Topological K-theory of GL(Z) at the Prime 2
Luke Hodgkin
Department of Mathematics, King's College, Strand, London WC2R 2LS
(email: luke.hodgkin@kcl.ac.uk)
16/7/97
Abstract
Recent results of Voevodsky and others have effectively led to the proof of the*
* Lichtenbaum-Quillen con-
jectures at the prime 2, and consequently made it possible to determine the 2-l*
*ocal homotopy type of the
K -theory spectra for various number rings. The basic case is that of BGL(Z); i*
*n this note we use these
results to determine the 2-local (topological) K-theory of the space BGL(Z), wh*
*ich can be described as a
completed tensor product of two quite simple components; one corresponds to a r*
*eal `image of J' space, the
other to BBSO .
1. Introduction.
As a result of Voevodsky's solution of the Milnor conjecture [V] and related wo*
*rk
by Bloch, Lichtenbaum, Voevodsky and Suslin [B-L] [V-S], Weibel in [W] has calc*
*u-
lated the algebraic K-theory of the integers ssi(BGL(Z)+ ) at the prime 2 in te*
*rms
which essentially confirm the appropriate version of the Lichtenbaum-Quillen co*
*njec-
tures [L,D-F]. This result, since it expresses the space BGL(Z)+ in terms of ra*
*ther
well-known spaces, makes it relatively easy to deduce other invariants. Arletta*
*z et al.
in [A-M-N-Y] have done this for the mod 2 cohomology; in this paper, I shall do*
* the
same for the (topological) 2-local K-theory; the result, which is, perhaps pred*
*ictably,
quite different from that for cohomology is stated in theorem 4.1 below.
While Weibel's results are more general in character, and could lead to similar*
* calcula-
tions for various other rings e.g. Z[i], I shall here confine my attention to t*
*he integers,
partly because of their `historical' interest, and partly because of the link w*
*ith the sta-
ble mapping class group B = lim!Bn via the composite B ! BSp(Z) ! BGL(Z),
which arises from the action of surface homeomorphisms on H1. It should be not*
*ed
that the corresponding decomposition of spectra for the p-adics BGL(Zp^) is als*
*o known
_ for arbitrary p _ through work of B"okstedt, Madsen and Rognes [B-M], [R]. (H*
*ere
the Milnor conjecture is not needed.)
Recall that the `etale K-spectrum' JK(Z) was defined in [B"o] as the (2-local) *
*homotopy
fibre of the composite map:
3-1 c
(1) c( 3 - 1) : BO -! BSpin -! BSU
This can be realized through a number of other fibrations, of which we shall no*
*te
particularly (cf [R], (2.3))
(2) JR2 ! JK(Z) ! BBSO
where JR2 is the real image of J space at 2, defined as the fibre of 3-1 : BO *
*! BSpin,
localized at 2. (See e.g. [Ma].) There has long been known to be a map (which, *
*properly
1
defined, extends to a map of ring spectra) from JK(Z) to BGL(Z)+ ; it is a cons*
*equence
of Voevodsky's theorem that this is an equivalence.
I shall suppose throughout that all spaces have been localized at 2. Since we s*
*hall be
dealing with 2-local theories, this will not affect the results, but it will gu*
*arantee the
accuracy of some statements which I would otherwise be unsure about.
2. The 2-complete theory.
The natural procedure is first to deal with the 2-completed theory, and proceed*
* to
integrate it with the rational to obtain a 2-local statement. However, as the *
*spaces
concerned are too large for Sullivan's fibre-square [S] to apply, this can't be*
* done
without some extra precautions. With this in mind, we make our first approach *
*via
the completed theory, and begin with the commutative diagram:
SU? -j! BGL(Z+?) -h! BO?
(3) ?yi ?yf3 ?yc
U -! F 3 -b! BU
Here the upper fibration is the one in (1) above (using the identification of B*
*GL(Z)+
which follows from Voevodsky's theorem). F 3 is the complex image of J space; *
*the
map b is the `Brauer lifting' [Q] which maps F 3 = BGL(F+3) to BU, and is its*
* fibre.
We know that the lower fibration is induced from the map 3 - 1 : BU ! BU, where
3 is the Adams operation. Since 3 commutes with complexification, the squares*
* are
commutative.
The most `natural' approach for such K-theory computations is usually via the g*
*e-
ometric spectral sequence of Rothenberg-Steenrod (see e.g. [A-H]), which gives*
* the
K-theory of a quotient of groups (for example) in terms of those of the group a*
*nd
subgroup. With this in mind, we consider the upper row of (3) a stage further t*
*o the
left, and compare it with the fibre sequence which defines the space JR2.
3-1) j
Ow c(-! SUx -! BGL(Z)+x
(4) ww ??c ??j
3-1
O - ! Spin -! JR2
Both rows in this diagram are fibrations. The two squares are commutative by c*
*on-
struction (compare the diagram on p.8 of [R]); and the right hand square is fib*
*red.
Hence, we can apply the Rothenberg-Steenrod spectral sequence to the fibre squa*
*re
provided that one of the Spin-actions is free. It will be convenient to suppos*
*e this
for the action on SU, by the usual device of replacing SU by SU x ESpin. We are
accordingly using a homotopy equivalence of BGL(Z)+ with (SU x ESpin) xSpinJR2.
Much of the argument can be simplified in this case , as we shall see, since we*
* are
dealing with a trivial comodule. To clarify the details of the application we w*
*ant, i.e.
to the 2-complete K-cohomology, one or two technical points should be made. Fir*
*st,
2
there is (as usual in K-cohomology of large spaces) the question of topology on*
* the
modules K*(X; Z2^). Second, the coefficients are not a field, and the modules *
*may
not be free or even projective. We need to deal with these objections together *
*so as
to obtain a reasonable cotensor product functor. To begin with, the category C*
*2 of
compact modules over Z2^is abelian, and countable products of Z2^'s are free ob*
*jects
in it. In particular, this applies to complete exterior algebras on a countable*
* number
of generators, and so to the Hopf algebras K*(Spin; Z2^) and K*(SU; Z2^). (Thi*
*s is
a consequence of [H], but the detail will be given later.) Hence K*( ; Z2^) tr*
*anslates
products of spaces into completed tensor products of Z2^-algebras, when one of *
*the
spaces is SU or Spin.
Next, since we are working in the category of compact Z2^modules, the completed
tensor product is right-exact; moreover, it is exact if we are tensoring with a*
* free
object. If A is a compact cocommutative Z2^-coalgebra which is free over Z2^, a*
*nd B; C
are compact A-comodules, we define the completed cotensor product B ^utAC to ma*
*ke
the sequence (cf [M-M])
0 -! B ^utAC -! B ^C 1-1-! B ^A ^C
exact. This is left exact on sequences of A-comodules which are split-exact ove*
*r Z2^. Its
A
derived functors will be written Cdotorp(B; C). Recall the spectral sequence _ *
*stated
here in the appropriate form for our purpose.
Proposition 2.1. Let G be a group, and let X, Y be G-spaces with either X or
Y free (all in a suitably small category, e.g. 2-local CW-complexes). If G, *
*X have
K*( ; Z2^) free in C2, then there is a strongly convergent spectral sequence w*
*ith
p * *
Ep2= CdotorK*(G;Z2^)(K (X; Z2^); K (Y ; Z2^))
E1 ~ K*(X xG Y ; Z2^)
Its edge homomorphism is the `standard' map
j : K*(X xG Y ; Z2^) ! K*(X; Z2^) ^ut K*(Y ; Z2^)
K*(G;Z2^)
(which follows from the definitions).
The proof is the usual geometric one, using the bar resolution. It should be no*
*ted that
the inverse limit is exact in C2, so there are no convergence problems.
Note. Since we are interested in the 2-local theory, we shall also need a loca*
*l ver-
sion of this. Here the arguments are more difficult; it will be preferable to *
*use the
corresponding sequence for K-homology, which involves the ordinary T or groups *
*over
K*(G; Z2) and the ordinary tensor product. Again (since homology theories beha*
*ve
well with respect to direct limits) the sequence is strongly convergent.
We are now ready to state the structure theorem for the map c : Spin ! SU; for
maximum generality we shall need the local version.
3
Proposition 2.2. (i) If PR resp. PC are the submodules of primitives in the c*
*om-
pleted exterior algebras K*(Spin; Z2) resp. K*(SU; Z2), the map c induces an e*
*pi-
morphism from PC to PR , whose kernel Q is a direct summand. Writing ^E( ) for *
*the
completed exterior algebra on primitive elements, we have:
K*(Spin; Z2) = ^E(PR )
K*(SU; Z2) = ^E(PC ) ~=^E(PR Q) = ^E(PR ) ^E(Q)
as a tensor product of Hopf algebras.
(ii) The same statement holds for K-theory with Z2^coefficients, and K*(Spin; Z*
*2^)
resp. K*(SU; Z2^) is isomorphic to K*(Spin; Z2) Z2 Z2^resp. K*(SU; Z2) Z2 Z2^
From this will follow:
Proposition 2.3. For any space X with an action of Spin, the edge homomorphism
of the spectral sequence defines a natural isomorphism
j : K*((SU x ESpin) x X; Z2^) ! ^E(Q) ^K*(X; Z2^)
Spin
We postpone the proof of proposition 2.2 to the next section, and show that it *
*implies
proposition 2.3. For this, it is sufficient to identify K*(SU; Z2^)^utK*(Spin;Z*
*2^)K*(X; Z2^).
From the splitting of proposition 2.2, we can deduce that
* : K*(SU; Z2^) ! K*(Spin; Z2^) ^K*(SU; Z2^)
is identified with
^E(PR ) ^E(Q) 1-!E^(PR ) ^^E(PR ) ^^E(Q)
We know that (E^(PR ) ^^E(Q)) ^ut^E(PR)K*(X; Z2^) ~= ^E(Q) ^K*(X; Z2^). This i*
*s the
dual of the well-known analogous formula for the tensor product, and the isomor*
*phism
is natural. However, ^E(Q) ^K*(X; Z2^) is exact in K*(X; Z2^), since ^E(Q) is f*
*ree, and
so its derived functors are trivial:
Cdotor0K*(Spin;Z2^)(K*(X; Z2^); K*(SU; Z2^)) = K*(X; Z2^) ^^E(Q)
CdotorpK*(Spin;Z2^)(K*(X; Z2^); K*(SU; Z2^)) = 0 (p > 0)
Using the edge homomorphism of the spectral sequence, proposition 2.3 follows.
3. Structure of K*(Spin); K*(SU).
We now proceed to the proof of proposition 2.2. Let irresp. icbe the ith `stabi*
*lized'
exterior power of the standard representation from Spin(2n + 1) resp. SU(2n + *
*1)
to U, considered as an element of the representation ring. That is, iris the re*
*sult of
applying the operation ito -(2n+1). Then it is obvious that under inclusion map*
*s of
Spin(2n + 1)'s and SU(2n + 1)'s the i's are preserved; and that c*(ic) = c*(ic)*
* = ir.
4
Let now fi be the operation (see [H]) which to any representation ae of G assig*
*ns its
class fi(ae) in K1(G) = [G; U] considered as a map from G to U. The basic theor*
*em of
[H] gives us that K*(SU(2n + 1); Z2) is the exterior algebra
EZ2(fi(1c); : :;:fi(nc); fi(1c); : :;:fi(nc ))
since these can be seen to be equivalent to the basic representations modulo a *
*little
manipulation. (The generators are also, as usual, the primitives for the Hopf a*
*lgebra
structure.) The similar result is not quite true for Spin(2n + 1), as is well k*
*nown, the
picture being complicated by the Spin representation n, of dimension 2n. We hav*
*e:
K*(Spin(2n + 1); Z2) = EZ2(fi(1r); : :;:fi(n-1r); fi(n))
However, there is a relation between nrand n, since (n)2 = nr+ a sum of terms in
1r; : :;:n-1r. Writing n = 2n + "n , and applying the usual relations for fi, w*
*e have
that fi(n)2 = 2n+1fi(n). Hence, fi(nr) = 2n+1fi(n) (mod fi(1r); : :;:fi(n-1r).
Write Mn for the Z2-module which generates the exterior algebra K*(Spin(2n+1); *
*Z2)
and Nn for the submodule generated by the fi(ir)'s. We can deduce a short exact
sequence
(En) 0 ! Nn ! Mn ! Z=2n+1 . fi(n) ! 0
The restrictions from En+1 to En are straightforward if we take into account th*
*at
n+1 restricts to 2:n. Hence the map from Z=2n+2 to Z=2n+1 in the above sequence
multiplies the generator by 2. It is easy to deduce that the inverse limit of t*
*he Z=2n+1's
is zero; and so (since they are finite) is the lim1. Hence the map from lim{Nn*
*} to
lim{Mn} = K*(Spin) is an isomorphism, and we have:
Proposition 3.1. The K-cohomology rings of Spin, SU are as follows:
K*(Spin; Z2) = ^EZ2(fi(1r); fi(2r); : :):
K*(SU; Z2) = ^EZ2(fi(1c); fi(2c); : :;:fi(1c); fi(2c) : :):
and the restriction c* from SU to Spin maps fi(ic); fi(ic) to fi(ir) (i = 1; 2;*
* : :):
From this, proposition 2.2 clearly follows.
We next deduce:
Proposition 3.2. The local K-theory of the quotient is given by
K*((SU x ESpin)=Spin; Z2) ~=^EZ2(fi(1c) - fi(1c); : :):~=^EZ2(Q)
in the terminology of proposition 2.2.
5
Proof. We can proceed in two ways (neither entirely satisfactory); either by de*
*veloping
the theory of the cohomology Rothenberg-Steenrod spectral sequence to extend to*
* nice
enough topological Z2-modules, or more convincingly by dualizing proposition 3.*
*1 to a
result on the homology K*, applying the well-behaved local homology spectral se*
*quence
to that, and then dualizing back.
4. The main result.
We are now in a position to put the pieces together. The key point is that JR2 *
*is a
2-adic space, so the local theory and the 2-adic theory coincide for it.
Theorem 4.1. There is a natural isomorphism:
K*(BGL(Z)+ ; Z2) ~=^EZ2(Q) ^K*(JR2; Z2)
~=K*(BBSO; Z2) ^K*(JR2; Z2)
Proof. We'd like to use a basepoint in JR2, but of course can't suppose there *
*is
one which is fixed under Spin. Consider instead the equivariant embedding of JR*
*2 in
the unreduced cone C+ JR2. If we can prove the result for C+ JR2 and for the p*
*air
P = (C+ JR2; JR2) separately, then it will follow for JR2 by the 5-lemma. Now *
*for
C+ JR2 it is already proved (by proposition 3.2). For P , we consider the commu*
*tative
diagram:
K*(SU xSpinP?; Z2) -j! K*(SU; Z2) ^utK*(Spin;Z2)K*(P?; Z2)
?yff ?yfi
K*(SU xSpin(P ); Z2^) -j! K*(SU; Z2^) ^utK*(Spin;Z2^)K*(P ; Z2^)
The arrow j in the lower row is an isomorphism by proposition 2.3. The two vert*
*ical
arrows are induced by the coefficient homomorphism. Since the homology of JR2 *
*is
finite in every dimension, the same is true for SU xSpinP ; so K*(SU xSpinP ; Z*
*2) is a
2-adic group. Hence the arrow marked ff is an isomorphism. On the other hand, w*
*e can
embed K*(SU; Z2) ^utK*(Spin;Z2)K*(P ; Z2) in K*(SU; Z2) ^K*(P ; Z2) and identif*
*y the
latter with
K*(SU; Z2) ^(Z2^^K*(P ; Z2))
(again because K*(P ; Z2) is 2-adic). Using this, and the definition of the co*
*tensor
product, we find that the right hand vertical arrow fi is also an isomorphism. *
*Hence
the upper arrow j is one.
Now by the argument used in proposition 2.3, this implies that K*(SU xSpinP ; Z*
*2)
is isomorphic to E^Z2(Q) ^K*(P ; Z2). This proves the first line of the theore*
*m. The
second results from the usual identification of SU=SO with BBO, and accordingly*
* of
SU=Spin with BBSO.
A comparison with the fibre sequence (2) shows that the sequence splits from the
viewpoint of 2-local K-theory. Finally, it is worth noting that the K-theory o*
*f JR2
has been known for a long time, see [H-S]; it is essentially the completed repr*
*esentation
ring of the infinite symmetric group 1 .
6
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7