ) mod oe ), d (5) = 0 by the proof of [Mu2, 5.4] (the formula g9i"2 !*j mod i), and d6(oe5) = *-1by the proof of [Mu2, 5.5] (the formula g6"oe00 x* mod j where x is odd). TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 17 ____________________________________________________________________________|||* *|||||||||| |___0___|______|_____|______|_____|_____|_____|______|_____|_____|____|____|_||* *||||||||||| |___0___|4(20_)|_____|______|_____|_____|_____|______|_____|_____|____|____|_||* *|*|||||||||| |2(-1|) |2(oe0)|2(21)|||| || || || || || || || || || || || || || || || || || || || || || || |_______|8(i0)_|_____|______|_____|_____|_____|______|_____|_____|____|____|_||* **||||||||||| |4(i-1)_|8(0)__|2(1_)|__0___|32(ae3)|___|_____|______|_____|_____|____|____|_|** *||*||2|||||||| |4(-1)| |2(0)| |2(j1)|16(2ae2)|2(oe3)|0|||||| || || || || || || || || 2 || || || || || || || || || || || |_______|2(j_ae0)|___|______|2(3)_|_____|_____|______|_____|_____|____|____|_||* *||||||||||| |2(-1__)|__0___|32(ae1)|0___|__0__|__0__|4(i5)|______|_____|_____|____|____|_|** *||2|||||||||| |2(j-1)||16(2ae0)|2(oe1)|0|||||0 |8(i4)|||0 |2(jffl6)||| || || || || || || || || || || || || || || || || || |_______|2(j0)_|_____|______|_____|_____|_____|______|_____|_____|____|____|_||* *2||||||||||| |32(ae-1)|2(oe0)|0___|__0___|4(i3)|__0__|2(5)_|__0___|16(oe7)|___|____|____|_|2* *|||||3||||||| |2(oe-1)|||0 ||0 |8(i2)||| 0 |2(4)||| 0 |8(2oe6)|0|| ||0 || || || || || || || || || || || || || || || || |2(-1)__|______|_____|______|_____|2(jffl4)|__|______|_____|_____|____|____|_||* *||||||||||| |___0___|__0___|4(i1)|__0___|2(3)_|__0__|16(oe5)|0___|_0___|_0___|_0__|____|_||* *|||||2|||||| |___0___|8(i0)_|_0___|2(jffl2)|2(3|)8(2oe4)|2(5)|0___|_0___|2(48)|_0__|_0__|_||* *||||||||||| |4(i-1)_|__0___|2(1)_|__0___|16(oe3)|0__|__0__|__0___|_0___|_0___|_0__|(210)|_|* *|3||||||||||| || 0 |2(0)| ||0 |8(2oe2)||0| || 0 || 0 |2(6)| ||0 ||0 |(49)||| || || || || || || || || || || || || || || |_______|2(jffl0)|___|______|_____|_____|_____|______|_____|_____|____|____|_||* *||||||||||| |2(-1)__|__0___|16(oe1)|0___|__0__|__0__|2(25)|__0___|_0___|(88)_|____|____|_||* *|2|||||||||| |___0___|8(2oe0)|2(1)|__0___|__0__|__0__|__0__|__0___|(7)__|_____|____|____|_||* *2||||||||||| |16(oe-1)|2(0)_|_0___|__0___|__0__|__0__|__0__|_(46)_|_____|_____|____|____|_||* *||||||||||| |___0___|__0___|_0___|__0___|__0__|__0__|(45)_|______|_____|_____|____|____|_||* *||||||||||| |___0___|__0___|_0___|__0___|__0__|(84)_|_____|______|_____|_____|____|____|_||* *||||||||||| |___0___|8(0)__|_0___|__0___|(23)_|_____|_____|______|_____|_____|____|____|_||* *||||||||||| |___0___|__0___|_0___|(22)__|_____|_____|_____|______|_____|_____|____|____|_||* *||||||||||| |___0___|__0___|(41)_|______|_____|_____|_____|______|_____|_____|____|____|_||* *||||||||||| |___0___|(20)__|_____|______|_____|_____|_____|______|_____|_____|____|____|_||* *||||||||||| |_(-1)__|______|_____|______|_____|_____|_____|______|_____|_____|____|____|_ Table 2.5. E4 in total degrees s + t 20. Lemma 2.9. The only nonzero d7-differential in (2:1) is d7(6) = j*-1. Proof. We have d7(6) = j*-1by [Mu2, 5.4] and its proof (the formula g7"00= !*). All other d7-differentials are zero by [Mo, 6.7] or bidegree reasons. 18 JOHN ROGNES Lemma 2.10. The nonzero d8-differentials in (2:1) are d8(167) = 2ae-1, d8(649) = 16ae1 and d8(6410) = 16ae2. Proof. These follow from [Mu1, 4.3] since 2ae generates the complex image of J * *in dimension 15, and from [Mo, 6.8]. Lemma 2.11. The remaining nonzero differentials in (2:1) are d9(2710) = 1 and d10(279) = i-1. Proof. These follow from [Mo, 6.9] and [Mu1, 4.3], since i generates the complex image of J in dimension 19. This leaves us with the E1 -term shown in Table 2.12, in total degrees s+t * *20. Recall the convention that n(fi) denotes a cyclic group of order n, generated b* *y fi. Theorem 2.13. The 2-primary homotopy groups of CP-11in dimensions * 20 are as follows: ss-2(CP-11)= Z(-1); ss-1(CP-11)= 0; ss0(CP-11)= Z(20); ss1(CP-11)= 0; ss2(CP-11)= Z(41); ss3(CP-11)= Z=8(0); ss4(CP-11)= Z(22); ss5(CP-11)= Z=2(oe-1); ss6(CP-11)= Z=2(20) Z(163); ss7(CP-11)= Z=2(-1) o Z=8(2oe0) ~=Z=16(2oe0); ss8(CP-11)= Z=2(21) Z(84); ss9(CP-11)= Z=2(30) Z=2(jffl0) Z=8(oe1); ss10(CP-11)= Z(325); ss11(CP-11)= Z=8(i0) Z=4(2oe2); ss12(CP-11)= Z(166); ss13(CP-11)= Z=2(ae-1) o Z=2(i1) o Z=2(jffl2) ~=Z=2(ae-1) o Z=4(jffl2); ss14(CP-11)= Z=2(oe20) Z=2(3 ) Z(287); ss15(CP-11)= Z=2(-1 ) o Z=16(2ae0) Z=2(j0) o Z=2(i2) o Z=4(2oe4) ~=Z=32(2ae0) Z=2(j0) o Z=2(i2) o Z=4(2oe4); ss16(CP-11)= Z=2(oe21) Z=2(25) Z(278); ss17(CP-11)= Z=2(0) Z=2(j2ae0) Z=16(ae1) o Z=2(34) Z=2(jffl4) ~=Z=2(0) Z=2(j2ae0) Z=32(jffl4) Z=2(34); ss18(CP-11)= Z=2(-1 ) o Z=8(*0) o Z=2(j*1) Z(299); ss19(CP-11)= Z=2(oe0) Z=8(i0) Z=8(2ae2) o Z=4(i4) o Z=2(48) TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 19 __________________________________________________________________________|||||* *|||||||| |__0___|______|_____|_____|_____|_____|_____|_____|____|_____|_____|_____|_||||* *||||||||| |__0___|4(20_)|_____|_____|_____|_____|_____|_____|____|_____|_____|_____|_||||* *||||||||| |2(-1|)|2(oe0)||0| || || || || || || || || || || || || || || || || || || || || || || || |______|8(i0)_|_____|_____|_____|_____|_____|_____|____|_____|_____|_____|_||*|* *|||||||||| |__0___|8(0)__|_0___|__0__|_____|_____|_____|_____|____|_____|_____|_____|_|||** *||2|||||||| || 0 |2(0)| |2(j1)|8(2ae2)|2(oe3)||||||| || || || || || || || || 2 || || || || || || || || || || || |______|2(j_ae0)|___|_____|2(3)_|_____|_____|_____|____|_____|_____|_____|_||||* *||||||||| |2(-1__)|_0___|16(ae1)|0__|_0___|_0___|_____|_____|____|_____|_____|_____|_|||2* *|||||||||| || 0 |16(2ae0)|2(oe1)|0|||0|| |4(i4)||0| || || || || || || || || || || || || || || || || || || || |______|2(j0)_|_____|_____|_____|_____|_____|_____|____|_____|_____|_____|_||2|* *|||||||||| |2(ae-1)|2(oe0)|0___|__0__|_0___|_0___|_0___|_0___|____|_____|_____|_____|_||||* *||3||||||| || 0 || 0 ||0 |2(i2)||0| |2(4)|||0 ||0 ||0 || || || || || || || || || || || || || || || || || |______|______|_____|_____|_____|2(jffl4)|__|_____|____|_____|_____|_____|_||||* *||||||||| |__0___|__0___|2(i1)|__0__|_0___|_0___|_0___|_0___|_0__|__0__|_____|_____|_||||* *|||2|||||| |__0___|8(i0)_|_0___|2(jffl2)|2(3|)4(2oe4)|2(5)|0_|_0__|2(48)|__0__|_____|_||||* *||||||||8| |__0___|__0___|_0___|__0__|_0___|_0___|_0___|_0___|_0__|__0__|__0__|(2_10)|_||3* *|||||||||9|| || 0 |2(0)| ||0 |4(2oe2)|0||||0 ||0 ||0 ||0 || 0 |(2|9)|| || || || || || || || || || || || || || || |______|2(jffl0)|___|_____|_____|_____|_____|_____|____|_____|_____|_____|_||||* *||||||7||| |2(-1)_|__0___|8(oe1)|_0__|_0___|_0___|_0___|_0___|_0__|(2_8)|_____|_____|_|||2* *||||||8|||| |__0___|8(2oe0)|2(1)|__0__|_0___|_0___|_0___|_0___|(2_7)|____|_____|_____|_||2|* *|||||||||| |2(oe-1)|2(0)_|_0___|__0__|_0___|_0___|_0___|(166)|____|_____|_____|_____|_||||* *||||||||| |__0___|__0___|_0___|__0__|_0___|_0___|(325)|_____|____|_____|_____|_____|_||||* *||||||||| |__0___|__0___|_0___|__0__|_0___|(84)_|_____|_____|____|_____|_____|_____|_||||* *||||||||| |__0___|8(0)__|_0___|__0__|(163)|_____|_____|_____|____|_____|_____|_____|_||||* *||||||||| |__0___|__0___|_0___|(22)_|_____|_____|_____|_____|____|_____|_____|_____|_||||* *||||||||| |__0___|__0___|(41)_|_____|_____|_____|_____|_____|____|_____|_____|_____|_||||* *||||||||| |__0___|(20)__|_____|_____|_____|_____|_____|_____|____|_____|_____|_____|_||||* *||||||||| |(-1)__|______|_____|_____|_____|_____|_____|_____|____|_____|_____|_____|_ Table 2.12. E1 in total degrees s + t 20. ~=Z=2(oe0) Z=8(i0) Z=64(48); ss20(CP-11)= Z=4(20 ) o Z=2(oe23) Z=2(3) Z(2810): 20 JOHN ROGNES Proof. Up to extensions, this can be read off from the E1 -term above. In dimensions * = 9; 11; 14; 17; 19 the subgroup in filtration s = 0 is spli* *t off by the composite map CP-11! CP+1 ! S0, followed by a retraction of ssS*onto the kernel of j :ssS*! ssS*+1. The extension in dimension 7 will follow from the proof of 2.21 below, in vi* *ew of h0-multiplications in the Adams spectral sequence for ss*(CP-11). The right hand extension in dimension 13 can be read off from ssS13(CP 2) ~=Z=8(jffl2) Z=2(32); see [Mu2, p.197]. The left hand extension in dimension 15 can be read off from ssS19(CP 2) ~= ssS15(CP-01), see [Mu3, p.133]. The splitting in dimension 16 can be deduced from the injection ss16(CP-11) ! ss16(CP 1) ~=(Z=2)3 Z, see [Mu2, 1(ii)]. The right hand extension in dimension 17 can be read off from ssS17(CP 4), s* *ee [Mu2, 4.7 and 4.8]. Note that j2oe = 3 + jffl, so twice the coextension of j2o* *e is twice a coextension of jffl. The middle and right hand extensions in dimension 19 follow from [Mu3, 3.2]. We proceed to compare these results with the Adams spectral sequence for ssS*(CP-11)^2. Let A = A(2) be the mod 2 Steenrod algebra, generated by the Ste* *en- rod squaring operations Sqi. For each sequence of natural numbers I = (i1; : :;* *:in) let SqI = Sqi1O . .O.Sqin be the composite operation. The sequence I, or the operation SqI, is said to be admissible if is 2is+1 for all 0 s < n. The set * *of admissible SqI form a vector space basis for A. Definition 2.14. Let C be the left ideal in A with vector space basis the set of admissible SqI such that I = (i1; : :;:in) has length n 2, or I = (i) with i o* *dd. Then A=C is a cyclic left A-module, with vector space basis the set of Sqi with i 0 even. Let us briefly write H*(X) for the mod 2 spectrum cohomology H*spec(X; F2) of a spectrum X. It is naturally a graded left A-module. Proposition 2.15. H*(CP-11) ~=-2A=C as graded left A-modules. Proof. It is clear that Hn(CP-11) ~= F2 for n -2 even, and 0 otherwise. In H*(1 CP+1) ~=F2{yj | j 0} with deg(y) = 2 the squaring operations are given by Sq2i-1(yj) = 0 and Sq2i(yj) = jiyi+j. By James periodicity and stability of the squaring operations the same formulas apply in H*(CP-11) ~=F2{yj | j -1} ; -1 also with j = -1. Then Sq2i(y-1) = yi-1 since i 1 mod 2. To prove the proposition it remains to show that SqI(y-1) = 0 when I = (i1; : :;:in) is admi* *ssible of length 2. Let z = Sqin(y-1). Then z has dimension (in - 2) and lifts to the ordinary cohomology H*(CP+1; F2) of the space CP+1, which is an unstable A- module. Thus Sqin-1(z) = 0 since in-1 > in - 2, and so SqI(y-1) = 0. TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 21 Lemma 2.16. In CP-11, the lowest k-invariant k1: -2HZ -! HZ is nontrivial, and has mod 2 reduction the class of Sq3 mod ASq1. Proof. The lowest homotopy group of CP-11is detected by a map CP-11! -2HZ. On cohomology it induces a surjection -2A=ASq1 ! -2A=C, whose kernel -2C=ASq1 begins with -2Sq3 mod ASq1 in degree 1. This is the cohomology operation represented by the lowest k-invariant k1. Consider the Adams spectral sequence (2.17) Es;t2= Exts;tA(H*(CP-11); F2) =) sst-s(CP-11)^2: Its E2-term can be computed in a range from a (minimal) resolution of -2A=C, either by hand or by Bruner's Ext-calculator program [Br]. The E2-term in ho- motopical degrees t - s 20 is shown in Tables 2.18(a) and (b). The notation sx represents a class arising in the Adams E2-term for CP-s1, mapping to the class named x in the Adams E2-term for CP-s1=CP-s-11~=2sS0. The distinction be- tween classes marked as `o' or as `O' will be explained in x5. The cofiber sequence of spectra CP-01-i!CP-11-j!CP 1 induces a short exact sequence in mod 2 spectrum cohomology, and thus gives a long exact sequence of Ext-groups relating the Adams E2-term (2.17) to the Adams E2-terms (2.19) 0Es;t2= Exts;tA(H*(CP-01); F2) =) sst-s(CP-01)^2 and (2.20) 00Es;t2= Exts;tA(H*(CP 1); F2) =) sst-s(CP 1)^2: Knowledge of the stable homotopy of CP-01' -4CP 2and CP 1 in a range allows us to determine the differentials in the spectral sequences 0E* and 00E* in a s* *imilar range. This is comparatively easy for CP-01, and was done for CP 1 by Mosher in [Mo]. Using the long exact sequence : :-:!0Es;t2i*-!Es;t2j*-!00Es;t2@-!0Es+1;t2-!: : : and the geometric boundary theorem [Ra, 2.3.4] we can transfer some of these differentials to (2.17). (The careful reader should come equipped with the Ext charts for CP-01and CP 1 to check the details in the following proof.) Proposition 2.21. In the Adams spectral sequence (2:17) the nonzero differentia* *ls landing in homotopical degree 20 are: (i) d1;82(2h2) = -1c0. (ii) d2;122(5h20) = h30. 1h3, d3;132(5h30) = h40. 1h3 = -1P h2 and d4;142(5h* *40) = h50. 1h3 = -1h0P h2. 22 JOHN ROGNES o o o O o O O | | | | | | | | | | | | | | o o o O o O O | | | | | | | | | | | | | | o o o O o O O | | | | | | | | | | | | | | o o o O o O O | | | | | | | | | | | | | | o o o O o O O | | | | | | | | | | | | | | o o o O o O O o | | | | | | | | | | | | | | | | o o o O o O j jOWW/ O o | | | | | j j j| |/// | | | | | | |j j j | | // | | o o o O o "O O j jO-1Ph2O/j/jOoWW/ | | | | | """ |j j j| |j/j/j|////| | | | | |""jjj| |j j j| / o| | o o o O o O O "OOWW//O/j/j O | | | | | | """|j/j/j|////| | | | | | |""jjj| // o| | o o o j j o0h20h2O j j O-1c0OWW/ O4h30OOjj//O 2h2O0h3O* * 4h31 | | j j j| """"| |j j j """|// |j j j """"| //|/"""| | | j j |"" | j j| """ | /j/j| "" | o |"" | o-1h20 o0h20j o1h20jo0h0h2jOj o j j Oj //O O1h22O O5h20 O2h0* *h3 | | j j j | |j"j"j"" //|/"""" | | | j j | j j|"" o|"" | o-1h0 o0h0j o0h2jo2h0 O2h2 O1h3 | | o-11 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 Table 2.18(a). The Adams E2-term for CP-11 (iii) d1;132(6h0) = h0 . 2h0h3 + h1 . 5h20, d2;142(6h20) = h20. 2h0h3, d3;15* *2(6h30) = h30. 2h0h3 and d2;152(5h0h2) = -1d0. (iv) d1;162(6h2) = h0. 0h23, d2;173(h0. 6h2) = 0h0d0, d3;183(h20. 6h2) = h0.* * 0h0d0 and d4;193(h30. 6h2) = h20. 0h0d0 = -1P c0. (v) d1;182(5h3) = h0 . 1h23, d2;192(h0 . 5h3) = h20. 1h23= -1f0 and d3;203(h* *20. 5h3) = 1h20d0. (vi) d4;222(x) = h20. 4h1c0 with h0 . x 6= 0, d5;232(h0 . x) = h30. 4h1c0, d* *6;242(9h60) = h70.5h3, d7;252(h0.9h60) = h80.5h3 = -1P 2h2 and d8;262(h20.9h60) = h90.5h3 = -* *1h0P 2h2. (vii) d4;2426= 0, d5;2526= 0, d6;2626= 0, d7;2726= 0, d1;222(3h4) 6= 0 and d* *6;2726= 0 all have rank 1. TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 23 O O o O O O (?) | | | | | | | | | | | | | | | | | | O O o O O o O (?) | | | | | | | | | | | | | | | | | | | | | O O o O j j OWW/ O o O Oo | | | jj|j |// | | | | | | | j j | | // | | | | | | |j j j | | // | | | | O O o O O j j O-1P2h2O/j/jOoWW/WWO/ O | | | """| jj|j |// jj/j|/ |// | || | | | "" j|j | j|j////| | // | || | | |""jjj | |j j j | / o| | // | || O O o O O OOWW///O/ j j OWW///OOWjjjOOjW/ | | | | """|// j/jj|/ ||//jjjjjj"""|//||* *// | | | | "" j|j////| jjjj||""|////||// | | | |""jjj | // o|jjjjjj||""/o|/// || O o O j j O-1Pc0OTT* O j jOOWW///O OOWW///O jjjOOj// | | | j jj """|** | j j j """||//// | """"""|jjjjjj|//|//// | | j|j "" | * j|j "" ||// //| """"jjjj|//|// /|/ | |j j j | "" |j*j*j | "" || //o |""""jjjjjj|o|// o| O o O o j j OT**T*O O1h20d0OOjj//O9h60WOOW/jj/OO/WW/OOTT* | | | j j j |**** | j"j"j"** ||//// j"j"j"|///||/"""| | | j|j |** ** j|j"" * ||// //j j"" | // //||"" | | | j j j| |j*j*j |"" ** ||jj/jo/"" | // o||"" | O OoWW/ O O0h0d0OO*****TT*** OO //OO O j j/OO/ j O O O | |// | ******| ** || """|// jj| j/j|j/""""""| | | // | *** *|* * ||"""/j|j/j j|j j //| """"| | | // | * **j| ** ||"jj o|j j | o|"""" | O OWW///O OW-1d0OjjW/ *jj*OO**jj O-1f0*O*Oj4h1c0OOjWWO/ OO O | |// j"j"j"|///|/ j j j *j"j"j"|**||*/j*j|j*/"""| | | | j|j""//|//// j|j j j"""*|*|** |jj// *|*"" | | | |j j j| "// o|j j/j/|j jj " **j|j|j j| / j|"" | | | O 2h2O0h3O/4h31Oj/jO//0h0Oh23Oj3c0OO** OWW///OO jjOO OO j jOWW/ jOO | "" WW///j|j // ""j jWjW**||// |// ""jjjjjj"|//| j|j ""jj|j// | | "" j|j/j//|/ / j"j" | // *|| |jjjjjj""|////j|jj | j"j""| // | |""j j| // o|/ j jjo""/ | // j|*|jjjj|""/ o|/ j j | j j j|"" | / | O5h20j O2h0h3/Oj/ O5h0h2 O0h23jOOjj//O O 1h23Oj///jO j O jO ///O ///| """ j j j///| "" /j|j/j j j j //| /|/"" j j j //| "" j j j /|/ j jj /|/ o|""j j o|""j j o| j j o| O6h0j O6h2j O5h3j O3h4 10 11 12 13 14 15 16 17 18 19 20 21 Table 2.18(b). The Adams E2-term for CP-11 Proof. We compare the Adams E2-term in Table 2.18 with its abutment 2.13. Each h0-torsion class in the E1 -term of (2.17) comes from an h0-torsion class in th* *e E2- term, and so is represented by a 2-torsion class in ss*(CP-11). (The proof of * *this assertion goes by induction over the subspectra CP-s1of CP-11.) In each degree t-s 5 the order of the 2-torsion in ss*(CP-11) equals the or* *der of 24 JOHN ROGNES the h0-torsion in Table 2.18, hence there are no nonzero differentials in this * *range. (i): In degree t - s = 6 the 2-torsion in the abutment is Z=2, while the E2-* *term has two h0-torsion generators, so one of these must be hit by a differential. * *For bidegree reasons the only possibility is d1;82(2h2) = -1c0, and then there is n* *o room for further differentials landing in degrees t - s 8. In degree 7 of the E1 -term there is then a nonzero multiplication by h30, s* *howing that the extension in ss7(CP-11) is cyclic. (ii) and (iii): We turn to degrees 9 t - s 13. The Adams spectral sequence for CP 1, denoted 00E* in (2.20), has differentials 00d2(5h20) = 1h30h3 and 00d* *2(6h0) = h0 . 2h0h3 + h1 . 5h20. This uses ssS9(CP 1) = Z=8 and ssS11(CP 1) = Z=4, see [* *Mo, 7.2]. The map of spectral sequences j*: E2 ! 00E2 is an isomorphism in bidegrees (2; 12) and (1; 13), so these differentials lift to E2. Regarding the first 00d2-differential, both basis elements in E4;132~=F2{h30* *.1h3; h1. 4h30} map to 1h30h3 in 00E4;132. Hence d2(5h20) equals one or the other of thes* *e basis elements. It cannot be h1 . 4h30, because then d2(5h30) = 0 by h0-multiplicati* *on, and more classes would survive to the E1 -term in degree 9 than the abutment ss9(CP-11) ~=Z=2 Z=2 Z=8 allows. Thus d2;122(5h20) = h30. 1h3. Multiplication* * by h0 implies d3;132(5h30) = h40. 1h3 and d4;142(5h40) = h50. 1h3 in (2.17). In bidegrees (2; 12), (2; 13) and (3; 14) the map j* is an isomorphism, so t* *he second 00d2-differential lifts to d1;132(6h0) = h0 . 2h0h3 + h1 . 5h20. Multip* *lication by h0, h20and h1 implies d2;142(6h20) = h20. 2h0h3, d3;152(6h30) = h30. 2h0h3 a* *nd d2;152(5h0h2) = -1d0, respectively. There is no room for further differentials * *landing in degree t - s 13. (iv): We turn to degrees 14 t - s 15. For bidegree reasons the class 0h232 E2;162survives to E1 , and the classes h0 . 0h23and 3c0 in E3;172can only be affected by a d2-differential from 6h2 2 E1;162. The 2-torsion in ss14(CP-1* *1) is (Z=2)2, so the class h0.0h23cannot survive to E1 , i.e., there is a nonzero dif* *ferential d2(6h2) = h0 . 0h23in E*. The Adams spectral sequence for CP-01, denoted 0E* in (2.19), has a differen* *tial 0d3(0h0h4) = 0h0d0. (This lifts the usual differential d3(h0h4) = h0d0 in the A* *dams spectral sequence for ssS*. Multiplying this by h20gives the differential 0d3(0* *h30h4) = -1P c0, arising from the hidden multiplicative relation j . {h30h4} = {P c0} in* * the stable 16-stem.) The map of spectral sequences i*: 0E2 ! E2 is injective in bidegree (2; 17), taking 0h0h4 to h0. 6h2. For in 0E2 we know that h2. 0h0h4 = h0. 0h2h4. Thus the image of 0h0h4 in E2 is such that h2 times it is divisible by h0, and by inspec* *tion this property characterizes h0 . 6h2 2 E2;172. Thus we have another nonzero differen* *tial d3(h0 . 6h2) = 0h0d0 in E*. Multiplication by h0 and h20leads to the differenti* *als d3(h20. 6h2) = h0 . 0h0d0 and d3(h30. 6h2) = h20. 0h0d0 = -1P c0, respectively.* * There is no room for further differentials landing in degrees 14 t - s 15. (v): Next we consider differentials landing in degree t - s = 16. For bide- gree reasons the two classes h1 . 6h2 and 1h23in E2;182survive to E1 , and since ss16(CP-11) ~=Z (Z=2)2, the remaining h0-torsion classes are hit by differenti* *als. Thus d1;182(5h3) = h0 . 1h23and d2;192(h0 . 5h3) = h20. 1h23= -1f0. To determine the last differential landing in degree 16, we compare once aga* *in with the Adams spectral sequence 00E* for ss*(CP 1). Comparing the 00E2-term and the 00E1 -term given in Table 7.2 of [Mo] we deduce that there are differen* *tials TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 25 00d2(1h4) = 1h0h23, 00d3(h0 . 1h4) = 1h0d0 and 00d3(h20. 1h4) = 1h20d0. In part* *icular, the cited table asserts that 00d2(4h1c0) = 0 does not interfere with the second* * 00d3- differential. Also 00d2(h2 . 71) = h2 . 00d2(71) = 0, and 00d3(h2 . 71) = 0 fol* *lows from ssS16(CP 1) ~=Z (Z=2)3. The map j*: E2 ! 00E2 is an isomorphism in degree t - s = 17 and Adams filtration s 4, while in degree t - s = 16 the kernel consists of the class h2* *0. 1h23= -1f0 only. Comparing d1;182to 00d1;182it follows that 5h3 2 E1;182maps to 1h4 mod h2 . 71 2 00E1;182. Thus d3(h205h3) maps under j* to 1h20d0, and we ha* *ve proven that d3;203(h20. 5h3) = 1h20d0 in E*. (vi): In degree t - s = 17, the abutment has order 28 and the E2-term has 16 classes. Hence there are five differentials landing in degree 17, in addition t* *o the three differentials we have just found leaving that degree. The 2-torsion in t* *he abutment in degree 18 has order 25, and the E2-term has seven h0-torsion classe* *s. Hence at most two differentials leave the h0-torsion in degree 18, and at least* * three differentials leave the h0-periodic part of the E2-term. For bidegree reasons * *this extreme case is precisely what occurs, so d6;242(9h60) 6= 0, d7;252(9h70) = -1P* * 2h2 and d8;262(9h80) = -1h0P 2h-2, and there are no nonzero differentials landing in de* *gree t - s = 18. To precisely pin down the differential d6;242we use the same argument as for d2;122. The map j*: E2 ! 00E2 is an isomorphism in bidegree (6; 24) and surject* *ive in bidegree (8; 25), so the relation h1 . 8h70= h70. 1h4 in 00E8;252and the dif* *ferential 00d2(9h60) = h70. 1h4 implies that d2(9h60) is either h1 . 8h70or h70. 5h3 in E* *8;252. Multiplying with h0 and comparing with d7;252eliminates the first possibility, * *so in fact d6;242(9h60) = h70. 5h3. Considering h0- and h2-multiplications in the E2-term, either d2 = 0 on all * *h0- torsion classes in degree t - s = 18, or d4;222(x) = h20. 4h1c0 on the classes * *x 2 E4;222 not divisible by h0, and d5;232(h0 . x) = h30. 4h1c0. In the former case, the * *d3- differential d2;173would propagate by h2- and h0-multiplications to three nonze* *ro d3-differentials from the h0-torsion in degree t - s = 18, which is incompatible with the abutment. Thus the two d2-differentials given above are correct, and t* *his accounts for all the differentials from degree t - s = 18. (vii): The proofs in degrees 19 t - s 21 are left as exercises for the rea* *der who needs these results. 3. The fiber of the cyclotomic trace map When localized at p = 2, the homotopy type of the spectrum K(Z) is known. This involves the Bloch-Lichtenbaum spectral sequence relating motivic cohomo- logy to algebraic K-theory, Voevodsky's proof of the Milnor conjecture, which r* *e- lates motivic cohomology toetale cohomology, and knowledge of theetale cohomo- logy of the rational 2-integers Z[1_2]. Similarly, the p-adic homotopy type of the spectrum T C(Z; p) is known for e* *ach prime p. They were determined by B"okstedt and Madsen in [BM1] and [BM2] for p odd, and by the author in [R2], [R3], [R4] and [R5] for p = 2. When p = 2 the homomorphisms induced by trcZ:K(Z) ! T C(Z; 2) on homotopy groups are known after 2-adic completion. In this chapter we use this to describe the homo* *topy fiber of the cyclotomic trace map as a spectrum. Let all spectra be implicitly completed at 2, throughout this chapter. 26 JOHN ROGNES 3.1. Some two-adic K-theory spectra. We say that a (-1)-connected spec- trum is connective, and a 0-connected spectrum is connected. Let KO and KU denote the real and complex topological K-theory spectra, let ko and ku denote their connective covers, and let bo and bu denote their connected covers, respe* *c- tively. Write bso and bspin for the 1- and 3-connected covers of KO, and bsu for the 3-connected cover of KU, as usual. Complex Bott periodicity provides a homotopy equivalence fi :2KU ! KU. There is a complexification map c: KO ! KU and a realification map r :KU ! KO. Smashing with the Hopf map j :1 S1 ! 1 S0 yields a map also denoted j :KO ! KO. We use the same notation for the various k-connected covers of these maps. There is a cofiber sequence of spectra 2rOfi-1 (3.2) ko j-!ko -c!ku -----! 2ko : This follows from R. Wood's theorem KO ^ CP 2' KU. See also [MQR, V.5.15]. Here we write 2r O fi-1 for a map @ :ku ! 2ko that satisfies @ O fi = 2r. This determines the map up to homotopy, even though fi :2ku ! ku is not exactly invertible. Theorem 3.3 (Quillen). There is a cofiber sequence of spectra 3-1 @3 K(F3) i3-!ku ---! bu -! K(F3) : This is the spectrum level statement of Quillen's computation in [Q1]. The computation in [RW] by Weibel and the author of the 2-primary algebraic * *K- groups of rings of 2-integers in number fields relies on Suslin's motivic cohom* *ology for fields [S2], Voevodsky's proof of the Milnor conjecture [Voe] and the Bloch- Lichtenbaum spectral sequence [BlLi]. In the case of the 2-integers Z[1_2] in Q* * the result implies that there is a 2-adic homotopy equivalence K(Z[1_2]) ' JK(Z[1_2* *]), where the latter spectrum was defined by B"okstedt in [B1]. This leads to the following statement: Theorem 3.4 (Rognes-Weibel). There is a cofiber sequence of spectra ko -! K(Z[1_2]) ss3-!K(F3) @-!2ko where ss3 is induced by the ring surjection ss3: Z[1_2] ! F3. The connecting ma* *p @ is homotopic to the composite 2rOfi-1 K(F3) i3-!ku -----! 2ko : Proof. B"okstedt's JK(Z[1_2]) can be defined as the homotopy fiber of the compo* *site 3-1 ko -c!ku ---! bu : By [B1, Thm. 2] there is a map : K(Z[1_2]) ! JK(Z[1_2]) inducing a split surjec* *tion on homotopy. By [RW], [We] these spectra have isomorphic homotopy groups, TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 27 hence is a homotopy equivalence. There is a square of horizontal and vertical cofiber sequences: JK(Z[1_2])_____//ko_____//bu | || |||| | |c || fflffl|i3 fflffl| 3||-1 K(F3) _______//_ku____//bu |@| |2rOfi-1||| fflffl| fflffl| fflffl| 2ko ________2ko _____//_* The left hand vertical yields the asserted cofiber sequence. B"okstedt's cited * *con- struction of the map identifies the composite K(Z[1_2]) ! JK(Z[1_2]) ! K(F3) with that induced by the ring homomorphism ss3. 3.5. The reduction map. Let us recall the Galois reduction map from [DwMi, x13] and [R5, x3]. Let OE3 2 Gal(Q2 =Q2) be a Galois automorphism of the algebr* *aic closure Q2 of the field Q2 of 2-adic numbers, such that OE3(i) = i3 when ipis_a 2-powerproot_of unity, i.e., in 21 Qx2. We may further assume that OE3(+ 3) = + 3. Then OE3 induces a self-map of K(Q2 ) which is compatible up to homotopy with 3: ku ! ku under Suslin's (implicitly 2-adic) homotopy equivalence K(Q2 )* * ' ku from [S1]. Hence the inclusion K(Q2) ! K(Q2 )hOE3to the homotopy equalizer of OE3 and the identity on K(Q2 ) yields a spectrum map K(Q2) ! (ku)h 3. The connective cover of the target is identified with K(F3) by Quillen's theorem, w* *hich defines the Galois reduction map red:K(Q2) -! K(F3) : Theorem 3.6 (Rognes). There are cofiber sequences of spectra Kred(Q2) -! K(Q2) red--!K(F3) @2-!Kred(Q2) and K(F3) fC-!Kred(Q2) -! ku @1-!2K(F3) : The former connecting map @2 is determined by its composite with Kred(Q2) -! 2ku, which up to a two-adic unit is homotopic to the composite -1 fi-1 K(F3) i3-!ku 1-----!bu --!'2ku : The latter connecting map @1 is homotopic to the composite -1) @3 ku (1-------!bu --! 2K(F3) : Both connecting maps induce the zero map on homotopy, and the extensions Kred*(Q2) -! K*(Q2) red*--!K*(F3) and ss*(K(F3)) fC*--!Kred*(Q2) -! ss*(ku) are both split. This is the conclusion of [R5, 8.1]. Consider the ring homomorphisms j :Z ! * *Z2 and j0:Z[1_2] ! Q2. 28 JOHN ROGNES Theorem 3.7 (Quillen). There is a map of horizontal cofiber sequences of spectra K(F2) _____//_K(Z)____//K(Z[1_2]) || | |0 || j| j| || fflffl| fflffl| K(F2) ____//_K(Z2)____//_K(Q2) inducing a homotopy equivalence hofib(j) '-!hofib(j0). This is the spectrum level statement of the localization sequences in K-theo* *ry from [Q2]. Theorem 3.8 (Hesselholt-Madsen). In the commutative square of spectra K(Z) __j__//_K(Z2) trcZ|| trcZ2|| fflffl|j fflffl| T C(Z)__'__//T C(Z2) the right hand map induces a homotopy equivalence on connective covers. The low* *er map is a homotopy equivalence, and there is a cofiber sequence of spectra hofib(j) -! hofib(trcZ) -! -2HZ : This is Theorem D of [HM], which uses McCarthy's theorem [Mc]. Theorem 3.9 (Rognes). The natural map j0:K(Z[1_2]) ! K(Q2) induces an isomorphism of 2-adic homotopy groups modulo torsion, in each positive dimension * 1 mod 4. This is the content of [R5, 7.7]. By a homotopy group modulo torsion we mean the quotient of the homotopy group by its torsion subgroup. Hence the assertion is stronger than just saying that j0 induces a homomorphism whose kernel and cokernel are torsion groups. Proposition 3.10. There is a map of horizontal cofiber sequences of spectra ss3 ko _______//K(Z[1_2])_//_K(F3) |jred| j0|| ' _|| fflffl| fflffl|red fflffl| Kred(Q2) _____//_K(Q2)____//_K(F3) such that the right hand map _is a homotopy equivalence. Hence there is a homo- topy equivalence hofib(jred) '-!hofib(j0). Proof. Suppose we have shown that the composite 0 red ko -! K(Z[1_2]) j-!K(Q2) --! K(F3) TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 29 is null homotopic. Then a choice of null homotopy defines an extension _:K(F3) ! K(F3) of redOj0 over ss3, as well as a lifting jred:ko ! Kred(Q2). By the cal- culations of [R5, x4 and x7] the composite redOj0 is surjective on homotopy in dimensions 0 * 7, hence in all dimensions by v41-periodicity. Thus _ induces surjections on homotopy in all dimensions, and must be a homotopy equivalence. This then completes the proof of the proposition. To show that the composite map ko ! K(F3) is null homotopic, it suffices to show that precomposition with the connecting map @ :K(F3) ! 2ko in 3.4 induces an injection between the groups of homotopy classes of maps to K(F3): # [2ko; K(F3)] @--![K(F3); K(F3)] : Here @ is the composite of i3: K(F3) ! ku with the map denoted 2r O fi-1 :ku ! 2ko. Thus it suffices to show that both homomorphisms i#3and (2r O fi-1)# are injective. There is an exact sequence 3-1)# i#3 [bu; K(F3)] (-----! [ku; K(F3)] -! [K(F3); K(F3)] : Any map bu ! K(F3) has the form @3 O OE, for some operation OE: bu ! bu. Thus its precomposition with ( 3-1) is null homotopic, because OE and ( 3-1) commute and @3 O ( 3 - 1) ' *. Hence the left hand map is null and i#3is injective. There is also an exact sequence # (2rOfi-1)# [ko; K(F3)] j--![2ko; K(F3)] --------! [ku; K(F3)] : From 3.3 we see that [ko; K(F3)] is zero, because postcomposition with ( 3- 1) acts injectively on the homotopy classes of maps ko ! ku, see [MST]. Thus also (2r O fi-1)# is injective, which completes the proof. Proposition 3.11. There is a cofiber sequence of spectra K(F3) -! hofib(jred) -! 2ko @-!K(F3) : The connecting map @ is homotopic to the composite 2c fi @3 2ko --! 2ku -!'bu -! K(F3) : Proof. The map ko ! K(Z[1_2]) induces an isomorphism on 2-adic homotopy mod- ulo torsion in dimensions * 1 mod 8, and multiplication by 2 times a 2-adic un* *it in dimensions * 5 mod 8. By 3.9 the same holds for the composite map from ko to K(Q2), and by 3.6 the same holds for the lift jred:ko ! Kred(Q2), as well as the composite map ko ! ku. Any such map factors as a self-map OE of ko followed by the suspended complexification map c: ko ! ku. Since the suspended complexification map induces the identity in dimensions * 1 mod 8, and multiplication by 2 in dimensions * 5 mod 8, it follows that OE is a 2-adic 30 JOHN ROGNES homotopy equivalence. We obtain the following diagram of horizontal and vertical cofiber sequences: K(F3) ________//*_____//_K(F3) | || | | | |fC fflffl| fflffl|jredfflffl| hofib(jred)____//ko_____//Kred(Q2) | | | | ' OE| | fflffl|j fflffl|c fflffl| 2ko _______//ko_______//ku The connecting map @ :2ko ! K(F3) is detected by its precomposition with 2r O fi-1 :ku ! 2ko, because [ko; K(F3)] = 0. By the diagram above, the composite @ O 2r O fi-1 is the desuspended connecting map -1@1 = @3O (1 - -1) from 3.6. Thus @ = @3 O fi O 2c in the stable category, by the calculation @3 O fi O 2c O 2r O fi-1 = @3 O (1 - -1) which uses c O r = 1 + -1, and k O fi = fi O 2(k k). Proposition 3.12. There is a cofiber sequence of spectra 3ko -! hofib(jred) -! ku @-!4ko : The connecting map @ is homotopic to the composite 3-1 fi-1 2(2rOfi-1) ku ---! bu --!'2ku --------! 4ko with the same notation as in 3:2. It is characterized by the following homotopy commutative diagram ku _____________@_____________//_4ko cov|| |cov| fflffl| 3-1 fi2 4r fflffl| KU _____//KUoo_'_ 4KU ____//_4KO : The maps labeled cov are k-connective covering maps, for suitable k. Proof. We use the factorization of the connecting map in 3.11 to form the follo* *wing diagram of horizontal and vertical cofiber sequences: 3ko _____//hofib(jred)____//ku || | | -1 3 || | fi| O( -1) || 2j fflffl|2c fflffl| 3ko _______//2ko ________//_2ku | |@ @3Ofi| | | | fflffl| fflffl| fflffl| * _______//_K(F3)_______K(F3) TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 31 The right hand column is a variant of the sequence in 3.3. It follows that the connecting map ku ! 4ko for the top row is the composite of 3-1 fi-1 ku ---! bu --!'2ku and the connecting map for the middle row, i.e., the double suspension of the connecting map @ :ku ! 2ko of 3.2. The covering maps induce an injection cov# :[ku; 4ko] ! [ku; 4KO] and a bijection cov# :[KU; 4KO] ~=[ku; 4KO], and the connecting map @ corresponds to 4r O fi-2 O ( 3 - 1) in [KU; 4KO]. Hence @ is characterized by the given diagram. The following theorem is the main result of this chapter. Theorem 3.13. There is a cofiber sequence of spectra 3ko -! hofib(trc) -! -2ku ffi-!4ko : The connecting map ffi is characterized by the following homotopy commutative d* *ia- gram _________ffi______//_ 4 -2ku ko |cov| cov|| fflffl|fi 3-1 fi2 4r fflffl| -2KU o'o__ KU _____//KU oo'__4KU _____//4KO : The maps labeled cov are suitable covering maps. Proof. Consider the following diagram of horizontal and vertical cofiber sequen* *ces of spectra, obtained by combining 3.7, 3.8, 3.10 and 3.12: 3ko _____//_hofib(j)_____//ku || | | || | | || fflffl| fflffl| 3ko _____//hofib(trc)____//_X | | | | | | fflffl| fflffl| fflffl| *_______//_-2HZ _______-2HZ Here X is the cofiber of the composite map 3ko ! hofib(j) ! hofib(trc). It is classified as an extension of -2HZ by ku by an element in [-2HZ; ku] ~=Z, whose mod 2 reduction is detected by the composite k :-2HZ ! ku ! HZ in [-2HZ; HZ] ~=Z=2. (Here ku ! HZ is the map inducing an isomorphism on ss1.) This composite k is the k-invariant of X relating the homotopy groups * *in dimensions -2 and 0. Since 3ko is 2-connected, this lowest k-invariant is the same for X as for hofib(trc). By combining 1.11 with 1.21 we obtain a cofiber sequence (3.14) CP-11-i!hofib(trc) -j!Wh Diff(*) 32 JOHN ROGNES whose connecting map is identified with ftrc. Since Wh Diff(*) is connected, it* * follows that the lowest k-invariants for hofib(trc) and CP-11are equal. By 2.16 the lat* *ter is nonzero. Hence k is the essential map. It follows that X is classified by a map u . @ where @ :-2HZ ! ku classifies -2ku and u is a 2-adic unit. We get a homotopy equivalence of cofiber sequences X _______//-2HZ u.@__//_ku |'| ' |u| |||| fflffl| fflffl|@ || -2ku _____//-2HZ _____//ku : Hence X ' -2ku, as claimed. To characterize ffi, we compare with the connecting map @ of 3.12. Precompo- sition with fi :ku ! -2ku, or its K-localization, induces the vertical map in t* *he commutative diagram cov# cov# [-2ku; 4ko] _____//[-2ku; 4KO] oo~=_ [-2KU; 4KO] fi#|| ~=|fi#| fi#|| fflffl|cov# fflffl| cov# fflffl| [ku; 4ko]________//_[ku; 4KO]oo~=___[KU;_4KO] Here the maps labeled cov# are injective, and the maps labeled cov# are bijecti* *ve. The class ffi in [-2ku; 4ko] maps to @ under fi# , which in turn maps to 4r O fi-2 O ( 3 - 1) in [KU; 4KO] by 3.12. The right hand fi# is bijective, so this characterizes the image of ffi in [-2KU; 4KO] as 4r O fi-2 O ( 3 - 1) O fi-1. T* *his characterizes ffi up to homotopy, by the injectivity claims above. Remark 3.15. By [6A] or 1.8 this theorem also determines the homotopy fiber of * *the cyclotomic trace map trcX:A(X) ! T C(X) completed at 2 for any 1-connected space X, since the natural map hofib(trcX) '-!hofib(trc) is a homotopy equivalence. Let v2(k) be the 2-adic valuation of k. Corollary 3.16. In positive dimensions (n > 0) the homotopy groups of hofib(trc) are 8 >>>0 for n = 0; 1 mod 8, >>>Z for n = 2 mod 8, >>< Z=16 for n = 3 mod 8, ssn(hofib(trc)) ~=> >>>Z=2 for n = 4; 5 mod 8, >>>Z for n = 6 mod 8, and >: Z=2v2(k)+4 for n = 8k - 1. Also ssn(hofib(trc)) ~=Z for n = -2 and n = 0. The remaining homotopy groups are zero. Proof. This is a routine calculation, given the action of 3 - 1 and 4r on homo- topy. TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 33 O O O O oWW/ o o | | | | |// | | | | | | | // | | O O O O oWW///o o | | | | |////|/ | | | | | | //o| | O O O O oWW///o o | | | | |////|/ | | | | | | //o| | O O O O oWW///o o | | | | |////|/ | | | | | | //o| | O O O O oWW///o o | | | | |////|/ | | | | | | //o| | O O O O oWW///o o j jO | | | | |////|/ jj j| """| | | | | | //o|j j j |"" | O O O O o // o o j jO | | | | | //|/ jj j | | | | | | o|j j j | O O O O o o O | | | | | | | | | | | | O O O O o o | | | | | | | | O O O j jO o | | j j j| """| | |j jj |"" | O O O j jO O | | j j j | """ | |j jj | "" O O O O | | """ | |"" O o -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 Table 3.18(a). The Adams E2-term for hofib(trc) The spectrum cohomology of hofib(trc) is given in 4.4 below. The Adams E2- term (3.17) Es;t2= Exts;tA(H*(hofib(trc)); F2) =) sst-s(hofib(trc))^2: is then easily deduced from the E2-terms in the Adams spectral sequences for ss*(ko)^2and ss*(ku)^2. Furthermore only one pattern of differentials is compat* *ible with 3.16: There is an infinite h0-tower of nonzero dr-differentials from colu* *mn t - s = 8k for all k 1, with r = v2(k) + 2, and no other differentials. The sp* *ectral sequence is displayed in Tables 3.18(a) and (b) below. 4. Cohomology of the smooth Whitehead spectrum We now determine the mod two spectrum cohomology of the smooth Whitehead spectrum of a point, as a module over the Steenrod algebra. 34 JOHN ROGNES o O oTT* o o | | |** | | | | |* | | o O o ** o oj j O | | | ** | j j| ""| | | | **|j j j |"" | o O o *o* oj j O O | | | *|* j j | "" | | | j|j j j | "" o O o o O o | | | | "" | | | |"" o O o o | | | | o O o | ""| |"" | o O O | "" | "" O o | "" |"" o 10 11 12 13 14 15 16 17 18 19 20 21 Table 3.18(b). The Adams E2-term for hofib(trc) Consider the following diagram: CP-11__________CP-11 |i| |ffl| fflffl| fflffl|ffi (4.1) 3ko _____//hofib(trc)___//_-2ku______//4ko || |j | || || | | || || fflffl| fflffl| || 3ko _____//WhDiff(*)___// hofib(ffl)_//4ko The middle row is the cofiber sequence from Theorem 3.13, and the left column is 3.14. We let ffl be the composite map CP-11! hofib(trf) ! -2ku. Then the right column and bottom row are cofiber sequences. Proposition 4.2. The map ffl induces the unique surjection of A-modules * -2A=A(Sq1; Sq3) ~=H*(-2ku) ffl-!H*(CP-11) ~=-2A=C : TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 35 Hence H*( hofib(ffl)) ~=-2C=A(Sq1; Sq3) as an A-module. Proof. We use that 4ko and Wh Diff(*) are connective spectra. Hence hofib(ffl) is connective, and so ffl induces an isomorphism in dimension -2. This determin* *es ffl* since H*(-2ku) is a cyclic A-module, and ffl* is surjective because H*(CP-* *11) is a cyclic A-module. We identify H*( hofib(ffl)) with ker(ffl*). Proposition 4.3. The connecting map ffi induces the zero homomorphism on co- homology. Proof. In fact, the group of A-module homomorphisms H*(4ko) ~=4A=A(Sq1; Sq2) -! -2A=A(Sq1; Sq3) ~=H*(-2ku) is zero. For A=A(Sq1; Sq3) is F2{Sq6; Sq4Sq2} in dimension 6, while Sq1 O Sq6 6* *= 0 and Sq2 O Sq4Sq2 6= 0 in this A-module. Theorem 4.4. The mod two spectrum cohomology of hofib(trc) is the unique non- trivial extension of A-modules -2A=A(Sq1; Sq3) -! H*(hofib(trc)) -! 3A=A(Sq1; Sq2) : Theorem 4.5. The mod two spectrum cohomology of Wh Diff(*) is the unique non- trivial extension of A-modules -2C=A(Sq1; Sq3) -! H*(Wh Diff(*)) -! 3A=A(Sq1; Sq2) : The mod two spectrum cohomology of A(*) is given by the splitting of A-modules H*(A(*)) ~=H*(Wh Diff(*)) F2: Here F2 = H*(S0) denotes the trivial A-module concentrated in degree zero. We prove these two theorems together. Proof of 4.4 and 4.5. We apply mod 2 spectrum cohomology to 4.1. By 4.2 the map ffl induces a surjection in each dimension, so -2ku ! hofib(ffl) induces an in* *jec- tion in each dimension. By 4.3 the map ffi induces the zero homomorphism in each dimension, and combining these facts we see that hofib(ffl) ! 4ko also induces the zero homomorphism in cohomology. Thus the long exact sequences in cohomo- logy associated to the middle and lower horizontal cofiber sequences in 4.1 bre* *ak up into short exact sequences. These express H*(hofib(trc)) and H*(Wh Diff(*)) * *as extensions of A-modules, as claimed. -2A=COO _____________-2A=COO i*|| ffl*|| | | (4.6) 3A=A(Sq1; Sq2) oo___ H*(hofib(trc))oo_O-2A=A(Sq1;OSq3)OO || *| | || j | | || | | 3A=A(Sq1; Sq2) oo___H*(Wh Diff(*))oo___-2C=A(Sq1; Sq3) 36 JOHN ROGNES It remains to characterize the extensions, which are represented by elements* * of Ext1A. Recall that H*(ko) = A=A(Sq1; Sq2) = A==A1 where A1 A is the sub-Hopf algebra generated by Sq1 and Sq2. Hence there are change-of-rings isomorphisms Ext1A(3A==A1; -2A=A(Sq1; Sq3)) ~=Ext1A1(3F2; -2A=A(Sq1; Sq3)) and Ext 1A(3A==A1; -2C=A(Sq1; Sq3)) ~=Ext1A1(3F2; -2C=A(Sq1; Sq3)) : An A1-module extension of -2A=A(Sq1; Sq3) by 3F2 is determined by the values of Sq1 and Sq2 on the nonzero element of 3F2, and these are connected by the Adem relation Sq1Sq2Sq1 = Sq2Sq2. By inspection of -2A=A(Sq1; Sq3) and -2C=A(Sq1; Sq3) as A1-modules, there are precisely two such A1-module extensions in both cases; one trivial (split) * *and one nontrivial (not split). Furthermore the map of extensions induced by 4.1 in* *duces an isomorphism ~= Z=2 ~=Ext1A1(3F2; -2A=A(Sq1; Sq3)) -! Ext1A1(3F2; -2C=A(Sq1; Sq3)) ~=Z=2 : Thus to prove that each extension is the unique nontrivial extension of its kin* *d, it suffices to show that H*(Wh Diff(*)) does not split as the sum of 3A=A(Sq1; Sq2) and -2C=A(Sq1; Sq3). Now -2C=A(Sq1; Sq3) is 3-connected, and by [BW, 1.3] the bottom homotopy group of Wh Diff(*) is ss3(Wh Diff(*)) ~=Z=2. Hence there is a nontrivial Sq1 a* *cting on the nonzero class in H3(Wh Diff(*)), which tells us that 3A=A(Sq1; Sq2) does not split off from H*(Wh Diff(*)). This proves that both extensions are nontrivial, and completes the proofs. Remark 4.7. By (4.6), we see that the lifted cyclotomic trace map ftrc:WhDiff(*) -! gT(C*) ' CP-11 induces the zero homomorphism on mod 2 spectrum cohomology. The map is nevertheless very useful. Question 4.8. The map ffl lives in the group [CP-11; -2ku] ~=[CP+1; ku] = KU0(CP 1) ~=Z[[fl1]] where the first isomorphism is the Thom isomorphism in complex topological K- theory for the virtual complex bundle -fl1 over CP+1. To which power series in * *fl1 does ffl correspond ? Proposition 4.9. The linearization map L: T C(*) ! T C(Z) and the suspended map ffl: CP-11! -1ku induce the same homomorphisms up to 2-adic units, on homotopy groups modulo torsion in dimensions * 3 mod 4. Proof. The suspended map ffl is the composite CP-11-! T C(*) L-!T C(Z) -! hofib(trc) -! -1ku : TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 37 The first map splits T C(*), the second is the linearization map, the third is * *the connecting map in the cofiber sequence generated by trcZ, and the fourth suspen* *ds a map that appears in 3.13. The first map induces an isomorphism on homotopy groups modulo torsion in all positive dimensions, since the other summand 1 S0 has finite homotopy groups in positive dimensions. The third and fourth maps also induce an isomorphism on homotopy groups modulo torsion in dimensions * 3 mod 4, by the calculation of trcZin [R5, 9.1], and the description of ffi in * *3.13 5. Two-primary homotopy of Wh Diff(*) Let -2 be the generator in dimension -2 of H*(-2ku) ~=-2A=A(Sq1; Sq3), and let 3 be the generator in dimension 3 of H*(3ko) ~=3A=A(Sq1; Sq2). By 4.2 the map ffl: CP-11! -2ku of (4.1) induces a surjection on cohomology, and we regard ker(ffl*) = -2C=A(Sq1; Sq3) -2A=A(Sq1; Sq3) = H*(-2ku) as a submodule of H*(-2ku). It is thus spanned by suitable monomials SqI-2 taken modulo A(Sq1; Sq3)-2. By inspection ker(ffl*) is 3-connected. The bottom cofiber sequence in (4.1) induces the nontrivial extension 0 ! ker(ffl*) -! H*(Wh Diff(*)) -! H*(3ko) ! 0 : We let 3 2 H3(Wh Diff(*)) denote the unique lift of 3 2 H3(3ko). With these notations we list a basis for H*(Wh Diff(*)) in dimensions * 14 in Table 5.1, together with generators for the A-module structure. We now consider the Adams spectral sequences associated with the spectra in the cofiber sequence of spectra (5.2) CP-11-i!hofib(trc) -j!Wh Diff(*) appearing vertically in (4.1). They are (5.3) cEs;t2= Exts;tA(H*(CP-11); F2) =) sst-s(CP-11)^2 (5.4) fEs;t2= Exts;tA(H*(hofib(trc)); F2) =) sst-s(hofib(trc))^2 (5.5) wEs;t2= Exts;tA(H*(Wh Diff(*)); F2) =) sst-s(Wh Diff(*))^2: The prefix `c' refers to the truncated complex projective space, `f' refers to * *the homotopy fiber of the cyclotomic trace map, and `w' refers to the Whitehead spe* *c- trum. The spectral sequence cE* was already studied in 2.17, 2.18 and 2.21, whi* *le the spectral sequence fE* appeared in 3.17 and 3.18. The spectral sequence wE* is displayed below, in Tables 5.7(a) and (b). The diagram (5.2) induces a short exact sequence of A-modules in cohomology, by (4.6), and thus a long exact sequence of Ext-groups (5.6) . .!.cEs;t2i*-!fEs;t2j*-!wEs;t2@-!cEs+1;t2! : ::: By the geometric boundary theorem [Ra, 2.3.4], the connecting map @ is induced by the spectrum map ftrc:WhDiff(*) ! CP-11extending (5.2), and so each map in 38 JOHN ROGNES _______________________________________________________________________________* *__||*Diff|1|2|4|8| |____x|2_H_(Wh____(*))_|Sq_(x)_______S|q_(x)________|Sq_(x)_______S|q_(x)______* *__|||||||| |__2_|________________|_____________|______________|______________|____________* *_|_|||42|7|4|8| |_3__3|______________|_Sq_Sq_-2_____|Sq_-2________|_Sq_3_________|Sq_3_________* *|_||42||62||842| |_4__S|q_Sq_-2________|0_____________|Sq_Sq_-2_____|0_____________|Sq_Sq_Sq_-2_* *_|_||7||9|11|132| ||5 S|q|-2 |0| ||Sq -27 2 | |Sq -2 |Sq| Sq1-214 * *|| |_____|_______________|_____________|___Sq_Sq_-2_|________________|_+Sq__Sq_-2_* *|_||62|9||102|1042| ||6 S|q|Sq -2 |Sq|-27 2 ||0 |Sq| Sq -2 |Sq| Sq Sq -2 * *|| |_____|_______________|___Sq_Sq_-2_|_______________|______________|____________* *_|_||9|||112|| ||7 S|q|-27 2 | |0 ||0 ||Sq Sq -2 || * * || || ||Sq4Sq -2 | | || 6 || 13 || * * || |____S|q_3____________|0_____________|Sq_3_________|Sq__-2________|____________* *_|_||82||102|122|| |_8__S|q_Sq_-2________|0_____________|Sq__Sq_-2____|Sq__Sq_-2____|_____________* *_|_||11||13|132|| ||9 S|q|6-2 |0|7 ||Sq -2 |Sq|1Sq3-22 || * * || || S|q|3 |Sq|3 |0| |Sq| Sq1-214 || * * || || || || || || +Sq10Sq -2 || * * || |_____|_______________|_____________|______________|__+Sq__3_____|_____________* *_|_||102|112|||| ||10 S|q|8Sq4-2 |Sq| Sq -2 |0| 10 4 |0|12 4 || * * || || S|q|Sq7-2 |0| |Sq| Sq -2 |Sq|1Sq1-2 | | * * || |____S|q_3____________|0_____________|0_____________|Sq__3_________|___________* *_|_||13||||| ||11 S|q|1-212 |0| |0|13 2 || || * * || || S|q|8Sq -2 |0|8 4 2 ||Sq10Sq -2 || || * * || |____S|q_3____________|Sq_Sq_Sq_-2__|Sq__3________|_______________|____________* *_|_||122|132|142||| ||12 S|q|1Sq0-24 |Sq|1Sq1-24 |Sq| Sq -2 || || * * || || S|q|8Sq4-22 |Sq| Sq -2 |0| 10 4 2 || || * * || |____S|q_Sq_Sq_-2_____|0_____________|Sq__Sq_Sq_-2_|______________|____________* *_|_||15||||| ||13 S|q|1-232 |0| || || || * * || || S|q|1Sq1-24 |0| || || || * * || || S|q|1Sq0-2 |0|11 || || || * * || |____S|q__3___________|Sq__3________|______________|______________|____________* *_|_||142||||| ||14 S|q|1Sq2-24 || || || || * * || || S|q|1Sq0-242 || || || || * * || || S|q|1Sq1Sq -2 || || || || * * || |____S|q__3___________|_____________|______________|______________|____________* *_|_ Table 5.1. H*(Wh Diff(*)) in dimensions 14. the long exact sequence is part of a map of spectral sequences. Furthermore the* *se maps are compatible with the maps in the long exact sequence in 2-completed homotopy induced by (5.2). The E2-term of the Adams spectral sequence (5.5) for Wh Diff(*) is displayed* * in dimensions t - s 21 in Tables 5.7(a) and (b). This was obtained from a minimal resolution of H*(Wh Diff(*)) in internal degrees t 14, using Table 5.1, and us* *ing TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 39 12 o OWW/ O o OWW/ o | |// | | |// | | | // | | |// | 11 o OWW//O/ o OWW//o/ | |// /|/ | |///|/ | | //o| | |//o| 10 o OWW//O/ o OWW//o/ | |// /|/ | |///|/ | | //o| | |//o| 9 o OWW//O/ o OWW//o/ | |// /|/ | |///|/ | | //o| | |//o| 8 o OWW//O/ o OWW//o/ | |// /|/ | |///|/ | | //o| | |//o| 7 o OWW//O/ o OWW//o/ | |// /|/ | |///|/ | | //o| | |//o| 6 o OWW//O/ o OWW//o/ | |// /|/ | |///|/ | | //o| | |//o| 5 o OWW//O/ o oWW//oj/j O | |// /|/ | |/jj|//""|/ | | //o| |j j j|/o|""/ | 4 o OWW//o/ o oWW/oOj/j/o | |// /|/ | |/jj""|//|/ | | //o| |j j j|""|o// | 3 o OWW//o/ o ooW//ojWj/o | |// /|/ | "" |/jj|/// | | | //o| |""jjj|/o|/ | 2 o oWW//o/ oj j oo //o oo | |// /|/j j "" | /|/ ""| | |j/joj|/ "" | o|"" | 1 o o //o o o o o /|/"" | o|"" | 0 O o o 3 4 5 6 7 8 9 10 11 12 Table 5.7(a). The Adams E2-term for Wh Diff(*) Bruner's Ext-calculator program [Br] in higher dimensions. The notation in these tables is that the maps in (5.6) take a class denoted `o' in one spectral seque* *nce to a class denoted `O' in the following spectral sequence, i.e., o 7! O. Proposition 5.8. The map i: CP-11! hofib(trc) induces a map i*: cEs;t2-!fEs;t2 of Adams E2-terms, which is surjective in dimensions t - s 2, t - s = 4 and t - s 5; 6 mod 8. In positive dimensions t - s 3 mod 8 its image equals the three h0-divisible classes. In other dimensions the map is zero. Proof. Note that fE2 has dimension 0 or 1 in each bidegree. In the range of bid* *e- grees displayed in Tables 2.18, 3.18 and 5.7, the claim follows by a dimension * *count using exactness in (5.6). Since hofib(trc) agrees with its Bousfield K-localiza* *tion in dimensions * 1 by 3.13, the result propagates to higher dimensions by applying suitable periodicity operators. 40 JOHN ROGNES 12 o o OTT* O o OWW/ o o | | |** | | |// | | | | |** | | | // | | 11 o o OTT**O o OWW//o/ o | | |****| | |// /|/ | | | |****| | | //o| | 10 o o OTT**O** o OWW//o/ o | | |****|** | |// /|/ | | | |***j|* | | //o| | 9 o o OTT**O** o oWW//oj/jOWW/ o | | |****|** | |/j/j|//""|// | | | |***j|* | jj j|//o|"" | //| 8 o o OTT**o** o oWW/oOj/jo/W//oW/ | | |****|** | |/j"j"|///|///|/ | | |***j|* | jj j|""o|// | //|o 7 o o OTT**o** o ooWW//oj/joWW//oo/ | | |****|** | ""|/j/j|// ||""|//// | | |***j|* |""jjj|//o| ||""|o// 6 o o oTT**o** oj jooWW//o/ ooW//oW/ | | |****|j*j* ""||///|/""""|//|// | | |j*jjj|***"" ||//o|""""| //|o 5 o OWW/ o oTT**o** ojTjooWW//o/T*oojj/oo/WW/ | "" |// | |****|j*j""***||//// j"j"|//||// |"" |// | |j*jjj|""*****||j/jjo""/| //||o 4 oO oWW//o/ oTT**oo** * oo /oo/ oj j/oo/ | |///|/ ****|** **|| ""|/j/j| /|/ | |//o| ***j|* **||""jjj|o | |o 3 o oWW/ooj/j/oWW/ j j*oojj*o*WWooj*joo*/ o oo | |/jj""|//|/j/j/ j"j"|**||j*j|**""|*// | |j j j|""|ojj/j|jj/j""/j|j|j|j/*j|""*/|/ | 2 o ooW//ojWj/o// oojj*oo*WWo/W/oojjjooj/Wooj/jo | "" |/jj|///// j"j"|//*|*| |/jjjj""|//|j/j| ""| |""jjj|/o|/jjoj"" | //j|j|jjjjj""||jjjo|//|"" | 1 o o //o o oj j/oo/ oo //oj j o o j jo /|/ "" j j //| "" j j|// j j o|""jjj o|""jjj o|j j j 0 o o o 11 12 13 14 15 16 17 18 19 20 21 Table 5.7(b). The Adams E2-term for Wh Diff(*) Proposition 5.9. In the Adams spectral sequence wE* the nonzero differentials landing in homotopical dimension 21 are (i) ds;s+826= 0 for s 0. (ii) ds;s+1126= 0 for s 1, with image divisible by hs+20. (iii) ds;s+1326= 0 for 0 s 3. The image of d0;132contains h0 . x + h1 . y * *for nonzero classes x; y. The image of ds;13+s2for 1 s 3 is divisible by hs+10. (iv) d1;1526= 0 has image divisible by h21. (v) d0;1626= 0 has image divisible by h0. (vi) ds;s+1636= 0 for s 1 is zero on the h0-torsion classes. (vii) ds;s+1826= 0 for s = 0; 1. (viii) d2;2036= 0 is zero on the h1-divisible classes. (ix) ds;s+1926= 0 for s = 3; 4 is zero on the h1-divisible classes, and take* *s h0- torsion values. (x) ds;s+1926= 0 for s 5, with image divisible by hs+20. (xi) d3;2426= 0, d4;2526= 0, d5;2626= 0, d6;2726= 0, d7;2826= 0, d0;2226= 0 * *and d5;2726= 0 all have rank 1. TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 41 Proof. The differentials in fE* given in Tables 3.18(a) and (b) induce differen* *tials in wE* by naturality with respect to the spectral sequence map j* in (5.6). Lik* *ewise the differentials in cE* given in 2.21 lift by the the connecting map @ in (5.6* *) to detect differentials in wE*. Taking the h0-multiplications in wE2 into account,* * this gives rise to all the differentials listed above. It remains to check that there are no further differentials in wE*. Any such would have to map from classes `o' detected by @ to classes `O' in the image of j*. For bidegree reasons the only possible targets are the h1-divisible classe* *s `O' in bidegree (s; t) = (4k; 12k + 3) with k 1. These classes are the image of ss8k+3(hofib(trc)) in ss8k+3(Wh Diff(*)). Now the generator of ss8k+3(hofib(tr* *c)) ~= Z=16 maps to the order 2 class j28k+1 in K8k+3(Z) ~=Z=16, which generates the kernel of the cyclotomic trace map trcZto ss8k+3(T C(Z)) ~=Z Z=8 by [R5, 9.1]. Hence by the diagram in 1.11, the image of ss8k+3(hofib(trc)) in ss8k+3(Wh Diff* *(*)) is nontrivial, and so the cited class in bidegree (4k; 12k + 3) must survive to* * the E1 -term. Hence it is not hit by a differential. Theorem 5.10. The 2-primary homotopy groups of Wh Diff(*) in dimensions * 21 are as follows: ssn(Wh Diff(*))= 0 for n 2, ss3(Wh Diff(*))= Z=2 ss4(Wh Diff(*))= 0 ss5(Wh Diff(*))= Z ss6(Wh Diff(*))= 0 ss7(Wh Diff(*))= Z=2 ss8(Wh Diff(*))= 0 ss9(Wh Diff(*))= Z=2 Z ss10(Wh Diff(*))= (Z=2)2 Z=8 ss11(Wh Diff(*))= Z=2 ss12(Wh Diff(*))= Z=4 ss13(Wh Diff(*))= Z ss14(Wh Diff(*))= Z=4 ss15(Wh Diff(*))= (Z=2)2 ss16(Wh Diff(*))= Z=2 Z=8 ss17(Wh Diff(*))= (Z=2)2 Z ss18(Wh Diff(*))= (Z=2)3 Z=32 ss19(Wh Diff(*))= Z=2 o Z=2 o Z=8 o Z=2 ss20(Wh Diff(*))= #27 ss21(Wh Diff(*))= #24 Z In the long exact sequence in homotopy induced by the cofiber sequence CP-11-i!hofib(trc) -j!Wh Diff(*) 42 JOHN ROGNES the image of j* is Z=2 in dimensions n 3 mod 8 and zero otherwise, for n 21. Proof. This follows by inspection of the E1 -term of the Adams spectral sequence for Wh Diff(*), and the long exact sequence tfrc* 1 . .!.ssn(CP-11) i*-!ssn(hofib(trc)) j*-!ssn(Wh Diff(*)) --! ssn-1(CP-1) ! : :* *:: The long exact sequence shows that ss18(Wh Diff(*)) ~= ss17(CP-11), which was found in 2.13. Next ss19(Wh Diff(*)) is an extension of the torsion in ss18(CP-* *11) ~= Z=2 o Z=8 o Z=2 Z by Z=2. Also ss20(Wh Diff(*)) is the kernel of a homomorphism from ss19(CP-11) ~=Z=2 Z=8 Z=64 with image Z=8. This is some group of order 27, denoted #27 in the statement of the theorem. Lastly ss21(Wh Diff(*)) is the* * sum of a group of order 24 and an infinite cyclic group, as can be read off from the E1 -term of wE*. Regarding multiplicative structure, we have the following addendum. Lemma 5.11. The homomorphism j# : ssn(Wh Diff(*)) ! ssn+1(Wh Diff(*)) has image (Z=2)2 for n = 9, image Z=2 for n = 10 and is zero for all other n 14. The homomorphism # :ssn(Wh Diff(*)) ! ssn+3(Wh Diff(*)) has image Z=2 for n = 7 and is zero for all other n 14. The homomorphism oe# :ssn(Wh Diff(*)) ! ssn+7(Wh Diff(*)) has image Z=2 for n = 5 and is zero for all other n 11. Proof. The nonzero multiplications listed are all detected by nontrivial h1, h2* * or h3-multiplications in the Adams spectral sequence (5.5) for Wh Diff(*). To see * *that there are no other nonzero multiplications in this range one can use Adams filt* *ra- tion arguments in this spectral sequence, combined with naturality with respect to the map ftrc:WhDiff(*) ! CP-11. For example, ss14(Wh Diff(*)) is detected in ss13(CP-11), but the image of ss7(Wh Diff(*)) in ss6(CP-11) is divisible by * * and oe = 0, so oe# = 0 for n = 7. 6. Cohomology of K(Z) and the linearization map We continue to implicitly complete all spectra at 2. B"okstedt's spectrum JK* *(Z) is the homotopy fiber of the composite 3-1 c ko ---! bspin -! bsu : It is also homotopy equivalent to the algebraic K-theory spectrum K(Z), by [RW], [We]. Hence there is a diagram of horizontal and vertical cofiber sequences of spectra: bso ________bso_____//_*____//bso j|| |t| || |j| fflffl|i fflffl| fflffl| 3fflffl|-1 spin_______//j______//ko____//bspin (6.1) c|| |i| |||| |c| fflffl| fflffl| || fflffl| su______//K(Z)_____//ko____//_bsu | | | | | | | | fflffl| fflffl| fflffl| fflffl| bso ______bso _____//_*___//_2bso TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 43 The right hand column is a connected covering of (3.2), and the second row defi* *nes the connective real image of J spectrum j. We let t = i O j be the composite of* * the Bott map j :bso ! spin and the connecting map i :spin ! j. Miller and Priddy [MP] define spectra g=o and ibo as the pullbacks in the following diagram: g=o _____//_ibo___//_S0 (6.2) || || |e| fflffl|j fflffl|i fflffl| bso______//spin____//_j (More precisely, they define the underlying infinite loop spaces G=O = 1 g=o and IBO = 1 ibo.) Here e: S0 ! j is the map representing the real Adams e-invariant. Its fiber c is the cokernel of J spectrum, which is K-acyclic. T* *hus the unit map i: S0 ! K(Z) factors, uniquely up to homotopy, as e composed with i: j ! K(Z). By (6.1) the cofiber of the bottom composite in (6.2) is K(Z). Hen* *ce there is a cofiber sequence (6.3) g=o -! S0 -i!K(Z) of 2-complete spectra. Thus there is a fiber sequence G=O ! QS0 ! K(Z) of underlying infinite loop spaces, and we might write G=O = IK(Z) as the `ideal' in QS0 = 1 S0 defining K(Z) (at the prime 2). We compute the mod 2 spectrum cohomology H*(K(Z)) by means of the cofiber sequence su ! K(Z) ! ko, where su ' 3ku. In view of (6.3) this also determines H*(g=o ). Miller and Priddy conjecture in [MP] that G=O ' G=O as infinite loop spaces. If confirmed, this would also lead to a calculation of the spectr* *um cohomology H*(g=o). It is known that G=O ' G=O as 2-complete spaces, and that H*(G=O; F2) ~=H*(G=O ; F2) as Hopf algebras over the Steenrod- and Dyer- Lashof algebras, by unpublished calculations of J. Tornehave. Theorem 6.4. The mod two spectrum cohomology of K(Z) is the unique nontrivial extension of A-modules A=A(Sq1; Sq2) -! H*(K(Z)) -! 3A=A(Sq1; Sq3) : The A-module H*(g=o ) is the connected cover of H*(K(Z)), i.e., the kernel of the augmentation H*(K(Z)) ! F2. Proof. We use the cofiber sequence K(Z) ! ko ! bsu where the right hand map is the composite of 3 - 1: ko ! bspin and c: bspin ! bsu. The induced map 4A=A(Sq1; Sq3) ~=H*(bsu) -! H*(ko) ~=A=A(Sq1; Sq2) is the zero homomorphism. For the complexification map c induces multiplication by 2 on ss4, and thus the zero map on H4. Thus the long exact sequence in spect* *rum cohomology decomposes as the A-module extension above. The second claim follows from the cofiber sequence S0 ! K(Z) ! g=o . It remains to characterize the extension. There are precisely two such A-mod* *ule extensions, since Ext1A(3A=A(Sq1; Sq3); A=A(Sq1; Sq2)) ~=Ext1E1(3F2; A=A(Sq1; Sq2)) ~=F2: 44 JOHN ROGNES ____________________________________________________________||*|1|2|4|8| |___|x_2_H_(K(Z))_|Sq_(x)___|Sq_(x)__Sq|(x)______|Sq_(x)____||||||4|8| |_0_0|___________|0________|0________S|q_0_______|Sq_0_____|_||||||| |_1__|___________|_________|_________|___________|_________|_||||||| |_2__|___________|_________|_________|___________|_________|_|||4|2|4|8| |_3_3|___________|Sq_0____|_Sq_3_____|Sq_3_______|Sq_3_____|_||4||6||84| |_4_S|q_0________|0________|Sq_0_____|0___________|Sq_Sq_0__|||2||7|42|82| |_5_S|q_3________|0________|Sq_0_____|Sq_Sq_3____|Sq_Sq_3__|_||6|7||10|104| |_6_S|q_0________|Sq_0____|_0________S|q__0_______|Sq__Sq_0_|||7|||11|| ||7 S|q|04 |0| |0|6 S|q|602 || || |___|Sq_3________|0________|Sq_3_____|Sq_Sq_3____|_________|_||8||10|12|| |_8_S|q_0________|0________|Sq__0____|Sq__0_______|________|_||6|7||10|| ||9 S|q|3 |Sq|3 ||0 S|q| 38 2 | | || || || 4 2 || || 6 2 ||+Sq1Sq33 | | || |___|Sq_Sq_3_____|0________|Sq_Sq_3__|Sq__0_______|________|_||10|11|||| ||10S|q|70 ||Sq 0 |0|8 4 0||11 || || || |Sq|3 |0| |Sq|Sq90| |Sq 3 || || |___|____________|_________|_+Sq_3_|_____________|_________|_||11||13||| ||11S|q|80 ||09 | |Sq100 || || || || |Sq|362 |Sq|384 ||Sq 3 || || || || |Sq|Sq 3 |Sq|Sq90 ||0 || || || |___|____________|__+Sq_3_|__________|___________|_________|_||12|13|14||| ||12S|q|804 ||Sq 0 |Sq|1004 || || || || |Sq|Sq90 |0| |Sq|1Sq004|| || || || |Sq|37 2 | |0 |Sq| Sq 0 || || || |___|__Sq_Sq_3_|___________|_________|___________|_________|_||13||||| ||13S|q|100 |0|11 || || || || || |Sq|832 ||Sq 3 || || || || |___|Sq_Sq_3_____|0________|_________|___________|_________|_||14||||| ||14S|q|1004 || || || || || || |Sq|1Sq10 || || || || || |___|Sq__3________|________|_________|___________|_________|_ Table 6.5. H*(K(Z)) in dimensions 14. Here E1 A is the exterior algebra generated by Sq1 and Sq3. We know that H*(K(Z)) is a nontrivial extension, because Hspec3(K(Z); Z2) ~=ss2(g=o ) ~=Z=2 so there is a nonzero Sq1 from dimension 3 to dimension 4 in H*(K(Z)). We list a monomial basis for H*(K(Z)) in dimensions 14 in Table 6.5. It differs from H*(g=o ) only in dimension 0. The notation is that 0 2 H0(K(Z)) pulls back from the generator of H0(ko), while 3 2 H3(K(Z)) is the unique lift of the generator in dimension 3 of H*(su) ~=3A=A(Sq1; Sq3). We have chosen Sq9(3) = Sq1Sq8(3) as the lift in H*(K(Z)) of Sq93 = Sq7Sq23 in H*(su). TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 45 The linearization map L: A(*) ! K(Z) from [Wa1] and 1.8 is compatible with the unit maps from S0. When combined with the pullback diagram (6.2) defin- ing g=o it yields the following spectrum level diagram with horizontal cofiber sequences: __i__//_ _____//_ Diff S0 A(*) Wh (*) || | | || |L L| || fflffl| fflffl| (6.6) S0 __i_//_K(Z)_____//_g=o |e| |||| || fflffl|i || fflffl| j_____//_K(Z)______//bso Proposition 6.7. The reduced linearization map L :Wh Diff(*) ! g=o is a ra- tional equivalence, but induces the zero homomorphism between the bottom homo- topy groups ss3(Wh Diff(*)) ~= Z=2 and ss3(g=o ) ~= Z=2. The induced map on spectrum cohomology L*:H*(g=o ) -! H*(Wh Diff(*)) is zero in all dimensions. Proof. The linearization map L: A(*) ! K(Z) is a rational equivalence between spectra of finite type, by [Wa1, 2.2], so its 2-adic completion is also a ratio* *nal equivalence. Comparison with (6.6) shows that also L is a rational equivalence. The homomorphism ss3(L ) is induced from the homomorphism ss3(L): ss3(A(*)) ~=Z=24 Z=2 -! K3(Z) ~=Z=48 by passage to the quotient with respect to subgroups ssS3~= Z=24 on both sides. Algebraically, the only possibility is that ss3(L ) = 0. In cohomology we have the following map of extensions of A-modules: * H*(g=o ) ______//_H*(K(Z))i__//_F2 |L*| L*|| |||| fflffl| fflffl|i* || H*(Wh Diff(*))____//_H*(A(*))___//_F2 The lower extension is split, as in 4.5. Here H*(K(Z)) is generated as an A-mod* *ule by classes 0 and 3, as in 6.4 and 6.5. The class 0 maps to the split summand F2 of H*(A(*)), hence the submodule it generates maps to zero in positive degrees. Likewise 3 maps to zero by the ss3-calculation above and the Hurewicz theorem. Thus L* is zero in positive degrees, and L* is zero in all degrees. Corollary 6.8. There is a long exact sequence in mod 2 spectrum cohomology *trc* trc*-L* @ . .!.H*(T C(Z)) L---Z--!H*(T C(*)) H*(K(Z)) --*---! H*(A(*)) -! : ::: 46 JOHN ROGNES Here L: A(*) ! K(Z) and trc*:A(*) ! T C(*) induce zero maps in positive dimen- sions, @ induces an injective map in positive dimensions, and L: T C(*) ! T C(Z) and trcZ:K(Z) ! T C(Z) both induce surjections in all dimensions. Proof. The sequence arises by applying mod 2 spectrum cohomology to the homo- topy cartesian square in 1.8 for X = *. The assertions for L: A(*) ! K(Z) and trc*follow from 4.7 and 6.7. The rest follows by exactness. In fact L* trc*Zwi* *ll be surjective in positive degrees, which is stronger than the stated conclusion. Remark 6.9. The rigid tubes map from [Wa3, x3] provides a space level map of horizontal fiber sequences G=O ________//BSO __j_//_BSG hw|| s|| |w| fflffl| fflffl|i fflffl| Wh Diff(*)_____//_QS0____//_A(*) : We call the left vertical map hw the Hatcher-Waldhausen map. It was proved in [R1] that this gives a diagram of infinite loop maps if one uses a multiplic* *ative infinite loop space structure on each of the spaces in the lower row. However, * *these are generally different from the additive infinite loop space structures we hav* *e been considering in this paper. Let Wh Diff(*) denote the spectrum with underlying infinite loop space given as the homotopy fiber of the unit map i: SG = Q(S0)1 ! A(*)1 with the multiplicative infinite loop space structures. It can be read off from Tables 5.1, 5.7 and 6.5 that the (space level) Hatch* *er- Waldhausen map hw :G=O ! Wh Diff(*) does not admit a four-fold delooping, when the target is given the additive infinite loop space structure. For by [Wa* *3], ss2(hw): Z=2 ~=Z=2 is an isomorphism, and a k-invariant argument (see 7.5 be- low) shows that ss4(hw): Z ! Z is a 2-adic equivalence. If hw admits a four-fold delooping then oe . hw(x) = hw(oe . x) for any x 2 ss4(G=O). But ss11(G=O) = 0, while the minimal resolution leading to Table 5.7 shows that there is a nonzero h3-multiplication from the class representing the generator of ss4(Wh Diff(*)) * *to the class representing the element of order 2 in ss11(Wh Diff(*)). See also 5.* *11. This contradicts the existence of the four-fold delooping. Note that we did not specify a choice of four-fold delooping of G=O in this argument, so it applies * *to both 1 (4g=o) and 1 (4g=o ), in case they are different. The spectrum map g=o ! Wh Diff(*) constructed geometrically in [R1] thus shows that the spectra Wh Diff(*) and Wh Diff(*) cannot be homotopy equivalent. 7. A spectrum map from Wh Diff(*) to g=o Observe by inspection of Tables 5.1 and 6.5 that H*(Wh Diff(*)) and H*(g=o ) are abstractly isomorphic as A-modules in dimensions * 9. In this chapter we construct a spectrum map M :Wh Diff(*) -! g=o inducing an isomorphism in these dimensions. As before, all spectra are implici* *tly 2-completed in this chapter. TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 47 Lemma 7.1. There is a spectrum map m: hofib(trc) ! K(Z) making the following diagram of horizontal cofiber sequences commute: hofib(trc)___//_-2ku__ffi//_4ko |m| rfi-1|| fi24c|| fflffl| fflffl|c( 3fflffl|-1) K(Z) ________//_ko_____//bsu : Proof. The maps in the right hand square are characterized (up to homotopy) by their K-localizations, and after K-localization we can compute fi24c O LK ffi = fi2 O 4c O 4r O fi-2 O ( 3 - 1) O fi-1 = c( 3 - 1) O rfi-1* * : Hence the right hand square commutes. We let m be the induced map of horizontal homotopy fibers. Lemma 7.2. There is a spectrum map M :Wh Diff(*) ! g=o making the fol- lowing diagram of horizontal cofiber sequences commute: CP-11___i_//hofib(trc)j_//WhDiff(*) | m| | | | M| fflffl|i fflffl| fflffl| S0 ________//K(Z)_______//g=o : Proof. We must show that the composite map CP-11-i!hofib(trc) m-!K(Z) -! g=o is null homotopic. Consider the diagram of horizontal and vertical cofiber sequ* *ences S0 _________S0 |i| i|| fflffl| fflffl|c( 3-1) K(Z) _______//ko______//bsu | | || | | || fflffl| fflffl| || g=o _____//ko=S0____//bsu We have [CP-11; su] = 0 by an application of the Atiyah-Hirzebruch spectral se- quence, so we can identify [CP-11; K(Z)] with the kernel of c( 3 - 1)# :[CP-11; ko] -! [CP-11; bsu] : By another calculation with the Atiyah-Hirzebruch spectral sequence using [Ad] and [AW], this kernel is isomorphic to Z, and is generated by the composite map CP-11-! CP+1 -!S0 -i!K(Z) : The left hand map pinches the bottom cell to a point; the middle map retracts CP 1 to a point. The composite maps to zero in [CP-11; g=o ], so m extends to a map M as claimed. 48 JOHN ROGNES Lemma 7.3. The map M :Wh Diff(*) ! g=o induces an isomorphism on ss3. Proof. Consider the maps of long exact sequences of homotopy groups induced by the diagrams in 7.1 and 7.2. The isomorphism ss4(fi24c): Z ~=Z passes to quotie* *nt isomorphisms ss3(m): Z=16 ~=Z=16 and ss3(M): Z=2 ~=Z=2. Theorem 7.4. There is a spectrum map M :Wh Diff(*) ! g=o inducing an isomorphism on mod 2 spectrum cohomology in dimensions * 9. So M is precisely 9-connected, and induces a map of spaces M :Wh Diff(*) -! G=O ' G=O such that ss*(M) is an isomorphism for * 8. Proof. The A-module homomorphism M* :H*(g=o ) ! H*(Wh Diff(*)) is an iso- morphism in degree 3 by 7.3. We can then compute M* in dimensions * 14 from Tables 5.1 and 6.5, finding that H*(hofib(M)) is 9-connected, has rank 1 in eac* *h di- mension 10 * 13, and has rank 1 in dimension 14. Thus M is 8-connected, and the surjection ss8(M) is in fact an isomorphism, since both its source and target are isomorphic to Z Z=2. Theorem 7.5. The Hatcher-Waldhausen map hw :G=O ! Wh Diff(*) induces an isomorphism on 2-primary homotopy in dimensions * 8, and an injection on 2-primary homotopy in dimensions * 13. Its 2-completion is thus precisely 8-connected. Proof. Let P nX denote the nth Postnikov section of a (simple) space X. The map P 2(hw): P 2G=O ! P 2Wh Diff(*) is a homotopy equivalence by 7.3. The k-invariants of G=O and Wh Diff(*) all li* *ft to spectrum cohomology, since these are infinite loop spaces, and are abstractly isomorphic for n 8 by 7.4. They can be partly read off from the minimal resolu- tion for H*(Wh Diff(*)) that was used to generate Table 5.7, yielding the follo* *wing facts: Let fi1: K(Z=2; n) ! K(Z; n + 1) be the mod 2 Bockstein map, and let i1: K(Z; n) ! K(Z=2; n) be the mod 2 reduction map. Then i1fi1 = Sq1. For m n let pmn:P mX ! P nX be a projection in the Postnikov system. Then k5: K(Z=2; 2) ' P 2Wh Diff(*) -! K(Z; 5) is fi1Sq2, while k7: P 4Wh Diff(*) -! K(Z=2; 7) factors as Sq5p42. The last k-invariant we consider is k9 = k91x k92:P 6Wh Diff(*) -! K(Z=2 Z; 9) ' K(Z=2; 9) x K(Z; 9) : Its projection k92to K(Z; 9) factors over p64, and the composite k92 K(Z; 4) -! P 4Wh Diff(*) -! K(Z; 9) TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES 49 is fi1Sq4i1. Here k92= k92O p64. Considering the maps of Postnikov sections P n(hw): P nG=O ! P nWh Diff(*) and comparing the k-invariants for G=O and Wh Diff(*), it follows that also P 4* *(hw) and P 6(hw) are homotopy equivalences, and that P 8(hw) induces an isomorphism on ss8 modulo the torsion subgroups. Hence ss*(hw) is an isomorphism for * 7. In particular the image of 2 2 ss6(SG) ~=ssS6in ss6(G=O) maps to the generator * *of ss6(Wh Diff(*)). The 2-torsion in ss8(G=O) is the image of 2 ss8(SG) ~=ssS8, satisfying j. * *= .2. The image of in ss8(Wh Diff(*)) is nonzero, because j . hw( ) = . hw(2) is nonzero, as can be seen from Table 5.7(a) or detected by M. Hence ss8(hw) is al* *so an isomorphism on the torsion in dimension 8. So hw is 8-connected, but cannot be 9-connected because ss9(G=O) = (Z=2)2 cannot surject to ss9(Wh Diff(*)) = (Z=2)2 Z=8. The nonzero multiplications by j in ss*(Wh Diff(*)) given in 5.11 then imply that ssn(hw) is injective for 9 n 11 and n = 13. Finally ss12(hw) is injective since ss12(G=O) = Z and hw is a rational equivalence [B1]. References [Ad] J.F. Adams, Vector fields on spheres, Ann. of Math. 75 (1962), 603-632. [AW] J.F. Adams and G. Walker, On complex Stiefel manifolds, Proc. Camb. Phil* *os. Soc. 61 (1965), 81-103. [At] M.F. Atiyah, Thom complexes, Proc. Lond. Math. Soc. 11 (1961), 291-310. [BlLi] S. Bloch and S. Lichtenbaum, A spectral sequence for motivic cohomology,* * Invent. Math. (to appear). [B1] M. B"okstedt, The rational homotopy type of WhDiff(*), Proc. Algebraic * *Topology, Aarhus 1982 (I. Madsen and B. Oliver, eds.), Lect. Notes Math., vol. 105* *1, Springer- Verlag, 1984, pp. 25-37. [B2] M. 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