TWOPRIMARY ALGEBRAIC
KTHEORY OF POINTED SPACES
John Rognes
Abstract. We compute the mod 2 cohomology of Waldhausen's algebraic Ktheo*
*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 2primary 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 HatcherWaldha*
*usen
map hw: G=O ! Wh Diff(*) does not admit a fourfold delooping, and there i*
*s a
2complete spectrum map M :WhDiff(*) ! g=o which is precisely 9connected.
Here g=o is a spectrum whose underlying space has the 2complete homotopy*
* type
of G=O.
Introduction
Let A(X) be Waldhausen's algebraic Ktheory of spaces functor evaluated on
the space X, see [Wa1]. When X is a manifold, A(X) provides the fundamental
link between algebraic Ktheory and the geometric topology of X _ in particular
with the concordance space, the hcobordism 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 2primary homotopy type of A(X) in
the case when X = * is the onepoint 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 Ktheory 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 CWcomplexes 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 STEX
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 Gspace 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
nonconnective spectra will appear.
Suspension of retractive spaces over X induces an equivalence on the level of
algebraic Ktheory, and so A(X) can also be considered as the algebraic Ktheory
of a category of spectra over X. It is simplest to make this precise for X = *,*
* when
A(*) is equivalent to the algebraic Ktheory of the category of finite CWspect*
*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 Smodules 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 Smodules. Then A(*) can be described as the algeb*
*raic
Ktheory of a category of Smodules subject to suitable finiteness conditions, *
*and
briefly A(*) is the algebraic Ktheory 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 Amodules, and form its algebraic Ktheor*
*y,
TWOPRIMARY ALGEBRAIC KTHEORY 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 Salgebr*
*as.
For each ring R in the algebraic sense, the EilenbergMac Lane spectrum HR is
an Salgebra whose algebraic Ktheory agrees with Quillen's K(R), see [Q2]. For*
* a
simplicial monoid G the unreduced suspension spectrum 1 (G+ ) is an Salgebra
whose algebraic Ktheory agrees with Waldhausen's A(X) for X = BG. Thus S
algebras encompass the previous examples of inputs for algebraic Ktheory, see *
*also
[Wa7]. Now S is a commutative Salgebra, so its algebraic Ktheory K(S) = A(*)
is itself a ring spectrum, and furthermore the algebraic Ktheory K(A) of any S
algebra is a module spectrum over A(*). Hence every algebraic Ktheory 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=3connected 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 upsidedown', 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
C2action 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 2primary completi*
*on
of the spectrum Wh Diff(*). Since the Whitehead spectrum is a homotopy functor
and preserves connectivity of maps, for any smooth nmanifold X which is roughly
n=3connected the map DiffXcomposed with the natural map
1 (EC2+ ^C2 Wh Diff(X)) ! 1 (EC2+ ^C2 Wh Diff(*))
is roughly n=3connected. Thus when our 2primary calculation is extended to a
calculation of the C2homotopy orbits of Wh Diff(*), we will have complete info*
*rma
tion about the 2primary homotopy type of the space of diffeomorphisms Diff(X)
of roughly n=3connected 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 Ktheory 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
fflfflL 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 2adic completion because the 2primary 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 padic completion of T C(Z) was computed in
[BM], and the 2adic completion was determined in [R5]. The 2adic completion of
K(Z) was found in [RW], by arguments based on Voevodsky's proof of the Milnor
conjecture [Voe] and the BlochLichtenbaum spectral sequence [BlLi]. The 2adic
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 pprimary
homotopy type of A(*) is the identification of the padic completion of K(Z), i*
*.e.,
a proof of the pprimary LichtenbaumQuillen 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 padic 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(*).
TWOPRIMARY ALGEBRAIC KTHEORY OF POINTED SPACES 5
Also in Chapter 1 we make precise a part of the calculation of T C(*) from [*
*BHM],
relating its padic completion to the Thom spectrum CP11= 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_CP11after
padic completion.
In Chapter 2 we analyze the 2primary homotopy type of CP11by classical
methods. We obtain its homotopy groups in dimensions * 20 in Theorem 2.13,
by use of the AtiyahHirzebruch spectral sequence for stable homotopy associated
to the skeleton filtration of CP11by the subspectra CPs1for s 1. The E1term
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 CP11*
*. 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 CP11cohomologically, and in Proposition 2.15 *
*we
find that the mod 2 spectrum cohomology of CP11is cyclic as an Amodule, 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 F2vector
space. Thus H*spec(CP11; F2) ~=2A=C as left graded Amodules. This allows us
to describe the E2term of the Adams spectral sequence for the 2adically compl*
*eted
homotopy of CP11in a range in Tables 2.18(a) and (b). Combined with the results
from the AtiyahHirzebruch 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 2completed) 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 fi2 O ( 3  1) O fi1 :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 LichtenbaumQuillen 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 2completed connective topological Ktheory 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 Ktheory, 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)
CP11i!hofib(trc) j!Wh Diff(*) ;
and the splitting A(*) ' 1 S0 _ Wh Diff(*), to reduce the identification of A(*)
to that of CP11, 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 CP11is cyclic as
an Amodule on a generator in degree 2, so because Wh Diff(*) is 2connected 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 Amodules
H*spec(A(*); F2) ~=H*spec(1 S0; F2) H*spec(Wh Diff(*); F2)
where H*spec(1 S0; F2) = F2 is the trivial Amodule in dimension zero, and there
is a unique nontrivial extension of left graded Amodules
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(CP11; 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
Amodules, 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) =) ssts(Wh Diff(*))^2:
Here we can in principle compute the E2term 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.
TWOPRIMARY ALGEBRAIC KTHEORY 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 E2terms of the Adams
spectral sequences for CP11, 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 E2terms to trans*
*late
the information from Chapter 2 about differentials in the Adams spectral sequen*
*ce
for CP11to 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 CP11mostly originate below this line. The only subtle
point concerns whether certain h1divisible 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 CP11, which is a wellexplored but not exhaustively analyzed subject.
The Adams E2term 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 2completed 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) HatcherWaldhausen map hw :G=O ! Wh Diff(*) constructed in [Wa3,
x3] induces an isomorphism on 2primary homotopy groups in dimensions * 8,
and an injection on 2primary homotopy groups in dimensions * 13. Thus the
better known homotopy groups of G=O ' BSO x CokJ account for much of the
lowdimensional 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 HatcherWaldhausen map hw does not admit a fourfold 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 2completed unit map 1 S0 ! K(Z). Its underlying space
G=O has the same 2adic homotopy type as the usual G=O. Although there is
no spectrum map g=o ! Wh Diff(*) inducing a ss3isomorphism, we construct in
Chapter 7 a 2complete 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 2completed 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 Ktheory and topological cyclic homology
We commence by discussing the cyclotomic trace map from algebraic Ktheory
to topological cyclic homology, and a special case of Dundas' theorem comparing
relative algebraic Ktheory to relative topological cyclic homology.
1.1. spaces and Salgebras. 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
basepoint 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 Salgebra. Its associated spectrum is an associative*
* ring
spectrum, conveniently thought of as an algebra over the sphere spectrum.
1.2. Examples of Salgebras. 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 Salgebra. 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 Rmodule on the nonbasepoint 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 Salgebra. Its associated ring spectrum is the
EilenbergMac 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
Salgebras 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.
TWOPRIMARY ALGEBRAIC KTHEORY OF POINTED SPACES 9
1.3. Algebraic Ktheory, topological Hochschild homology and topolog
ical cyclic homology. Let A be an Salgebra. 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 Ktheory 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 padic 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 Ktheory spec
trum A(X) of the space X [Wa1], i.e., the algebraic Ktheory 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 Ktheory spectrum K(R) of the ring R [Q2], i.e., the algebraic Ktheo*
*ry
of the category of finitely generated projective Rmodules.
We recall from [BHM, 3.7] that there are Cequivariant homotopy equivalences
(1.4) T HH(X) 'C 1C(X+ )
for each finite subgroup C S1. Here 1C denotes the Cequivariant 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 Salgebras 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 Salgebras a theorem of McCarthy [Mc] valid for maps of simplicial ring*
*s.
Both results are analogous to an older theorem about rational algebraic Ktheory
due to Goodwillie [Go].
10 JOHN ROGNES
Theorem 1.7 (Dundas). Let OE: A ! B be a map of Salgebras, 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
fflfflOE 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
fflfflL 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 hcobordisms, 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.
TWOPRIMARY ALGEBRAIC KTHEORY 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 fflfflfi 
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 Segaltom Dieck splitting
Yn
1Cpn(X+ )Cpn ' 1 (ECpkxCpk XCpnk)+
k=0
and the power map homeomorphisms nkp:X ~=XCpnk 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 (ECpn1xCpn1X)+
be the BeckerGottlieb transfer for the principal Cpbundle ECpn1xCpn1X !
ECpn xCpn X. There are restriction and Frobenius maps R; F :T HH(X)Cpn !
T HH(X)Cpn1. Up to homotopy these are given by the formulas:
R(x0; x1; : :;:xn)= (x0; x1; : :;:xn1)
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 1F :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 (1T )O(1D): 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)1F//_T R(X; p)
ffX 1D 
fflffl fflffl1T 
C(X; p)_____//_T R(X;_p)__//T R(X; p) :
TWOPRIMARY ALGEBRAIC KTHEORY 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 1T 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+
1D 1p
fflfflpr 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 BeckerGottlieb
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
fflffl1p 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 dimensionshifting S1transfer map
trfnS1:1 ((CP+1)) ! 1 (BCpn+)
associated to the S1bundle 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 padic completion. Hence we can identify *
*the
map pr0 above with the S1transfer map trf0S1, briefly denoted trfS1, after pa*
*dic
completion. Combined with the padic 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 padic 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 CP11, 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
CP11' 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
(CP11)^p' gT C(*)^p
of padically completed spectra.
2. Twoprimary homotopy of CP11
In this chapter we study the 2primary homotopy type of the Thom spectrum
CP11of minus the canonical complex line bundle over CP 1. We first use a rein
dexed AtiyahHirzebruch spectral sequence for stable homotopy to compute the
2completed homotopy groups ss*(CP11)^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 AtiyahHirzebruch spectral sequence in question is derived from
the stable homotopy exact couple associated to the filtration of CP11by the su*
*b
spectra CPs1, for s 1. Its E1term is
(2.1) E1s;t= sss+t(CPs1=CPs11) ~=ssSts
for s 1, and zero elsewhere. Here ssSk= ssk(1 S0) is the kth stable stem.
To determine the differentials in the reindexed AtiyahHirzebruch 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 E1term
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 :CP11! 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.
TWOPRIMARY ALGEBRAIC KTHEORY 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 S2 ! CP11!
CP+1. This is the wedge sum of the (desuspended) S1transfer map CP 1 ! S1,
and the (desuspended) multiplication by j map S0 ! S1. 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 2primary components, and all spectra and groups
are implicitly 2completed. Differentials are mostly given only up to multiplic*
*ation
by a 2adic 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 drdifferentials 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) = jfis1 for s even.
Proposition 2.3. d2(fis) = fis2 for s 0; 1; 4; 5 mod 8, d2(fis) = 2fis2 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) = fls3, 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 d1differentials 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 d2differentials 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 d3differentials 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 E4term 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 d4differentials in (2:1) are d4(23) = 2oe1, d4(45) =
8oe1, d4(46) = 8oe2, d4(7) = 2oe3, d4(88) = 8oe4, d4(49) = 4oe5, d4(210) = 2oe6*
* and
d4(oe3) = oe21.
Proof. The d4differentials 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 S1transfer in [Mu1] and [Mu2]. Thus d4(23) = 2oe1 by [Mu1, 13.1(iii)],
d4(oe3) = oe21by 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 d5differentials in (2:1) are d5(86) = 1, d5(168) = 3
and d5(1610) = 5.
Proof. The d5differentials 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 d6differentials in (2:1) are d6(85) = i1, 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) = i1 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).
TWOPRIMARY ALGEBRAIC KTHEORY OF POINTED SPACES 17
____________________________________________________________________________*
*
___0____________________________________________________________*
*
___0___4(20_)___________________________________________________*
**
2(1) 2(oe0)2(21)         
            
_______8(i0)____________________________________________________*
**
4(i1)_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(j1)16(2ae0)2(oe1)00 8(i4)0 2(jffl6)    
            
_______2(j0)____________________________________________________*
*2
32(ae1)2(oe0)0_____0___4(i3)__0__2(5)___0___16(oe7)____________2*
*3
2(oe1)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(i1)___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(oe1)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 d7differential 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 d7differentials are zero by [Mo, 6.7] or bidegree reasons.
18 JOHN ROGNES
Lemma 2.10. The nonzero d8differentials in (2:1) are d8(167) = 2ae1,
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) = i1.
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 2primary homotopy groups of CP11in dimensions * 20
are as follows:
ss2(CP11)= Z(1);
ss1(CP11)= 0;
ss0(CP11)= Z(20);
ss1(CP11)= 0;
ss2(CP11)= Z(41);
ss3(CP11)= Z=8(0);
ss4(CP11)= Z(22);
ss5(CP11)= Z=2(oe1);
ss6(CP11)= Z=2(20) Z(163);
ss7(CP11)= Z=2(1) o Z=8(2oe0)
~=Z=16(2oe0);
ss8(CP11)= Z=2(21) Z(84);
ss9(CP11)= Z=2(30) Z=2(jffl0) Z=8(oe1);
ss10(CP11)= Z(325);
ss11(CP11)= Z=8(i0) Z=4(2oe2);
ss12(CP11)= Z(166);
ss13(CP11)= Z=2(ae1) o Z=2(i1) o Z=2(jffl2)
~=Z=2(ae1) o Z=4(jffl2);
ss14(CP11)= Z=2(oe20) Z=2(3 ) Z(287);
ss15(CP11)= 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(CP11)= Z=2(oe21) Z=2(25) Z(278);
ss17(CP11)= 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(CP11)= Z=2(1 ) o Z=8(*0) o Z=2(j*1) Z(299);
ss19(CP11)= Z=2(oe0) Z=8(i0) Z=8(2ae2) o Z=4(i4) o Z=2(48)
TWOPRIMARY ALGEBRAIC KTHEORY 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)00 4(i4)0      
            
______2(j0)___________________________________________________2*
*
2(ae1)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)00 0 0 0  0 (29) 
            
______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(oe1)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(CP11)= 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 CP11! 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 h0multiplications in the Adams spectral sequence for ss*(CP11).
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(CP01), see [Mu3, p.133].
The splitting in dimension 16 can be deduced from the injection ss16(CP11) !
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*(CP11)^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 Amodule, 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 Amodule.
Proposition 2.15.
H*(CP11) ~=2A=C
as graded left Amodules.
Proof. It is clear that Hn(CP11) ~= 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 Sq2i1(yj) = 0 and Sq2i(yj) = jiyi+j. By James periodicity and stability of
the squaring operations the same formulas apply in
H*(CP11) ~=F2{yj  j 1} ;
1
also with j = 1. Then Sq2i(y1) = yi1 since i 1 mod 2. To prove the
proposition it remains to show that SqI(y1) = 0 when I = (i1; : :;:in) is admi*
*ssible
of length 2. Let z = Sqin(y1). 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 Sqin1(z) = 0 since in1 > in  2, and so SqI(y1) = 0.
TWOPRIMARY ALGEBRAIC KTHEORY OF POINTED SPACES 21
Lemma 2.16. In CP11, the lowest kinvariant
k1: 2HZ ! HZ
is nontrivial, and has mod 2 reduction the class of Sq3 mod ASq1.
Proof. The lowest homotopy group of CP11is detected by a map CP11! 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 kinvariant k1.
Consider the Adams spectral sequence
(2.17) Es;t2= Exts;tA(H*(CP11); F2) =) ssts(CP11)^2:
Its E2term can be computed in a range from a (minimal) resolution of 2A=C,
either by hand or by Bruner's Extcalculator program [Br]. The E2term 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 E2term for CPs1, mapping to the class
named x in the Adams E2term for CPs1=CPs11~=2sS0. The distinction be
tween classes marked as `o' or as `O' will be explained in x5.
The cofiber sequence of spectra
CP01i!CP11j!CP 1
induces a short exact sequence in mod 2 spectrum cohomology, and thus gives a
long exact sequence of Extgroups relating the Adams E2term (2.17) to the Adams
E2terms
(2.19) 0Es;t2= Exts;tA(H*(CP01); F2) =) ssts(CP01)^2
and
(2.20) 00Es;t2= Exts;tA(H*(CP 1); F2) =) ssts(CP 1)^2:
Knowledge of the stable homotopy of CP01' 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 CP01, 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 CP01and 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 jO1Ph2O/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 O1c0OWW/ O4h30OOjj//O 2h2O0h3O*
* 4h31
  j j j """" j j j """// j j j """" ///"""
  j j ""  j j """  /j/j ""  o "" 
o1h20 o0h20j o1h20jo0h0h2jOj o j j Oj //O O1h22O O5h20 O2h0*
*h3
  j j j  j"j"j"" ///"""" 
  j j  j j"" o"" 
o1h0 o0h0j o0h2jo2h0 O2h2 O1h3


o11
2 1 0 1 2 3 4 5 6 7 8 9 10 11
Table 2.18(a). The Adams E2term for CP11
(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.
TWOPRIMARY ALGEBRAIC KTHEORY 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
   jjj //    
   j j   //    
  j j j   //    
O O o O O j j O1P2h2O/j/jOoWW/WWO/ O
   """ jjj // jj/j/ //  
   "" jj  jj////  //  
  ""jjj  j j j  / o  //  
O O o O O OOWW///O/ j j OWW///OOWjjjOOjW/
    """// j/jj/ //jjjjjj"""//*
*//
    "" jj//// jjjj""//////
   ""jjj  // ojjjjjj""/o/// 
O o O j j O1Pc0OTT* O j jOOWW///O OOWW///O jjjOOj//
   j jj """**  j j j """////  """"""jjjjjj//////
  jj ""  * jj "" // // """"jjjj//// //
 j j j  "" j*j*j  ""  //o """"jjjjjjo// 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"////"""
  jj ** ** jj"" * // //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/jj/""""""
  //  *** ** * """/jj/j jj j // """"
  //  * **j ** "jj oj j  o"""" 
O OWW///O OW1d0OjjW/ *jj*OO**jj O1f0*O*Oj4h1c0OOjWWO/ OO O
 // j"j"j"//// j j j *j"j"j"***/j*jj*/"""  
 jj""////// jj j j"""**** jj// **""   
j j j "// oj j/j/j jj " **jjj j / j""   
O 2h2O0h3O/4h31Oj/jO//0h0Oh23Oj3c0OO** OWW///OO jjOO OO j jOWW/ jOO
 "" WW///jj // ""j jWjW**// // ""jjjjjj"// jj ""jjj// 
 "" jj/j/// / j"j"  // * jjjjjj""////jjj  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/// "" /jj/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 E2term for CP11
Proof. We compare the Adams E2term in Table 2.18 with its abutment 2.13. Each
h0torsion class in the E1 term of (2.17) comes from an h0torsion class in th*
*e E2
term, and so is represented by a 2torsion class in ss*(CP11). (The proof of *
*this
assertion goes by induction over the subspectra CPs1of CP11.)
In each degree ts 5 the order of the 2torsion in ss*(CP11) equals the or*
*der of
24 JOHN ROGNES
the h0torsion in Table 2.18, hence there are no nonzero differentials in this *
*range.
(i): In degree t  s = 6 the 2torsion in the abutment is Z=2, while the E2*
*term
has two h0torsion 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(CP11) 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 00d2differential, 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 h0multiplicati*
*on,
and more classes would survive to the E1 term in degree 9 than the abutment
ss9(CP11) ~=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 00d2differential 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 d2differential from 6h2 2 E1;162. The 2torsion in ss14(CP1*
*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 CP01, 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 16stem.)
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(CP11) ~=Z (Z=2)2, the remaining h0torsion 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 00E2term
and the 00E1 term given in Table 7.2 of [Mo] we deduce that there are differen*
*tials
TWOPRIMARY ALGEBRAIC KTHEORY 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 E2term 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 2torsion in t*
*he
abutment in degree 18 has order 25, and the E2term has seven h0torsion classe*
*s.
Hence at most two differentials leave the h0torsion in degree 18, and at least*
* three
differentials leave the h0periodic part of the E2term. 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 2h2, 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 h2multiplications in the E2term, 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 h0multiplications to three nonze*
*ro
d3differentials from the h0torsion in degree t  s = 18, which is incompatible
with the abutment. Thus the two d2differentials 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 BlochLichtenbaum spectral sequence relating motivic cohomo
logy to algebraic Ktheory, Voevodsky's proof of the Milnor conjecture, which r*
*e
lates motivic cohomology toetale cohomology, and knowledge of theetale cohomo
logy of the rational 2integers Z[1_2].
Similarly, the padic 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 2adic 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 twoadic Ktheory spectra. We say that a (1)connected spec
trum is connective, and a 0connected spectrum is connected. Let KO and KU
denote the real and complex topological Ktheory 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 3connected covers of KO, and bsu for
the 3connected 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 kconnected covers of
these maps. There is a cofiber sequence of spectra
2rOfi1
(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 fi1 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
31 @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 2primary algebraic *
*K
groups of rings of 2integers 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 2integers Z[1_2] in Q*
* the
result implies that there is a 2adic 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 (RognesWeibel). 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
2rOfi1
K(F3) i3!ku ! 2ko :
Proof. B"okstedt's JK(Z[1_2]) can be defined as the homotopy fiber of the compo*
*site
31
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,
TWOPRIMARY ALGEBRAIC KTHEORY 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 
fflffli3 fflffl 31
K(F3) _______//_ku____//bu
@ 2rOfi1
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 2adic numbers, such that OE3(i) = i3 when ipis_a
2powerproot_of unity, i.e., in 21 Qx2. We may further assume that OE3(+ 3) =
+ 3. Then OE3 induces a selfmap of K(Q2 ) which is compatible up to homotopy
with 3: ku ! ku under Suslin's (implicitly 2adic) 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 twoadic unit is homotopic to the composite
1 fi1
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 Ktheo*
*ry
from [Q2].
Theorem 3.8 (HesselholtMadsen). In the commutative square of spectra
K(Z) __j__//_K(Z2)
trcZ trcZ2
fflfflj 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 2adic 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 fflfflred 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)
TWOPRIMARY ALGEBRAIC KTHEORY 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 v41periodicity. 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 fi1 :ku !
2ko. Thus it suffices to show that both homomorphisms i#3and (2r O fi1)# are
injective.
There is an exact sequence
31)# 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 ( 31) is null homotopic, because OE and ( 31) commute
and @3 O ( 3  1) ' *. Hence the left hand map is null and i#3is injective.
There is also an exact sequence
# (2rOfi1)#
[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 fi1)# 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 2adic homotopy mod
ulo torsion in dimensions * 1 mod 8, and multiplication by 2 times a 2adic 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 selfmap 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 2adic
30 JOHN ROGNES
homotopy equivalence. We obtain the following diagram of horizontal and vertical
cofiber sequences:
K(F3) ________//*_____//_K(F3)
  
  fC
fflffl fflffljredfflffl
hofib(jred)____//ko_____//Kred(Q2)
  
 ' OE 
fflfflj fflfflc fflffl
2ko _______//ko_______//ku
The connecting map @ :2ko ! K(F3) is detected by its precomposition with
2r O fi1 :ku ! 2ko, because [ko; K(F3)] = 0. By the diagram above, the
composite @ O 2r O fi1 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 fi1 = @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
31 fi1 2(2rOfi1)
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 31 fi2 4r fflffl
KU _____//KUoo_'_ 4KU ____//_4KO :
The maps labeled cov are kconnective 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 fflffl2c fflffl
3ko _______//2ko ________//_2ku
 @ @3Ofi
  
fflffl fflffl fflffl
* _______//_K(F3)_______K(F3)
TWOPRIMARY ALGEBRAIC KTHEORY 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
31 fi1
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 fi2 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
fflfflfi 31 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 kinvariant of X relating the homotopy groups *
*in
dimensions 2 and 0.
Since 3ko is 2connected, this lowest kinvariant is the same for X as for
hofib(trc). By combining 1.11 with 1.21 we obtain a cofiber sequence
(3.14) CP11i!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 kinvariants for hofib(trc) and CP11are 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 2adic 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 Klocalization, induces the vertical map in t*
*he
commutative diagram
cov# cov#
[2ku; 4ko] _____//[2ku; 4KO] oo~=_ [2KU; 4KO]
fi# ~=fi# fi#
fflfflcov# 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
fi2 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 fi2 O ( 3  1) O fi1. 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 1connected
space X, since the natural map
hofib(trcX) '!hofib(trc)
is a homotopy equivalence.
Let v2(k) be the 2adic 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.
TWOPRIMARY ALGEBRAIC KTHEORY 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 """
     //oj j j "" 
O O O O o // o o j jO
     /// jj j 
     oj 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 E2term 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) =) ssts(hofib(trc))^2:
is then easily deduced from the E2terms 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 h0tower of nonzero drdifferentials 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  ""
   jj 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 E2term for hofib(trc)
Consider the following diagram:
CP11__________CP11
i ffl
fflffl fflfflffi
(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 CP11! hofib(trf) ! 2ku. Then the
right column and bottom row are cofiber sequences.
Proposition 4.2. The map ffl induces the unique surjection of Amodules
*
2A=A(Sq1; Sq3) ~=H*(2ku) ffl!H*(CP11) ~=2A=C :
TWOPRIMARY ALGEBRAIC KTHEORY OF POINTED SPACES 35
Hence
H*( hofib(ffl)) ~=2C=A(Sq1; Sq3)
as an Amodule.
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 Amodule, and ffl* is surjective because H*(CP*
*11)
is a cyclic Amodule. 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 Amodule 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 Amodule.
Theorem 4.4. The mod two spectrum cohomology of hofib(trc) is the unique non
trivial extension of Amodules
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 Amodules
2C=A(Sq1; Sq3) ! H*(Wh Diff(*)) ! 3A=A(Sq1; Sq2) :
The mod two spectrum cohomology of A(*) is given by the splitting of Amodules
H*(A(*)) ~=H*(Wh Diff(*)) F2:
Here F2 = H*(S0) denotes the trivial Amodule 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 Amodules, as claimed.
2A=COO _____________2A=COO
i* ffl*
 
(4.6) 3A=A(Sq1; Sq2) oo___ H*(hofib(trc))oo_O2A=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 subHopf
algebra generated by Sq1 and Sq2. Hence there are changeofrings 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 A1module 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 A1modules, there
are precisely two such A1module 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 3connected, 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*) ' CP11
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
[CP11; 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: CP11! 1ku induce the same homomorphisms up to 2adic units, on
homotopy groups modulo torsion in dimensions * 3 mod 4.
Proof. The suspended map ffl is the composite
CP11! T C(*) L!T C(Z) ! hofib(trc) ! 1ku :
TWOPRIMARY ALGEBRAIC KTHEORY 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. Twoprimary 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: CP11! 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 SqI2
taken modulo A(Sq1; Sq3)2. By inspection ker(ffl*) is 3connected. 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 Amodule structure.
We now consider the Adams spectral sequences associated with the spectra in
the cofiber sequence of spectra
(5.2) CP11i!hofib(trc) j!Wh Diff(*)
appearing vertically in (4.1). They are
(5.3) cEs;t2= Exts;tA(H*(CP11); F2) =) ssts(CP11)^2
(5.4) fEs;t2= Exts;tA(H*(hofib(trc)); F2) =) ssts(hofib(trc))^2
(5.5) wEs;t2= Exts;tA(H*(Wh Diff(*)); F2) =) ssts(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 Amodules in cohomology,
by (4.6), and thus a long exact sequence of Extgroups
(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(*) ! CP11extending (5.2), and so each map in
38 JOHN ROGNES
_______________________________________________________________________________*
*__*Diff1248
____x2_H_(Wh____(*))_Sq_(x)_______Sq_(x)________Sq_(x)_______Sq_(x)______*
*__
__2______________________________________________________________________*
*__42748
_3__3_______________Sq_Sq_2_____Sq_2_________Sq_3_________Sq_3_________*
*_4262842
_4__Sq_Sq_2________0_____________Sq_Sq_2_____0_____________Sq_Sq_Sq_2_*
*__7911132
5 Sq2 0 Sq 27 2  Sq 2 Sq Sq1214 *
*
____________________________________Sq_Sq_2__________________+Sq__Sq_2_*
*_6291021042
6 SqSq 2 Sq27 2 0 Sq Sq 2 Sq Sq Sq 2 *
*
_______________________Sq_Sq_2__________________________________________*
*__9112
7 Sq27 2  0 0 Sq Sq 2  *
* 
 Sq4Sq 2    6  13  *
* 
____Sq_3____________0_____________Sq_3_________Sq__2____________________*
*__82102122
_8__Sq_Sq_2________0_____________Sq__Sq_2____Sq__Sq_2_________________*
*__1113132
9 Sq62 07 Sq 2 Sq1Sq322  *
* 
 Sq3 Sq3 0 Sq Sq1214  *
* 
     +Sq10Sq 2  *
* 
_________________________________________________+Sq__3__________________*
*__102112
10 Sq8Sq42 Sq Sq 2 0 10 4 012 4  *
* 
 SqSq72 0 Sq Sq 2 Sq1Sq12   *
* 
____Sq_3____________0_____________0_____________Sq__3____________________*
*__13
11 Sq1212 0 013 2   *
* 
 Sq8Sq 2 08 4 2 Sq10Sq 2   *
* 
____Sq_3____________Sq_Sq_Sq_2__Sq__3___________________________________*
*__122132142
12 Sq1Sq024 Sq1Sq124 Sq Sq 2   *
* 
 Sq8Sq422 Sq Sq 2 0 10 4 2   *
* 
____Sq_Sq_Sq_2_____0_____________Sq__Sq_Sq_2___________________________*
*__15
13 Sq1232 0    *
* 
 Sq1Sq124 0    *
* 
 Sq1Sq02 011    *
* 
____Sq__3___________Sq__3________________________________________________*
*__142
14 Sq1Sq224     *
* 
 Sq1Sq0242     *
* 
 Sq1Sq1Sq 2     *
* 
____Sq__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 2completed
homotopy induced by (5.2).
The E2term 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
TWOPRIMARY ALGEBRAIC KTHEORY 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 E2term for Wh Diff(*)
Bruner's Extcalculator 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: CP11! hofib(trc) induces a map
i*: cEs;t2!fEs;t2
of Adams E2terms, 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 h0divisible 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 Klocaliza*
*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* **""jjjo  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/jjj/j""/jjjj/*j""*// 
2 o ooW//ojWj/o// oojj*oo*WWo/W/oojjjooj/Wooj/jo
 "" /jj///// j"j"//** /jjjj""//j/j ""
""jjj/o/jjoj""  //jjjjjjj""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 oj j j
0 o o o
11 12 13 14 15 16 17 18 19 20 21
Table 5.7(b). The Adams E2term 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 h0torsion classes.
(vii) ds;s+1826= 0 for s = 0; 1.
(viii) d2;2036= 0 is zero on the h1divisible classes.
(ix) ds;s+1926= 0 for s = 3; 4 is zero on the h1divisible 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.
TWOPRIMARY ALGEBRAIC KTHEORY 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 h0multiplications 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 h1divisible 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 2primary 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
CP11i!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(CP11) i*!ssn(hofib(trc)) j*!ssn(Wh Diff(*)) ! ssn1(CP1) ! : :*
*::
The long exact sequence shows that ss18(Wh Diff(*)) ~= ss17(CP11), 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(CP11) ~=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
h3multiplications 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(*) ! CP11. For example, ss14(Wh Diff(*)) is detected
in ss13(CP11), but the image of ss7(Wh Diff(*)) in ss6(CP11) 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
31 c
ko ! bspin ! bsu :
It is also homotopy equivalent to the algebraic Ktheory 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
fflffli fflffl fflffl 3fflffl1
spin_______//j______//ko____//bspin
(6.1) c i  c
fflffl fflffl  fflffl
su______//K(Z)_____//ko____//_bsu
   
   
fflffl fflffl fflffl fflffl
bso ______bso _____//_*___//_2bso
TWOPRIMARY ALGEBRAIC KTHEORY 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
fflfflj fflffli 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
einvariant. Its fiber c is the cokernel of J spectrum, which is Kacyclic. 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 2complete 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 2complete 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 Amodules
A=A(Sq1; Sq2) ! H*(K(Z)) ! 3A=A(Sq1; Sq3) :
The Amodule 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 Amodule 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 Amod*
*ule
extensions, since
Ext1A(3A=A(Sq1; Sq3); A=A(Sq1; Sq2)) ~=Ext1E1(3F2; A=A(Sq1; Sq2)) ~=F2:
44 JOHN ROGNES
____________________________________________________________*1248
___x_2_H_(K(Z))_Sq_(x)___Sq_(x)__Sq(x)______Sq_(x)____48
_0_0___________0________0________Sq_0_______Sq_0______
_1____________________________________________________
_2____________________________________________________4248
_3_3___________Sq_0_____Sq_3_____Sq_3_______Sq_3______4684
_4_Sq_0________0________Sq_0_____0___________Sq_Sq_0__274282
_5_Sq_3________0________Sq_0_____Sq_Sq_3____Sq_Sq_3___6710104
_6_Sq_0________Sq_0_____0________Sq__0_______Sq__Sq_0_711
7 Sq04 0 06 Sq602  
___Sq_3________0________Sq_3_____Sq_Sq_3______________81012
_8_Sq_0________0________Sq__0____Sq__0________________6710
9 Sq3 Sq3 0 Sq 38 2   
  4 2   6 2 +Sq1Sq33   
___Sq_Sq_3_____0________Sq_Sq_3__Sq__0________________1011
10Sq70 Sq 0 08 4 011  
 Sq3 0 SqSq90 Sq 3  
_________________________+Sq_3________________________1113
11Sq80 09  Sq100   
 Sq362 Sq384 Sq 3   
 SqSq 3 SqSq90 0   
_________________+Sq_3________________________________121314
12Sq804 Sq 0 Sq1004   
 SqSq90 0 Sq1Sq004  
 Sq37 2  0 Sq Sq 0   
_____Sq_Sq_3__________________________________________13
13Sq100 011    
 Sq832 Sq 3    
___Sq_Sq_3_____0______________________________________14
14Sq1004     
 Sq1Sq10     
___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).
TWOPRIMARY ALGEBRAIC KTHEORY 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  
fflffli  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 2adic 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 Amodules:
*
H*(g=o ) ______//_H*(K(Z))i__//_F2
L* L* 
fflffl fflffli* 
H*(Wh Diff(*))____//_H*(A(*))___//_F2
The lower extension is split, as in 4.5. Here H*(K(Z)) is generated as an Amod*
*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 ss3calculation 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)) LZ!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 fflffli fflffl
Wh Diff(*)_____//_QS0____//_A(*) :
We call the left vertical map hw the HatcherWaldhausen 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 fourfold 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 kinvariant argument (see 7.5 be
low) shows that ss4(hw): Z ! Z is a 2adic equivalence. If hw admits a fourfold
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
h3multiplication 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 fourfold delooping. Note that we did not
specify a choice of fourfold 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 Amodules 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
2completed in this chapter.
TWOPRIMARY ALGEBRAIC KTHEORY 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 rfi1 fi24c
fflffl fflfflc( 3fflffl1)
K(Z) ________//_ko_____//bsu :
Proof. The maps in the right hand square are characterized (up to homotopy) by
their Klocalizations, and after Klocalization we can compute
fi24c O LK ffi = fi2 O 4c O 4r O fi2 O ( 3  1) O fi1 = c( 3  1) O rfi1*
* :
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:
CP11___i_//hofib(trc)j_//WhDiff(*)
 m 
  M
fflffli fflffl fflffl
S0 ________//K(Z)_______//g=o :
Proof. We must show that the composite map
CP11i!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 fflfflc( 31)
K(Z) _______//ko______//bsu
  
  
fflffl fflffl 
g=o _____//ko=S0____//bsu
We have [CP11; su] = 0 by an application of the AtiyahHirzebruch spectral se
quence, so we can identify [CP11; K(Z)] with the kernel of
c( 3  1)# :[CP11; ko] ! [CP11; bsu] :
By another calculation with the AtiyahHirzebruch spectral sequence using [Ad]
and [AW], this kernel is isomorphic to Z, and is generated by the composite map
CP11! 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 [CP11; 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 9connected, and induces a map of spaces
M :Wh Diff(*) ! G=O ' G=O
such that ss*(M) is an isomorphism for * 8.
Proof. The Amodule 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 9connected, has rank 1 in eac*
*h di
mension 10 * 13, and has rank 1 in dimension 14. Thus M is 8connected,
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 HatcherWaldhausen map hw :G=O ! Wh Diff(*) induces
an isomorphism on 2primary homotopy in dimensions * 8, and an injection
on 2primary homotopy in dimensions * 13. Its 2completion is thus precisely
8connected.
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 kinvariants 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 kinvariant 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)
TWOPRIMARY ALGEBRAIC KTHEORY 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 kinvariants 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 2torsion 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 8connected, but cannot
be 9connected 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].
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Department of Mathematics, University of Oslo, Norway
Email address: rognes@math.uio.no