19/1/1998
K-theory of mapping class groups III:
Odd torsion
Luke Hodgkin
King's College, London
1. Introduction.
Let n be the mapping class group of a 2-sphere with n punctures (n 3), an inte*
*r-
esting infinite group of finite virtual cohomological dimension. The aim of thi*
*s series
of papers is to compute the (topological) K-theory of the classifying spaces Bn*
*. It
is known (see [6], [10]) that the reduced K-theory of Bn is profinite. In [10]*
* the
torsion-free part of K*(Bn) was found for all n, but it was also shown that the*
*re is
sometimes torsion. The aim of this paper is to find when this torsion occurs (f*
*or odd
primes), and to describe it.
Where the work of [10] was based on a formula of A. Adem using elements of fini*
*te
order in the group, and hence on the relation between the group's structure and*
* its
geometry, for this one we use homotopy theoretic methods. These were applied to*
* find
the cohomology of Bn by B"odigheimer, Cohen and Peim in [7]; we use them to find
the rank of mod p K-theory and by comparing with [10] we locate the torsion. We*
* are
also able to apply similar methods in the p-local case to compute its order (wh*
*ich is
an improvement over the cohomology result). The basic tool is the spectral sequ*
*ence
in K-theory of a fibration relating the whole family of Bn's, BSO(3), and the s*
*pace
2S2q+2 of maps from a 2-sphere to a (2q +2)-sphere. The computation of this spe*
*ctral
sequence is straightforward on the whole, if a bit lengthy. Our main result is:
Theorem 1.1. Let p be an odd prime. Then the p-torsion in K*(Bn; Z) is all in
K1, and is as follows:
Suppose that either n = q 3; : :;:p - 1 (mod p), or n = qpr + ff where ff 2 {*
*0; 1; 2}
and q is prime to p. Let vp(t) denote the p-adic value of t. Then for each inte*
*ger t such
that 2pt < q, Tors(K*(Bn; Z)) has one finite cyclic p-power summand C(t); and t*
*he
order of C(t) is p1+vp(t). The p-torsion is therefore a sum of q_2pcomponents.
In particular, the K-theory is p-torsion free if and only if q < 2p.
On the way, we shall find the additive structure of K*(Bn; Z=p), indeed, it wil*
*l be
necessary for the proof of theorem 1.1. We shall also find a considerably more *
*detailed
description of the mod p and p-local K-theories of Bn, with generators and some
extra structure. The detailed results will be given in xx5, 6.
In order to describe the layout of this paper more fully, I shall first recall *
*the way in
which results for odd primes are arrived at in [7]. We begin with the `fibred c*
*onfigura-
tion space' for a fibration ss : E ! B with fibre Y . This is defined as
E(ss; k) = {(e1; :::; ek) 2 Ek|ei6= ej andss(ei) = ss(ej) ifi 6= j}
The symmetric group k acts freely on E(ss; k), and the important case for us is*
* that
of the fibration j : BSO(2) ! BSO(3) whose fibre is S2. Here we have:
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19/1/1998
Fact 1. For k 3, E(j; k)=k is a model for Bk.
This is proposition 1.1 of [7], and it gives a relation between mapping class g*
*roups
and configuration spaces. To find the K-theory of the configuration spaces the*
*y in
their turn need first to be combined into a construction which contains all val*
*ues of
k at once, and then to be related to loopspaces. We accordingly define for any *
*based
CW-complex X: ha i
E(j; X) = E(j; k) xk Xk =(~)
k0
where ~ is the natural contraction (compare reduced product spaces) which forge*
*ts the
basepoint when it occurs. This space has a filtration coming from k, and if we *
*define
Dk(j; X) to be the filtration quotient Ek(j; X)=Ek-1(j; X), then there is a sta*
*ble
splitting generalizing the Snaith splitting [17]:
_
(1) E(j; X) 's Dk(j; X)
k1
(This is proposition 1.3. of [7].)
We are particularly interested in Dk(j; Sm ) for appropriate values of m, since*
* it is
closely related to E(j; k)=k and so to the mapping class group. For this we fol*
*low the
paper [8] _ [7] uses a homology argument at the chain level which is inappropri*
*ate for
K-theory. Consider the vector bundle
: E(j; k) x Rk ! E(j; k)=k
k
By the argument of Milgram [15] quoted in [8] 2.6, the Thom space T (m:) can be
identified with Dk(j; Sm ) We cannot, as in [8], choose m so that m: is trivial*
*, since
the group K0 of the base contains, as we know, numerous elements of infinite or*
*der in
general. (The base is equivalent to Bk, and not a finite complex.) However, it *
*is easy
to choose m so that n: is at least orientable for K-theory with any coefficient*
*s; in fact
we only need that the bundle admits a complex structure (see [4]), so that m ev*
*en is
sufficient. For such m, the reduced K-theory of T (m:) is the same as the K-the*
*ory of
the base. We deduce:
Proposition 1.1. There is an isomorphism, for m even,
(2) K*(Bk; R) ! "K*(Dk(j; Sm ); R)
for any ring of coefficients R.
Corollary. K"*(E(j; Sm ); R) splits as a direct sum corresponding to the filtra*
*tion quo-
tients Dk; and for k 3 the kth summand is isomorphic to the K-theory of Bk.
Note 1. In the cohomology version of this result (cf. [7] proposition 3.1.), *
*there is
a shift in dimension coming from the Thom isomorphism. Since the bundle is even
dimensional, the shift is not perceived by K-theory, and we needn't bother with*
* it.
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19/1/1998
Note 2. We can make the isomorphism (2) canonical, since the natural complex
structure on m: gives a preferred choice of Thom class.
The proposition and its corollary shift the focus of interest to the space E(j;*
* Sm ), and
to find the K-theory here we have our last important result, Proposition 1.2 of*
* [7] in
our particular case:
Fact 2. There is a homotopy equivalence
(3) E(j; Sm ) ! ESO(3) xSO(3)2Sm+2
where 2X denotes the free space of all maps from S2 to X, and SO(3) acts on the
mapping space via the usual action on S2. In consequence, there is a fibration
(4) 2Sm+2 ! E(j; Sm ) ! BSO(3)
Note. We shall also want a related result `over a point'. Namely, consider th*
*e one-
point S2-bundle ss : S2 ! *. The space E(ss; Sm ) can be naturally identified *
*with
2Sm+2 itself. Using the decomposition of E(ss; Sm ) into filtration quotients,*
* and the
identification of those quotients as Thom spaces, we find that for m even:
M1
(5) K"*(2Sm+2 ; R) ~=K"*(E(ss; Sm ); R) ~= K*(E(ss; k)=k; R)
k=1
The spaces E(ss; k)=k are the configuration spaces of unordered sets of points *
*on S2;
we shall abbreviate them to Fk(S2).
From the above summary of the theory of [7], the following plan of attack emerg*
*es.
First in section 2 we recall the (known) mod p K-theory of the loopspace 2Sm+2 *
* for
m = 2n even, and deduce it for the free mapping space 2S2n+2. We discuss the na*
*ture
of the torsion in the p-local theory in each case. In section 3 we begin the ca*
*lculation
of the spectral sequence of the fibration (4) in K-theory mod p, with the aim o*
*f finding
K*(E(j; S2n); Z=p) and so (by the corollary to proposition 1.1) K*(Bk; Z=p) for*
* all
k. We begin with the cases where the weight k is pr or 2pr, which are particul*
*arly
easy, find the main differentials in section 4, and deduce the K-theory in thes*
*e cases.
We then in section 5 compute the mod p spectral sequence in general _ this has *
*to
be divided into cases according to the residue class of k mod p. Along the way*
* it is
necessary to compare our results with those of [10] so as to know which element*
*s are to
be identified as torsion. Finally in section 6, we calculate the p-local spectr*
*al sequence
and find the order and location of the torsion summands identified in section 5.
In what follows, p will denote a fixed odd prime and the letter q, wherever it *
*occurs,
will be used for a number which is prime to p (not necessarily odd).
2. K-theory of 2S2n+2.
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We begin _ again following closely the lines of [7] _ by recalling that, locali*
*zed at an
odd prime, an odd sphere is an H-space, and there is a decomposition
(6) S2n+1 x 2S4n+3 '!2S2n+2
This is in fact derived from the `Hopf fibration' of S4n+3 and so deloops. We c*
*onsider
the simpler K-homology for the present; with coefficients in a field it is of c*
*ourse a Hopf
algebra. However, any homology theory applied to S2n+1 is very simple using the
standard reduced product space decomposition [12]. We find here that K*(S2n+1; *
*R)
is the polynomial algebra R[x], where x is the fundamental class of the 2n-sphe*
*re
mapped in by the inclusion.We can deduce
Lemma 2.1. Over any ring of coefficients R, the K-homology K*(2S2n+2; R) is
naturally isomorphic to the tensor product R[x] K*(2S4n+3; R).
On the other hand, the K-theory of second loopspaces presents more complication*
*s.
To begin with we concentrate on the mod p theory.
Proposition 2.1. There are generators y0; y1 2 K1(2S2q+1; Z=p) and z1; z2; z3*
*; : : :
2 K0(2S2q+1; Z=p) so that
(7) K*(2S2q+1; Z=p) = E(y0; y1) Z=p [z1; z2; z3; : :]:=((z1)p; (z2)p; : :):
Proof. The computation has been done for the Morava K-theories K(n), for all n *
*and
all odd primes, by Yamaguchi [18]; see also Langsetmo [13] for K-theory in part*
*icular.
The diagonal _ and so the full product structure in K-cohomology _ are not given
in [18]; however, they have recently been determined, in our case by Ravenel [1*
*6]. (For
the general space kSn, see Langsetmo [14].) The results are as follows (again g*
*eneral
for the first loopspace and mod p for the second):
Proposition 2.2. (i) For any ring of coefficients R, K*(S2q+1; R) is a `compl*
*eted
divided polynomial algebra on one generator'; more precisely, there are generat*
*ors
1; 2; : :2:K0(S2q+1; R) such that
(8) K*(S2q+1; R) = R[[1; 2; : :]:]=(ij = (i; j)i+j; i; j = 1; 2; : :):
as an algebra. (Here (i; j) is the binomial coefficient as usual.)
(ii) There are generators j0; j1 2 K1(2S2q+1; Z=p) and i1 2 K0(2S2q+1; Z=p) so
that
(9) K*(2S2q+1; Z=p) = E(j0; j1) Z=p [[i1]]
_ the tensor product of an exterior and a power series algebra.
Note. Here and henceforth it is important that Snaith's stable splitting [17] *
*of the
spaces 2S2q+1 etc. gives a grading by `weight' on the K-theory algebras. Spec*
*ifi-
cally, i has weight i and ji; ii have weight 2pi when defined. (These are well *
*known
verifications.)
Proposition 2.2 (i) is well known; while part (ii) is proved in [16].
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19/1/1998
Corollary. K*(2S2n+2; Z=p) is a completed tensor product of:
(i) a divided power series algebra on 1
(ii) an exterior algebra on j0,j1
(iii) a power series algebra on i1.
It is also convenient, as an aid in calculating the p-local theory, to know the*
* Bockstein
spectral sequence (see [2]) of the second loopspace. Here the essential initial*
* result is
to be found in [14], although there it is stated for K-homology. In our terms i*
*t is:
r-1-1
Proposition 2.3. For r = 2; 3; : :,:let jr = j1 . ip1 2 K1(2S2n+1; Z=p). *
*Then
in the Bockstein spectral sequence of 2S2n+1 we have
r-1 pr-1
Efir= E(j0; jr) Z=p [[ip1 ]]; dr(i1 ) = jr
In other words, j0 is an integral class, while jr is the reduction of an elemen*
*t of order
pr in K1(2S2n+1; Zp). Clearly, essentially the same relations hold in 2S2n+2.
It is obviously more convenient to use the K-cohomology once we start investiga*
*ting
the free mapping space 2S2n+2, since it is not an H-space, and the Hopf algebra
structure is not available. We have as usual the Serre type spectral sequence *
*of the
fibration
(10) 2S2n+2 ! 2S2n+2 ! S2n+2
with E2 = H*(S2n+2; K*(2S2n+2; Z=p)) and E1 ~ K*(2S2n+2; Z=p). The only
differential is d2n+2. Let be the generator of H2n+2(S2n+2; Z=p).
Lemma 2.2. In the Serre spectral sequence of the fibration (11), we have:
d2n+2(j0) = 1
d2n+2(i) = d2n+2(i1) = d2n+2(j1) = 0
Proof. The essential point is that we can compare the spectral sequence with t*
*he
similar one for cohomology which is computed in [7]. By the same method, first *
*of all,
we can establish that the i's are cycles (the argument for this involves no pec*
*uliarities
of cohomology). Now we use the splitting (5) to identify K*(2S2n+2; Z=p) with t*
*he
direct sum for all k of K*(Fk(S2); Z=p) _ and similarly for the cohomology. We *
*have
that j0 has weight 2 in the corresponding decomposition of 2S2n+2 (into braid s*
*paces,
[CLM]); and so d2n+2(j0) can be found by looking at the mod p K-theory of F2(S2*
*).
This, like the cohomology, is trivial (the space is homotopy equivalent to RP2)*
*. It
follows that j0; 1 (the only non-trivial generators of weight 2) must cancel *
*under
d2n+2 as stated.
i1 and j1 on the other hand have weight 2p and to investigate them we need to l*
*ook at
H*(F2p(S2); Z=p). An inspection of the results and methods of [7] section 7 (in*
* partic-
ular lemma 7.2) shows that this has Poincare polynomial 1 + t3 + t2p-2+ t2p-1. *
*There
is therefore no possibility of non-zero differentials in the Atiyah-Hirzebruch *
*spectral
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sequence (the first one would be d2p-1). The rank of K*(F2p(S2); Z=p) over Z=p*
* is
therefore 4 and it follows immediately that d2n+2(i1) = d2n+2(j1) = 0.
The structure of K*(2S2n+2; Z=p) is a direct consequence of this, although it i*
*s not as
simple to state as one could wish. (In particular, the topology of the K-groups*
* has to be
included.) Imitating precisely the statement in [7], we let Ad be the (Z Z=2)-*
*graded
Z=p-module with generators:
j; j 0
j0 . j; j -1 (mod p)
. j; j 0 (mod p)
j0 . . j; j 0
where the notional bidegrees assigned are (j; 0) to j, (2; 1) to j0, and (1; 0)*
* to . The
first grading is the weight, the second that of K-theory. There is a natural w*
*eight
filtration on Ad, and A is the completion of Ad with respect to the correspondi*
*ng
topology.
Proposition 2.4. K*(2S2n+2; Z=p) is the completed tensor product of A as defined
above and B = E(j1) Z=p[[i1]]; where j1 has bidegree (2p; 1) and i1 has bidegr*
*ee
(2p; 0).
The proof is immediate from lemma 2.2.
Note. Unlike the cohomology case, not all the generators as given above are obv*
*iously
unique. The i's are, of course (as units in the components of the decompositio*
*n).
The arguments of lemma 2.2 show that the classes we have called j0 . (which is*
* not
in fact a product) and j1, i1 are well defined; hence so are all powers of i1 a*
*nd all
products of any of them with a . We can also fix j0 . p-1 in the K-theory of t*
*he
(p + 1)-component, since it's low-dimensional; and this enables us to define j0*
*qp-1
for all q by multiplication. On the other hand, we have a certain freedom in ch*
*oosing
the classes . qpr in the (qpr + 1)-component, which by the arguments of x1 can*
* be
identified with K0(Bqpr+1; Z=p), which will be useful.
We now proceed to study the p-local theory. First we look at 2S2n+1. Since clea*
*rly i
and j0 are integral classes, we consider them as elements of the p-local K-theo*
*ry using
(without too much confusion)rthe-same1names. If q is prime to p, then it follow*
*s from
proposition 2.3 that j1. iqp1 -1is the reduction mod p of a class in K1(2S2n+1*
*; Zp),
whose weight is 2qpr and whose order is pr. Choose one such class and call it *
*qpr.
Let M be the topological Zp module generated by 1 (in degree zero) and the qpr'*
*s (in
degree 1), completed with respect to the weight filtration topology. Then,
(11) K*(2S2n+1; Zp) = E(j0) M
And the p-local K-theory of 2S2n+2 follows from this using Proposition 2.2:
Proposition 2.5. With the above notation,
K*(2S2n+2; Zp) = R[[1; 2; : :]:]=(ij = (i; j)i+j; i; j = 1; 2; : :):^(E(j0) *
* M)
(completed tensor product).
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We next proceed to the free loopspace 2S2n+2. Here we have:
Lemma 2.3. In the p-local spectral sequence of the fibration (10), the differ*
*entials
are given by:
d2n+2(j0) = 1
d2n+2(i) = d2n+2(qpr) = 0
for all i; r (> 0) and q prime to p.
The only difficult point concerns the 's, which is essentially the question of *
*whether
qpr is in the image of the restriction i* : K1(2S2n+2; Zp) ! K1(2S2n+2; Zp). In
any case, we have integral classes i; and an integral class of weight 3 which c*
*orresponds
to j0. . Since there is no way of expressing this as a product, we shall call i*
*t fl3 (which
clarifies the weight). To check that qprsurvives in the spectral sequence of (1*
*0), it is
enough to know that the Bockstein spectral sequencerrelations-of1proposition 2.*
*3 still
hold in 2S2n+2, since then the class j1 . iqp1 -1 in 2S2n+2 will be the reduct*
*ion
mod p of a local class of order pr which restricts to qpron the based loopspace.
To derive the Bockstein relations, we can calculate the Bockstein spectral sequ*
*ence as
before for the free loopspace, weight by weight, forgetting (at least for the m*
*oment)
the difficult weights which are 1 (mod p).r-In1Efirwe find that the only terms*
* of
weight 2pr are those we have called jr, ip1 , and integral classes. Since i* s*
*ends the
Bockstein spectral sequence of 2S2n+2 to that of 2S2n+2, the required relation *
*for
dr follows. There is therefore an element in K1(2S2n+2; Zp), which we still cal*
*l pr,
which has order pr and weight 2pr, and which reduces to jr mod p and must restr*
*ict
to pr in K1(2S2n+2; Zp). This implies the result for all qpr, q prime to p.
Lemma 2.3 gives straightforward computations of the spectral sequence as before*
* in
weights 6 1 (mod p), the other case having already appeared as `exceptional' in*
* propo-
sition 2.4. However, we need still more work to clarify what happens in that ca*
*se. We
have in K1(2S2n+2; Zp) classes of the form q0ps-1j0qpr, with q0; q prime to p; *
*and
these are torsion classes of order pr. We also have that:
d2n+2(q0ps-1j0qpr) = 1q0ps-1qpr= (qps) q0psqpr
(using the formula (m; 1) = m+1). A simple computation now shows that if s r, *
*the
differential is zero and we have contributions Z=pr to K0 and K1; while if s < *
*r, the
differential is non-zero, and we have two Z=ps's. In the latter case, the odd c*
*omponent
is generated by the class of pr-sq0ps-1j0qpr, and the even component by q0psq*
*pr
as before.
Since the multiplicative structure is not so useful for the p-local K-theory as*
* for the
mod p, we state our structural result differently from proposition 2.4.
Proposition 2.6. K*(2S2n+2; Zp) is generated (as a topological module, complete
with respect to the weight filtration as usual) by the following elements:
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19/1/1998
j; j 0
fl3 . j; j 0
. qps; q; s > 0, q prime to p (of order ps)
j . qpr; j 0, q; r > 0, q prime to p (of order pr)
fl3 . j . qpr; j 0, q; r > 0, q prime to p (of order pr)
and also by elements
. qpr
j0 . q0ps-1. qpr
. q0ps. qpr
of order pr, with the usual conventions, where s r, and
pr-sj0 . q0.ps-1. q.pr
. q0.ps. q.pr
of order ps, where s < r.
The bidegrees of generators are as in proposition 2.5; and in addition the bide*
*gree of
qpris (2qpr; 1).
By analogy with the previous notation, we write A0 for the topological module g*
*ener-
ated by the first three families in the above statement, i.e. by the elements j*
*; fl3 . j
and . qps.
3. The auxiliary spectral sequence.
We are now in a position to begin studying the spectral sequence in K-theory (m*
*od p
and local) of the fibration
2S2n+2 ! E(j; S2n) ! BSO(3)
which, according to the theory explained in x1, will give the direct sum of the*
* groups
K*(Bk; R) for R = Z=p; Zp. In this section we shall call the sequence {Er(R)}. *
*We
have, of course:
(12) Er;s2(R) = Hr(BSO(3); Ks(2S2n+2; R)) = Hr(BSO(3); R) Ks(2S2n+2; R)
(recall that BSO(3) has no torsion in its p-local cohomology) and the main task*
* is to
find what happens to the generators for K*(2S2n+2) which we have just found. Let
x be the standard generator of H4(BSO(3); R) (the pontryagin class); we shall a*
*im for
relations of form
d4r(1 u) = xr v
when u is one of the generators found in x2. Our first theorem introduces an a*
*ux-
iliary spectral sequence which gives useful information about the generators in*
* the
components we have called A, A0.
Theorem 3.1. If A K*(2S2n+2; Z=p), A0 K*(2S2n+2; Zp) are the submod-
ules described in Propositions 2.4, 2.6, then there exist
(i) spectral sequences {E0r}, {E0r(0)} with E02= H*(BSO(3); Z=p) A, E02(0) =
H*(BSO(3); Zp) A0
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(ii) homomorphisms of spectral sequences j* : {Er(Z=p)} ! {E0r} and {Er(Zp)} !
{E0r(0)} which are the natural epimorphisms on E2.
Furthermore, the mod p sequence {E0r} has the following structure:
(iii) d4(1 j0 . . k) = x k+3 if k 6 -3; -2; -1 (mod p)
(iv) If k = qpr - ff where r 1, ff 2 {1; 2; 3} and q is prime to p, then the f*
*irst
non-vanishing differential on j0 . . k is
pr+1_
d2(pr+1)(1 j0 . . k) = x 2 k+3
(v) If k = qpr- 1, with r; s as in (iv), then the first non-vanishing different*
*ial on j0. k
pr-1_
is d2(pr-1)(1 j0 . k) = x 2 . k+1
We shall leave the differentials in the p-local spectral sequence (which are on*
* the whole
simpler to calculate) until x6. Let us first note (it is important for the stru*
*cture) the
following fact about the modules A; A0:
Lemma 3.1. The component of A in weight k is generated by:
(i) k, j0 . . k-3 if k 6 1 (mod p)
(ii) k; . k-1; j0 . k-2; j0 . . k-3 if k 1 (mod p) and k > 1
(iii) 1; . 0 if k = 1.
The component of A0 in weight k is generated by:
(i) k, fl3 . k-3 if k 6 1 (mod p)
(ii) k; . k-1 (of order as specified above), fl3 . k-3 if k 1 (mod p) and k *
*> 1
(iii) 1; . 0 if k = 1.
The proof is immediate from the definitions. What happens in weights 1, 2 is ir*
*relevant
to our main purpose, since the groups are not then defined; but the spectral s*
*equence
{E0r} is defined nonetheless, and it is neater to work with it consistently.
The key to theorem 3.1 is the embedding in Fk(S2) _ a component of 2S2n+2 by (5)
_ of special point sets which are symmetrical under cyclic and dihedral groups;*
* it will
be in terms of such geometrical subsets that {E0r} will be described. Let k ff*
* (mod p)
where ff is 0, 1 or 2, k p. Consider the configuration q(k) of k - ff points a*
*rranged
symmetrically around the equator of S2, with a further ff at the poles. The poi*
*nt is
that k - ff is positive and divisible by p _ this situation is discussed in som*
*e detail in
[10]. Then q(k) 2 Fk(S2), and the stabilizer C(q(k)) of q(k) in SO(3) is the di*
*hedral
group Dk-fffor ff = 0; 2 and the cyclic group Ck-1 for ff = 1. Consequently we *
*have a
map from SO(3)=C(q(k)) to the orbit of q(k), say OE or OEk : SO(3)=C(q(k)) ! Fk*
*(S2).
If k = 0; 1 or 2, the same construction gives more trivial sets q(k) _ the empt*
*y set, the
north pole, and the pair of poles respectively. We find that SO(3)=C(q(k) is a *
*point
(k = 0), an S2 (k = 1) and a projective plane (k = 2). More generally, if k is*
* not
as above but k 3, we can define OEk : SO(3) ! Fk(S2) to be the map sending g to
g:q(k) for some fixed `maximally unsymmetrical' basepoint q(k) 2 Fk(S2); in thi*
*s case
C(q(k)) = 1. The classes which we are interested in, as appears from the next r*
*esult,
are restrictions under OE*k.
9
19/1/1998
Proposition 3.1. For k 0, OE*kmaps K*(Fk(S2); R) onto K*(SO(3)=C(q(k)); R).
Proof. We can first easily dispose of the cases where k < 3. From the remarks a*
*bove,
OEk is a homeomorphism for k = 0; 1; while it is easy to see that the image of *
*OE2 in
F2(S2) is the set of all antipodal point-pairs, which is a deformation retract.
We now turn to the main case where k 3. Note that the spaces SO(3)=C(q(k)) as
defined are orientable 3-manifolds and their K-theory and cohomology are the sa*
*me.
Hence, comparing the cohomology and K-theory of 2S2n+2, it will be enough to pr*
*ove
the corresponding result for cohomology in R, or the stronger:
Proposition 3.2. For k 3, the induced homomorphism
OE*k: Ht(Fk(S2); R) ! Ht(SO(3)=C(q(k)); R)
is an isomorphism (t = 1; 2; 3).
Proof. We begin with the case of H3; H1 and H2 vanish for SO(3)=C(q(k)) unless
C(q(k)) is cyclic, i.e. ff = 1. Both the H3 groups in question are isomorphic*
* to R,
the first by [7], the second because the space is an orientable 3-manifold. Mor*
*eover, in
each case the generator is an integral class. Our method is to study the compos*
*ition
* ss*
(13) H3(Fk(S2); Z) OEk!H3(SO(3)=C(q(k)); Z) ! H3(SO(3); Z)
where ss is the quotient map. We prove, for each k 3,
Lemma 3.2. The composite homomorphism ss* O OE*kdefined above is multiplication
by k(k - 1)(k - 2)2 Z.
From the lemma, Proposition 3.2 follows when t = 3. In fact, if k = qpr + ff w*
*here
r 1 and ff 2 {0; 1; 2}, then k(k - 1)(k - 2) is a p-local unit times pr; if no*
*t then it is
a p-local unit. The same is true of |C(q(k))|. Since ss* is multiplication by |*
*C(q(k))|,
we find that, p-locally, ss* = (a unit times) ss* O OE*k; so p-locally, OE*kis *
*an isomorphism.
To prove the lemma, we start with the case k = 3. From the fibration SO(3) !
F3(S2) ! B3 and the identification of 3 with 3 = D3 SO(3), we obtain that
OE : SO(3)=D3 ! F3(S2) is a homotopy equivalence. Hence OE*3is an isomorphism a*
*nd
ss* O OE*3is multiplication by 6.
Now we have an action of SO(3) on 2S2n+2, say : SO(3) x 2S2n+2 ! 2S2n+2.
This with the identifications of x1 gives the usual action (say k) on each Fk(S*
*2). We
therefore turn our attention to H2kn+3(2S2n+2; Z) which is generated (for n lar*
*ge)
by the class which corresponds to our j0 . . k-3. (Remember that this is integ*
*ral.) In
the notation of [7] this is ffl . i . flk-3.
It is easy to see that OEk O ss = k O jk where jk is the inclusion of SO(3) = S*
*O(3) x
* in SO(3) x Fk(S2); the basepoint being one of the appropriate symmetry type.
Considering H*(Fk(S2; Z) as a summand in H*(2S2n+2; Z) via the identifications *
*we
have the obvious formulae in (Fk)-dimensions 0, 3:
*(flk) = 1 flk
10
19/1/1998
(14) *(ffl . i . flk-3) = 1 ffl . i . flk-3 + N(k) . x3 flk
where N(k) is some integer. It follows (since j*kis 1 ", where " is the augmen*
*tation in
H*(Fk(S2)), which kills all elements in positive Fk-dimension) that ss*OOE*k(ff*
*l.i.flk-3) =
j*kO *k(ffl . i . flk-3) = N(k) . x3, and ss* O OE* is multiplication by N(k). *
*We know that
N(3) = 6, i.e. *(ffl . i) = 1 ffl . i + 6x3 fl3. Now use the 2S2n+2-product to*
* compute
*(ffl . i . flk-3) in two ways and compare; we find
6x3 fl3flk-3 = N(k) . x3 flk
Now from the formula fliflj = (i; j) fli+jit is easy to deduce that N(k) = k(k-*
*1)(k-2),
and hence the lemma.
It remains to consider the cases of t = 1; 2, when (as we have seen) we must ha*
*ve
k = qpr + 1 with q prime to p. In this case, H1(Fk(S2); Zp) is Z=pr (see [6]).*
* It
follows easily that H1(Fk(S2); Z=p) = Z=p generated by the element we have call*
*ed
flk-1 . i, while H2(Fk(S2); Z=p) = Z=p generated by flk-2 . ffl. Moreover, the *
*two classes
are connected by the Bockstein operator of order r, and their product in the mod
p cohomology of Fk(S2) is the generator of H3. This, however, is also clearly *
*the
structure of the cohomology of SO(3)=C(q(k)). If OE*kwere zero on H1, it would *
*follow
that it was zero on H2; H3. Lemma 3.2 implies therefore that it is non-zero, he*
*nce an
isomorphism, on all three. It must therefore be an isomorphism with any of the *
*rings
of coefficients R.
It follows that the class which we have called . qpr restricts to the generato*
*r of
K"0(SO(3)=C(q(k)); R), for k = qpr + 1. While this does not specify it complete*
*ly (see
note following Proposition 2.4), it goes some way towards doing so. For the p-*
*local
theory we can do better, since we have a map H2(Fqpr(S2); Zp) ! K0(Fqpr(S2); Zp)
induced by the inclusion of BU(1) in BU. We can define . qpr to be the image *
*of
a generator under this map; it will then restrict to a generator in SO(3)=C(q(k*
*)) as
before.
Note. A more `geometric' proof of proposition 3.2 might proceed by considering *
*the
Leray cohomology spectral sequence of the quotient map from Fk(S2) to Fk(S2)=SO*
*(3),
and showing that (a) under our hypotheses E0;32= H3(SO(3)=C(q(k))) (this is eas*
*y),
(b) this is the whole of E1 and so of H3(Fk(S2)). Since I have failed to do (b*
*), I pass
this idea on to anyone who would like to try it.
Now for each k 1 define {E00r}(k) to be the spectral sequence in K-theory with
coefficients in Z=p or Zp of the fibration
(15) SO(3)=C(q(k)) ! BC(q(k)) ! BSO(3)
and let 1
M
{E00r} = {E00r}(k)
k=1
Since OEk is a map of SO(3)-spaces for each k, we have a homomorphism OE* from
{Er(Z=p)} resp. {Er(Zp)} to {E00r} (the sum of the components OE*k); and from p*
*ropo-
sition 3.1 we can deduce
11
19/1/1998
Corollary. OE* induces an isomorphism from the quotients H*(BSO(3); Z=p) A,
H*(BSO(3); Zp) A0 of E2(Z=p), E2(Zp) to the relevant E002. Hence, we can define
quotient spectral sequences of the {Er}'s called {E0r}, {E0r}(0) which are isom*
*orphic
to {E00r} and have E02= H*(BSO(3); Z=p) A resp E02(0) = H*(BSO(3); Zp) A0
Writing j* for the natural quotient map, we have dealt with parts (i) and (ii) *
*of
theorem 3.1. It remains to compute the differentials in {E0r}, or equivalently *
*in {E00r},
the spectral sequences of the fibrations (15). Because we are dealing with fini*
*te groups,
these are comparatively simple. (I omit the rather trivial cases k 2.)
Lemma 3.3. Let n = qpr, where r > 0 and q is prime to p, and let u be a generat*
*or of
K1(SO(3)=D(n); Z=p) = Z=p. Then, in the mod p K-theory spectral sequence of the
fibration (15) with C(q(k)) = Dn, the only non-vanishing differential is d2(pr+*
*1)(1u) =
pr+1_
x 2 1.
Note 1. Because we have allowed r = 0, this includes the case n = 1, and so in *
*the
schema above (setting D(1) = 1), accounts for all weights k which are not congr*
*uent
to 1 mod p. For n = 1 of course we have the spectral sequence of the universal *
*SO(3)-
bundle, with d4(1 u) = x 1.
Note 2. Here and elsewhere, by `the only non-vanishing differential is ~' I me*
*an,
following the usual convention, that the only non-vanishing differentials are t*
*he obvious
consequences of the relation ~.
Proof. We have
E002= H*(BSO(3); K*(SO(3)=Dn; Z=p)) = Z=p [x] (u)
where u and x are as defined. For dimensional reasons, all differentials vanish*
* on x 1.
On the other hand, if some differential, necessarily d4t(1 u), is non-zero it *
*must be
equal to xt 1 up to units; and in this case, we find immediately that E4t+1= E1*
* is
all even and has rank t over Z=p. Now we know that the rank of E1 is the rank *
*of
K0(BDn) Z=p,rwhich (cf [11]) is the number of p-power order conjugacy classes *
*in
Dn, i.e.p_+1_2. This implies lemma 3.3.
We now turn to the more complicated case of the cyclic groups Ck.
Lemma 3.4. Let n = qpr, where r > 0 and q is prime to p; and let 1, v, w,
v . w generate K*(SO(3)=Cn; Z=p) (v odd, w even). Then the only non-vanishing
differentials in the spectral sequence of the fibration (15) with C(q(k)) = Cn *
*(i.e.
k = n + 1) are:
pr-1_
(i) d2(ps-1)(1 v) = x 2 w
pr+1_
(ii) d2(ps+1)(1 v . w) = x 2 1
Proof. The inclusion of Cn in Dn induces a map of spectral sequences, which tak*
*es
1 u in E0;12(Dn) to 1 v . w in E0;12(Cn). It therefore follows from lemma 3.2*
* that
dk(1 v . w) = 0 for k < 2(ps+ 1). (We can't deduce formula (ii) immediately, t*
*hough,
since the right hand side might have become zero.) Since K*(BCn; Z=p) must be f*
*inite
12
19/1/1998
and all in K0, xi w must be a d4i-boundary for some i, while 1 v must have some
non-zero differential on it.
If d4t(1 v) = xt 1, then necessarily d4q(1 v . w) = xt w. This would imply
that rk(E4q+1) = rk(E1 ) is 2t, while in fact by the preceding argument rk(E1 )*
* is
the number of p-power order conjugacy classes in Cn, i.e. ps, which is odd. We *
*must
therefore necessarily have d4t(1 v) = xt w for some t, implying d4t(1 v . w) *
*= 0.
pr+1_
From this information, we canrdeduce that d2(pr+1)(1 v . w) = x 2 1 as clai*
*med,
and {xi 1; i = 0; 1; : :;:p_-1_2} are linearly independent in E1 . Counting,rwe*
* see that
to make up a basis of the required rank we need {xi w : i = 0; 1; : :;:p_-3_2},*
* which
implies formula (i).
Finally, putting together lemmas 3.3 and 3.4 and identifying the isomorphic spe*
*ctral
sequences {E0r} and {E00r}, we deduce parts (iii)-(v) of theorem 3.1.
4. The differentials.
We can now state the main theorem about the differentials in the spectral seque*
*nce
mod p; in this and the next section we shall write this simply as {Er}. Here, a*
*nd in
what follows, all relations are up to units of Z=p.
Theorem 4.1. The generators of K*(2S2n+2; Z=p) behave in the following way in
{Er}:
(i) If k 6 -3; -2; -1 (mod p), then d4(1 j0 . . k) = x k+3.
(ii) If k = qpr - ff where r 1, ff 2 {1; 2; 3} and q is prime to p, then the f*
*irst
non-vanishing differential on j0 . . k is
pr+1_
d2(pr+1)(1 j0 . . k) = x 2 k+3
(iii) If k = qpr- 1, with q; r as in (ii), then the first non-vanishing differe*
*ntial on j0. k
pr-1_
is d2(pr-1)(1 j0 . k) = x 2 . k+1
r
(iv) The first non-vanishing differential on ip1 (where r 0) is
r pr+1-1_ pr-1
d2(pr+1-1)(1 ip1) = x 2 i1 . j1
(v) All the generators k and all ik1. j1's are permanent cycles, for k 0.
Clearly the relations (i)-(iii) are identical to the relations (iii)-(v) of the*
*orem 3.1.
They do not follow immediately from that result, however, since the map j* of s*
*pec-
tral sequences is not split. Our strategy will be to use what we know from [10*
*] of
K*(Bk; Z=p) _ that is, the p-adic part. This must be part of the E1 term in we*
*ight
k, and the rest, if any, must come from torsion. We can use induction on k to *
*deal
with each new generator of weight k as it arises and find what the first non-va*
*nishing
differential must be (if there is one).
To start with we have:
13
19/1/1998
Lemma 4.1. Every element 1 k is a permanent cycle.
Proof. Interpret the component of weight k as coming from the fibration defined*
* in
x1 with fibre Fk(S2) and total space Bk. Then k corresponds to the unit element
and so is a permanent cycle.
We next dispose of the easy cases of parts (i)-(iii).
Lemma 4.2. (a) The relation (i) in theorem 4.1 holds for all relevant values *
*of k.
(b) The relations (ii), (iii) hold when r = 1.
(c) If (ii) holds for k = pr - 3 (if (iii) holds for k = pr - 1), then it holds*
* for all
k = qpr - ff, (for all k = qpr - 1), where q is prime to p.
Proof. Again consider the spectral sequence, and the homomorphism j* of theorem
3.1 as graded by weight. In weight 2p (and of course 3), j* is an isomorphism
(there are no generators of form i; j), and so we can deduce that the relations*
* (i)-(iii)
are true. In particular, (i) and (ii) hold on j0 . . k 1 for k = 0 and for k *
*= p - ff
respectively; and (iii) holds on j0 . k 1 for k = p - 1.
To prove (a), we now use the relation which follows from lemma 4.1,
d4(1 j0 . . k)= (d4(1 j0 . )) . (1 k)
= 1 3 . k = (3; k) x k+3
(The last equality is the relation in the divided algebra.) As we have seen in *
*x3, (3; k)
is non-zero mod p precisely when k is as specified in (i). Hence (a) is proved.
It remains to prove (c); then (b) will follow using what we have just shown abo*
*ut the
cases p - ff. Now, (qpr; t) 6= 0 in Z=p provided that the sequence of numbers
qpr + 1; : :;:qpr + t contains no multiple of pr+1. Hence qpr-ff= pr-3 . (q-1):*
*pr+3-ff
up to non-zero multipliers. Applying the same argument as before, suppose d4t(1*
* j0.
. pr-3) = xt pr. Then,
d4t(1 j0 . . qpr-ff)= d4t(1 j0 . . ps-3):(q-1)pr-ff+3
= (pr; (q - 1)pr - ff + 3)(xt qpr-ff+3)
And the binomial coefficient is non-zero if q is prime to p. This proves the ar*
*gument
for formula (ii); that for (iii) is similar.
We now come to the first of two propositions which are the main inductive steps*
* in the
proof of theorem 4.1.
Proposition 4.1. If relations (ii), (iv), (v) of theorem 4.1 hold in weights le*
*ss than pr,
then relation (ii) holds for k = pr - 3 (weight pr). That is, d2(pr+1)(1 j0 . *
* . pr-3) =
pr+1_
x 2 pr
Proof. We consider the component of the spectral sequence {Er} in weight pr. The
fibre generators (generators of K*(Fpr(S2); Z=p)) are:
(a) pr; j0 . . pr-3
*
* r-1-1
(b) The products of pr-2ps; j0 . . pr-2ps-3with is1; is-11j1 for s = 1; 2; : :*
*;:p_____2.
14
19/1/1998
The generators in (a) are those involved in the proposition's statement, while *
*the
differentials on those in (b) are given by our induction hypotheses. In fact, *
*for any
given s = qpt-1 (q prime to p, 1 t < r) we have that the first non-vanishing
differentials on j0 . . pr-2ps-3(respectively on is1) are d2(pt+1)(respectivel*
*y d2(pt-1))
by parts (ii) and (iv) of the theorem. The important point is that pr - 2ps and*
* ps are
divisible by the same power pt of p under our conditions on r. The other two el*
*ements
involved in the products, i.e. pr-2psand is-11j1 are cycles by (v).
We shall now and henceforth use notation of form (m; n) for the bigraded rank (*
*in the
lattice ZxZ) of modules such as K*(X) or the Er's with coefficients in a domain*
*; such
`biranks' can and will be added and multiplied by integers where it's helpful.
We now have immediately, using the Leibniz formula:
Lemma 4.3. Let s = qpt-1; then the first non-vanishing differentials involvin*
*g the
four products in (b) above are
pt-1_ r-1
d2(pt-1)(1 ir1fl) = x 2 i1 j1fl
where fl is either pr-2psor j0 . . pr-2ps-3. Hence, the contribution of such e*
*lements
t-1 pt-1
to E1 is of rank at most (p___2; ____2).
We also have:
Lemma 4.4. The differential d2(pr-1+1)vanishes on j0 . . pr-3.
Proof. As in the proof of lemma 4.2, we have that pr-3 = pr-1-3. (p-1):pr-1up to
non-zero multipliers; hence,
d2(pr-1+1)(1 j0 . . pr-3)= (1 pr-1)(1 (p-1):pr-1)
= (pr-1; (p - 1)pr-1) 1 pr = 0
since the binomial coefficient is zero.
We now consider the term E2pr-1+2 in the weight pr part of the spectral sequenc*
*e.
The contribution from part (a) above is just Z=p [x] {pr; j0 . . pr-3} by lem*
*ma 4.4.
On the other hand the part (b) contribution vanishes in base degree 2(pr-1 - 2*
*).
This enablesrus-to1complete the proof of proposition 4.1. In fact, let us consi*
*der d4m
for m > p___-1_2. For such r, E4m;04mis generated by xm pr (the (a) part) onl*
*y, so
that d4m (1 j0 . . pr-3) is necessarily some multiple of xm pr. Eventually *
*the
differential must hit a multiple which is non-zero (in particular, it cannot ha*
*ve been hit
by an earlier differential); since otherwise j0 . . pr-3 would be an infinite *
*cycle, and
give rise to an infinite summand in E1 . But the limit of the spectral sequence*
* is equal
to K*(Bk; Z=p) = K*(F"k(S2) x Ek; Z=p), which is a finitely generated module ov*
*er
k
K*(Bk; Z=p) by the finiteness of orbit types, and so finite over Z=p. Now we ca*
*n use
the auxiliary spectral sequence {E0r} of x3; d4m (1 j0 . . pr-3) is determine*
*drby
its image under the map of spectral sequences j*, and so is zero for r < p_+1_2*
*, while
pr+1_
d2(pr+1)is x 2 pr as claimed.
To complete the induction for theorem 4.1, we need the result which corresponds*
* to
proposition 4.1 for the products of i1; j1.
15
19/1/1998
Proposition 4.2. If relations (ii), (iv), (v) of theorem 4.1 hold in weightsrle*
*ss-than1
*
* pr-1_
2pr, then relations (iv), (v) hold for weight 2pr. That is, d2(pr-1)(1 ip1 ) *
*= x 2
r-1-1 pr-1-1
ip1 j1; 1 i1 j1 is a permanent cycle.
Proof. As before, we list the generators of K*(F2pr(S2); Z=p)). These are:
(a) 2pr; j0 . . 2pr-3
r-1 pr-1-1
(b) ip1 ; i1 j1
(c) All products of 2pr-2ps; j0 . . 2pr-2ps-3with is1; is-11j1 for 1 s < pr-1
We know the first differential on (c) by the induction hypothesis; and the same*
* applies
(since we have proved Proposition 4.1) to (a). The analogous result to lemma 4*
*.3,
describing the differentials, is:
Lemma 4.5. (i) The first non-vanishing differential on j0 . . 2pr-3 is
pr+1_
d2(pr+1)(1 j0 . . 2pr-3) = x 2 2pr
(ii) Suppose r = q:pt-1 with q prime to p, 1 t < s. Then the first non-vanish*
*ing
differentials on the four products in (c) above are
pt-1_ s-1
d2(pt-1)(1 is1fl) = x 2 i1 j1fl
where fl is either pr-2psor j0 . . pr-2ps-3. Hence, the contribution of such e*
*lements
t-1 pt-1
to E1 is of rank at most (p___2; ____2).
To proceed further, we can't use the auxiliary spectral sequence as before. Ins*
*tead, we
need to recall the results of [10] on the p-adic K-theory of Bn. Because the K-*
*theory
is profinite, it is enough to give (following [1]) the tensor product K*(Bn) C*
*p; and
to know this additively, we only need the Cp-ranks of K0, K1. Writing these aga*
*in as
an ordered pair of integers it is convenient to use the notation
Rank (n) = (rank(K0(Bn) Cp); rank(K1(Bn) Cp))
The main results of [10] can be conveniently compressed into the following stat*
*ement:
Theorem 0. Let N(m) denote the ordered pair which is (k; k) when m = 2k and
(k + 1; k) when m = 2k + 1, and let OE be Euler's function. Then
(i) For q prime to p and r 1,
Xr
(16) Rank (qpr) = (1; 0) + 1_2 OE(pi)N(qpr-i)
i=1
(ii) Rank (qpr + 2) is the same, while Rank (qpr + 1) is the same with the fact*
*or 1_2
deleted.
(iii) In all other cases, Rank (n) = (1; 0) is trivial.
16
19/1/1998
For our present purpose we need only the case n = 2pr of the formula. Using the
description of N(2pr-i), this gives:
Xr
(17) Rank (2pr) = (1; 0) + 1_2 OE(pi)(pr-i; pr-i)
i=1
Returning to proposition 4.2, we now consider E2pr-1+2as before. This has an in*
*finite
part generated by the elements (a) and (b) tensored with Z=p [x] and a finite p*
*art
coming from the generators (c). Lemma 4.5 gives a bound on the rank of this fi*
*nite
part, as follows. There are OE(pr-t)tnumbersts < pr-1 of form qpt-1, and each, *
*as we
have seen, contributes at most (p_-1_2; p_-1_2) to the total dimension. Hence *
*the finite
part has Z=p-rank (n0; n0) where
r-1X pr-i- 1
(18) n0 = OE(pi)(________)
i=1 2
(We have written i = r - t to bring the formula in line with (17).) Using the f*
*ormula
OE(pi) = (p - 1)pi-1,
Xr 1 pr + 1 pr - 1
Rank (2pr) - (n0; n0) = (1; 0) + __(OE(pi); OE(pi)) = (______; _____*
*_)
i=12 2 2
Now the rank of E1 is equal to the rank of K*(B2ps; Z=p) whichris Rank (2pr).
Clearly, therefore, there must be an extra summand of rank m p_-1_2in the odd
r-1-1
part of E1 , and this can only arise from ip1 j1 and its products by 1; x; :*
* :;:xm-1 .
r-1-1
Hence, xm ip1 j1 is a d4m -boundary, and since we know what happens to all o*
*ther
r-1
generators, it must be d4m (1 ip1 ).
r-1
We know therefore that 1 ip1 contributes nothing torthe even part of E1 ; so
the even rank over Z=p, using lemma 4.5 (i), is n0 + p_+1_2, i.e. is exactly eq*
*ual to that
obtained fromrthe formula (17). However, this means that there cannot be any p-*
*torsion
in K*(B2p ), since torsion gives rise (by the universal coefficient theorem) to*
* equal
summands in K0 and K1 mod p. The rank of E1 is therefore exactly Rank (2pr) =
r+1 pr-1 pr-1 p*
*r-1
(n0; n0) + (p___2; ____2). And this implies that m = ____2, i.e. d2pr-2(1 i1*
* ) =
pr-1_ pr-1-1
x 2 i1 j1. This completes the proof of proposition 4.2.
There is, however, a problem which we have not encountered before in deducing p*
*art
(v) of theorem 4.1 from proposition 4.2. This is, that the general statement th*
*at the
generators ik-11j1 are permanent cycles does not follow immediately from the ca*
*ses
k = pr. (For the -generators the analogous statement was true for other reason*
*s.)
To analyse the general situation we consider a k of form qpr, where q is prime *
*to p.
Inductively we can easily deduce, using proposition 4.2 and the Leibniz formula,
17
19/1/1998
s s+1
Lemma 4.6. In the above situation, d4m (iqp1-1j1) = 0 for all m p___-1_2.
The important remaining step is the following:
Lemma 4.7. In the spectral sequence of B2qpr+1,r+1
(a) if q < p, then E4m;*4m= 0 for all m > p___-1_2
r+1+1
(b) in general, E4m;*4m= 0 for all m > p_____2.
We defer the proof of this to the next section (since it comes most naturally w*
*ith the
detailed calculation of the spectral sequence).
Now for any q writerq = lp + s where 1 s < p. If l = 0, then lemmas 4.6 and 4.*
*7(a)
imply that iqp1-1j1 is an infinite cycle. If not, then
r-1 lpr+1spr-1
d4m (iqp1 j1) = d4m (i1 i1 j1)
r+2*
*-1
Using lemma 4.6 we can deduce that this element is a d4m -cycle for all m p___*
*__2;
and by lemma 4.7 (b) this means that it is a d4m -cycle for all m.
5. Computing the sequence.
We now have found the necessary differentials; the computation of the spectral *
*sequence
{Er} for K*(Bk; Z=p) using these is more or less routine. We shall divide it in*
*to three
cases:
(1) k 0 or 2 (mod p)
(2) k 1 (mod p)
(3) Not as in (1), (2);
and we consider these three in turn. First, a general result:
Lemma 5.1. Let k = q:pr + ff where ff = 0 or 2 ff < p. Then the submodule of
weight k in K*(2S2n+2; Z=p),that is, K*(Fk(S2); Z=p), is generated by all produ*
*cts
of form
(19) u . im1; u . im-11j1; (m = 1; 2; : :;:[1_2qpr-1])
where u = qpr-2mp+ffor u = qpr-2mp+ff-3. j0 . ;
and by qpr+ff, qpr+ff-3. j0 .
The proof is straightforward from our previous description of the basis. It sho*
*uld be r-1
noted that if q = 2t is even and ff = 0; 2, the terms of highest weight in i1 a*
*re ff.it:p1
r-1-1
and ff. it:p1 j1; there is no term here involving j0 . .
Now we restrict attention to case (1). For each element of the basis, we know w*
*hich
is the first non-vanishing differential. The key point will be, of course, to f*
*ind whether
there are any others.
To begin with, consider a product of type (19). Then if m = spt and t < r, by t*
*heorem
4.1 the first differential on the part is d2(pt+1+1); and the first one on the*
* i part is
d2(pt+1-1). The latter therefore comes first, and accounts for all four terms, *
*giving:
t 1_(pt+1-1) spt-1
(20) d2(pt+1-1)(1 u . isp1) = x2 u . i1 j1
18
19/1/1998
for both possible values of u. Hence the four products contribute a submodule o*
*f rank
(1_2(pt+1- 1); 1_2(pt+1- 1)) to the next stage in the spectral sequence.
However, a different situation arises if m is divisible by pr. In this case, q*
*pr - 2mp
must have p-value exactly r, while the p-value of 2mp is necessarily greater; a*
*nd the
first non-vanishing differential on the four products is :
r+1)
(21) d2(pr+1)(1 qpr-2mp-3+ff. j0 . . v) = x1_2(p (qpr-2mp+ff. v)
where v = im1 or im-11j1. In this case the products contribute a submodule of *
*rank
(1_2(pr + 1); 1_2(pr + 1)) to the next stage in the spectral sequence.
From these considerations, we can prove lemma 4.7. In fact we see that if ff = *
*0; 2, the
products of type (19) are all accounted for by the differentials up to d2(pr+1)*
*; while if
q < p, d2(pr-1)will do. This is enough to prove the lemma.
What possibilities remain for non-vanishing differentials in the spectral seque*
*nce?
These can only arise when one of the products (19) which is annihilated under t*
*he
rules (20), (21) has a non-vanishing later differential. In the case of the pro*
*ducts from
(21), i.e. qpr-2mp+ff. v this is impossible; we have already seen that d2(pr+1)*
*which
vanishes on these is the last possible non-vanishing differential. It remains t*
*o deal with
the products in (20), that is to show:
Lemma 5.2. Supposetu is as in lemma 5.1, m = spt, s prime to p and t < r. Then
di(1 u . isp1-1j1) = 0 for all i.
Proof. We begin by noting that by theorem 4.1(v), half the given generators are*
* per-
manent cycles anyway (those where u = q:pr-2mp+ff). The remainder are necessari*
*ly
of form:
t-1
(22) 1 (jp-s)pt+1+ff-3. j0 . . isp1 j1
where j > 0. If j = 1, the generators are of weight exactly pt+2; the result i*
*s true
for these by proposition 4.1. In general, though, the generator (22) (call it z*
*j) is the
product (j-1)pt+2.z1 by the usual rules for multiplication of the i's. Since, a*
*gain from
theorem 4.1, all differentials vanish on (j-1)pt+2, the result follows.
Our investigation of case (1) concludes with the following result:
Theorem 5.1. Let q be prime to p, r 1, and ff = 0; 2. Then
r+ff * qpr+ff
(23) rkZ=p(K*(Bqp ; Z=p)) = rkQp(K (B ; Qp)) + (m; m)
where m is the integral part of _q_2p(note that _q_2pcannot be an integer).
Accordingly, there are m p-torsion summands in the integral K-theory of Bqpr+ff
(cyclic groups of p-power order).
19
19/1/1998
Note that this result does not say how the summands are distributed as between *
*K0
and K1, nor what their order is.
Proof. Our strategy is a simple one, already employed in the proof of propositi*
*on 4.2:
namely, to find the rank of the E1 term in the spectral sequence, and to compa*
*re it
with the rank obtained from Theorem 0 above. By the observation following lemma
5.1, the cases of odd and even q need different treatment; let us first suppose*
* q odd.
We have the following types of contribution to E1 , by lemmas 5.1 and 5.2:
(A) xi qpr+ff (i = 0; : :;:1_2(pr - 1))
t-1-1 1 t
(Bt) xi qpr-2spt+ff. isp1 . j1(i = 0; : :;:__2(p - 3))
t-1-1 1 t
(Ct) xi qpr-2spt+ff-3. j0 . . isp1 . j1(i = 0; : :;:__2(p - 3))
where 0 < t r and s is prime to p, 2s < qpr-t; and
r 1 r
(D) xi qpr-2spr+1+ff. isp1 (i = 0; : :;:__2(p - 1))
r-1 1 r
(E) xi qpr-2spr+1+ff. isp1 . j1 (i = 0; : :;:__2(p - 1))
where now the only restriction on s is that 2ps < q. (The strict inequalities h*
*ere and
above are because q is odd.)
The cases (D) and (E) correspond to the situation of formula (21); and it is th*
*ese ones
which contribute the torsion. The elements (A) are entirely in even degree, whi*
*le the
others come in pairs ((B) with (C) and (D) with (E)) of equal parts in even and*
* odd.
Reckoning up the contributions to E1 , and so to K*(Bqpr+ff; Z=p), we find:
(A): contributes a submodule of rank (1_2(pr + 1); 0)
(Bt), (Ct), where t < r: there are 1_2qOE(pr-t) = p-1_2qpr-t-1 choices of s, a*
*nd each
contributes a submodule of rank (1_2(pt- 1); 1_2(pt- 1) to E1 .
(Br), (Cr): the calculation is the same, except that there are only q_2- q_2p*
*choices
of s. Note that in fact q_2= q-1_2.
(D), (E): there are q_2pchoices of s, and each contributes a submodule of rank*
* (1_2(pr+
1); 1_2(pr + 1)).
None of these terms corresponds exactly to those in the sum of theorem 0; howev*
*er,
they are sufficiently near to make a term by term comparison worthwhile. We ha*
*ve
from equation (16) of theorem 0,
r+ff 1 Xr i r-i
(24) rkQp(K*(Bqp ; Qp)) = (1; 0) + __2 OE(p )N(qp )
i=1
Xr qpr-i+ 1 qpr-i- 1
= (1; 0) + 1_2 OE(pi)(_________; _________)
i=1 2 2
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19/1/1998
while the calculations above (setting i = r - t) give
r+ff pr + 1 1 Xr i pr-i- 1 pr-i- 1
(25) rkZ=p(K*(Bqp ; Z=p)) = (______2; 0) + __2 q:OE(p )(________; ________)
i=1 2 2
r - 1 pr - 1 q q
+ (q_-_1_2)(p_____2; ______2) + ( ___2p; ___2p)
Here we have rearranged the formula by noting that the terms (D), (E) are just *
*one
more than the terms (B)r, (C)r; we have grouped the main part of the terms toge*
*ther
(last term but one in (25)) and separated out the excess as the last term.r
Now, the difference between the even and odd ranks in (25) is clearly p_+1_2, w*
*hile that
in (24) is
Xr p - 1
1 + 1_2 OE(pi) = 1 + _____(1 + p + p2 + : :+:pr-1)
i=1 2
r+1
which is also equal to p___2. Next, the last term (the excess) in formula (25) *
*is equal to
the difference (m; m) between the Z=p-rank and the Qp-rank as claimed in theore*
*m 5.1.
Hence, to establish the theorem, it is sufficient to show that the odd degree c*
*omponent
of formula (24) is equal to the odd degree component of formula (25) after the *
*last
term has been left out; that is,
Lemma 5.3. For q odd, we have
1_Xr OE(pi)(qpr-i-_1_) = 1_Xr q:OE(pi)(pr-i-_1_) + (q_-_1_)(pr_-_1_)
2 i=1 2 2 i=1 2 2 2
Proof. The difference between the two sides in the formula is (subtracting the *
*terms
which cancel immediately):
1_- Xr OE(pi) + q: Xr OE(pi) - (q - 1)(pr - 1)
4 i=1 i=1
Using the familiar formula OE(p) + : :+:OE(pr) = pr- 1, the lemma follows immed*
*iately;
and so therefore does theorem 5.1 in the odd case.
We next consider the case where q is even, say q = 2l. In that case, the list (*
*A) - (E)
of contributions to E1 needs to be supplemented by one other. In fact, in (Br)*
*, but
not in (Cr), we have the possibility that 2s = q, giving a term which is differ*
*ent from
the others:
r-1-1 1 r
(F ) xi ff. ip1 . j1 (i = 0; : :;:__2(p - 3))
This contributes an odd module, of rank (0; 1_2(pr - 1). And, in enumerating th*
*e con-
tribution from type Br, we must replace q_2(= l) by l - 1, since the case s = *
*l is
different.
21
19/1/1998
Now we have a similar comparison to make between these results and those of the*
*orem
0. They give:
r+ff 1 Xr i r-i
(26) rkQp(K*(Bqp ; Qp)) = (1; 0) + __2 OE(p )N(qp )
i=1
Xr qpr-i qpr-i
= (1; 0) + 1_2 OE(pi)(_____ ; _____)
i=1 2 2
in the even case, while
r+ff pr + 1 1 Xr i pr-i- 1 pr-i- 1
(27) rkZ=p(K*(Bqp ; Z=p)) = (______2; 0) + __2 q:OE(p )(________; ________)
i=1 2 2
r - 1 pr - 1 pr - 1
+ (_q_2- 1)(p_____2; ______2) + (0; ______2)
q q
+ ( ___2p; ___2p)
Once again, we can compare the two; we find (using, basically, the calculation *
*of lemma
5.3.) that the rhs in (27) exceeds that in (26) by
-_q_4(OE(p) + : :+:OE(pr)) . (1; 1) + q_4(pr - 1; pr - 1) + (m; m) = (m*
*; m)
q
where as before m = __2p. This completes the proof of theorem 5.1.
Having found the spectral sequence in case (1) (ff = 0; 2), we proceed to deal *
*more
briefly with the other two cases. Case (2), or ff = 1 is more complicated (comp*
*are case
(iii) of theorem 4.1). Let us first give the result which corresponds to lemma *
*5.1.
Lemma 5.4. Suppose k = qpr + 1. Then the submodule of elements of weight k in
K*(2S2n+2; Z=p),that is, K*(Fk(S2); Z=p), is generated by all products of form
(28) u . im1; u . im-11j1; (m = 1; 2; : :;:[1_2qpr-1])
where u = qpr-2mp+1 or qpr-2mp . j0 or qpr-2mp-1 . or qpr-2. j0 . and by qpr+*
*1,
qpr. j0, qpr-1. , qpr-2. j0 . .
If we now use theorem 4.1 as before to find the first non-vanishing differentia*
*l, we find
that exactly the same results as before hold, for the same reasons, for the ele*
*ments
where u = qpr-2mp+1 or qpr-2mp-2 . j0. . We have to be more careful with the ot*
*her
two types of u, since in the `generic' case the non-vanishing differentials on *
*the part
and the i part occur at the same time. If we compare vp(qpr - 2mp) and vp(mp), *
*we
find that
(i) when vp(qpr - 2mp) > vp(mp) = t + 1, the first non-vanishing differential o*
*n all
products is d2(pt+1-1)and the homology is generated by products qpr-2mp.j0.im-1*
*1j1,
qpr-2mp-1 . . im-11j1;
22
19/1/1998
(ii) when the two p-values are both equal to t + 1, the first non-vanishing dif*
*ferential
on all products is again d2(pt+1-1), and this time the homology is generated by*
* the two
cycles qpr-2mp . j0 . im-11. j1 - qpr-2mp-1 . . im1 and qpr-2mp-1 . . im-11. *
*j1 (even
and odd respectively);
(iii) when r = vp(qpr - 2mp) < vp(mp) then the first non-vanishing differential*
* on all
products is d2(pr-1), and the homology generators are qpr-2mp. . im1 and qpr-2*
*mp-1 .
. im-11. j1.
We next prove, following the methods of lemma 5.2, that there are no other diff*
*erentials;
this follows essentially the same lines and we shall omit the proof. And final*
*ly, it
remains to relate the E1 which results to the torsion-free K-theory as given by*
* theorem
0. Here the result is perhaps a surprise: the torsion part, unlike the rest of *
*the K-theory,
is not doubled.
Theorem 5.2. Let q be prime to p, r 1. Then
r+1 * qpr+1
(29) rkZ=p(K*(Bqp ; Z=p)) = rkQp(K (B ; Qp)) + (m; m)
where m is the integral part of _q_2p.
Proof. We shall just give the case where q is odd, that where q is even being s*
*imilar.
If we look at the list of terms (A); : :;:(E) in E1 , we see that each needs to*
* be sup-
plemented by a term involving or j0 (or some more complicated expression). From
our computations above, each of the new terms matches the corresponding old term
exactly, except that: r
(a) in (A), the number of terms of form xi qpr-1. is p_-1_2, i.e. one less tha*
*n the
number of terms of form xi qpr+1;
(b) in each exceptional case (D), (E), (m is a multiple of pr) the rank of the *
*new module
is smaller by (1; 1). r
From these tworfacts, we see that the rank of K*(Bqp +1; Z=p) is obtained by do*
*ubling
rkZ=p(K*(Bqp ; Z=p) and subtracting (1; 0) (from the (A)-terms), together with *
*(1; 1)
for each s which occurs in the definition of the (D), (E) terms. That is:
r+1 * qpr+1 q q
rkZ=p(K*(Bqp ; Z=p)) = 2:rkZ=p(K (B ; Z=p)) - (1; 0) - ( ___2p___2p)
From this, and from the statement of theorem 0 in the case ff = 1, theorem 5.2 *
*follows
immediately.
The last case of the spectral sequence to be dealt with _ case (3) where k = qp*
* + ff
and 2 < ff < p _ is the easiest. (It of course only occurs when p > 3.) Here th*
*ere is,
as we shall see, only one differential, d4, to consider. The basis of K*(Fk(S2)*
*; Z=p) is
given by lemma 5.1 (formula (19)) and we have the following simple relations:
(30) d4(1 qp-2mp+ff-3. j0 . . v)= x qp-2mp+ff. v
d4(1 qp+ff-3. j0 . )= x q:p+ff
(Recall that v is either im1 or im-11 . j1, one even and the other odd.) It is *
*easy from
this to deduce that E4 is generated by all elements of form 1 q:p-2mp+ff. v an*
*d by
1 q:p+ff; and that there are no further differentials. Hence, (since, by theor*
*em 0, the
p-adic K-theory is trivial) we have the very simple parallel to theorems 5.1 an*
*d 5.2:
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19/1/1998
Theorem 5.3. Let 2 < ff < p. Then
(31) rkZ=p(K*(Bqp+ff; Z=p)) = rkQp(K*(Bqp+ff; Qp)) + (m; m)
where m is the integral part of q_2. In consequence, the p-torsion part of the*
* group
K*(Bqp+ff; Zp) is a sum of q_2cyclic components.
6. The p-local calculations.
We now embark on the p-local case, which will give us the size and location of *
*the torsion
components as well as some information about generators. For this, besides the *
*additive
structure, it will be important to look at the structure of K*(Bn; Zp) as an al*
*gebra
over K*(BSO(3); Zp) (via the identification of K*(Bn; Zp) with K*(Fn(S2) xSO(3)
ESO(3); Zp)). We shall denote the `ground ring' K*(BSO(3); Zp) by S in this sec*
*tion.
We consider the analogue over Zp of the spectral sequence used in the preceding*
* sec-
tions. Recall that the structure of K*(2S2n+2; Zp) was found in section 2 (prop*
*osition
2.6). If we concentrate on a particular weight, then we have:
Lemma 6.1. (i) Let k = q:pr + ff where 0 ff < p and ff 6= 1. Then the submodu*
*le
of weight k in K*(2S2n+2; Zp), that is K*(Fk(S2); Zp), is generated by all prod*
*ucts
of form
(32) u . m ; (m = 1; 2; : :;:[1_2q:pr-1]:p)
where u = q:pr-2m+ffor u = q:pr-2m+ff-3. fl3 and the products are torsion class*
*es of
the appropriate order (depending on the p-value of m);
and by q:pr+ff; q:pr+ff-3. fl3 which are torsion-free classes.
(ii) Let k = q:pr + 1, and for m = q0ps, let a(m; r) = ps-r if s > r, a(m; r) =*
* 1
otherwise. Then the submodule of weight k in K*(2S2n+2; Zp) is generated by the
terms in (32) and by
(33) q:pr-2m. . m (m = 1; 2; : :;:[1_2q:pr-1]:p)
a(m; r)q:pr-2m-1. j0 . m
of order pmin(r;s);
and by q:pr+1; q:pr-2. fl3, (torsion-free) and q:pr-1. (of order pr).
If q = 2t is even, then the `top ', t:pr-1appears only multiplied by u = ff(in *
*case
(i), ff = 0; 2) and only by 1 or (in case (ii)). (Compare lemma 5.1.)
It should be noted that the classes u . m are in K1 (resp. K0) when u is of fo*
*rm i
(resp. i. fl3).
Now we begin consideration of the spectral sequence
{En(Zp)} : Hs(BSO(3); Kff(2S2n+2; Zp)) ) K*(2S2n+2 xSO(3)ESO(3); Zp)
which will give the p-local K-theory of the Bn's. We start with a relatively si*
*mple
result which describes the behaviour of the torsion-free classes:
24
19/1/1998
Lemma 6.2. The first non-trivial differential d4 in the spectral sequence is *
*non-
vanishing on the class k . fl3 for every k, and satisfies:
d4(qpr-3+ff. fl3)= pr:x qpr+ff (ff = 0; 1; 2)
= x qpr+ff (2 < ff < p)
(up to units); and dn(k) = 0 for all n; k.
Also, d4(qpr. ) = 0 (case ff = 1).
Proof. This is analogous to lemmas 4.1 and 4.2 (a). As before, we have that k i*
*s a
permanent cycle; while d4(fl3) = x 3 (the result was in fact proved over Zp). *
*Hence,
d4(k . fl3) = x 3:k. Once again, 3:k = (3; k)k+3, which implies the formulae of
the lemma as regards fl3. In the case ff = 1, we have of course d4() = 0 in the*
* sequence
of weight 1, from which the general case follows.
Accordingly, k.fl3 is killed in E4, and fails to provide any generators for the*
* p-local K-
theory. For each i > 0, we have an infinite cycle xik of order at most pr; howe*
*ver, this
is a priori an upper bound, since it is possible that some of these elements (f*
*or k > 1)
could be d4k-boundaries. k survives to E1 , and there is a generator in K0(Bk; *
*Zp)
which corresponds to it _ it is of course again the unit in the K-theory of Bk.*
* Let
us call it "k2 K0(Bk; Zp).
To clarify the situation further, let us consider K*(2S2n+2 xSO(3)ESO(3); Zp) as
a module over S = K*(BSO(3); Zp). The latter is, as we know [5] R(SO(3))^
Zp = R(Spin(3))^ Zp (completed representation ring). Set equal to the basic
representation of Spin(3) (of rank 2) and let " = - 2; then S = Zp[["]]. We se*
*e from
the spectral sequence that the S-module S:k is filtered, with corresponding gra*
*ded
module coming from the component (Zp[x]=(prx) k)/(boundaries) of E1 . Our
crucial next step is to determine the structure of this module _ (and, in passi*
*ng,
to eliminate the question of non-zero boundaries). We shall consider the case *
*ff = 1
(which is special as usual) separately.
Proposition 6.1 (i) Let k = qpr + ff as before, where ff 6= 1. Then the module *
*S:k
is isomorphic to K*(BDk-ff; Zp) = R(Dk-ff)^ Zp if ff = 0; 2, and is trivial (=Z*
*p:1) if
2 < ff < p.
(ii) No non-zero element ps:xi k is a boundary in the spectral sequence, for i *
*> 1.
(iii) If ff = 0; 2, therannihilator of k in S is (the ideal generated by) a mon*
*ic polynomial
Fr(") in " of degree p_+1_2whose lowest non-vanishing term is pr:"; if 2 < ff <*
* p, it is
just ".
(iv) K*(Bk; Zp) is an algebra over the ring S:k as described in (i), (iii); in *
*particular,
the polynomial Fr annihilates every element of K*(Bk; Zp) for ff = 0; 2, while *
*if
2 < ff < p, every element of K*(Bk; Zp) is annihilated by ".
Proof. By theorem 3.1 (ii), we have a map j* of spectral sequences from Er(Zp) *
*to
the spectral sequence of the fibration
(33) SO(3)=C(q(k)) ! SO(3)=C(q(k)) xSO(3)ESO(3) ! BSO(3)
25
19/1/1998
which converges to K*(BC(q(k)); Zp). Now, if for some least i > 0 there is a re*
*lation
d4i(u) = ps:xi k, (u 2 K1(Fk(S2); Zp)), then d4i(j*(u)) = ps:xi 1 in the spectr*
*al
sequence of BC(q(k)). However, j*(u) is necessarily 0, so ps must be. This pr*
*oves
(ii). Hence, the submodule over H*(BSO(3); Zp) which k generates in E1 is exac*
*tly
Zp[x]=(prx) k.
We see therefore that the map of spectral sequences which j* defines is an isom*
*orphism
not only on the part of E2 generated by k; k-3 . fl3 (the `A0-component' of sec*
*tion
3), but on the corresponding part of E1 as well. Hence, the map of S-modules f*
*rom
S:k to K*(BC(q(k)); Zp) is an isomorphism on the associated graded modules, and*
* so
an isomorphism. Since C(q(k)) = Dk-fffor ff = 0; 2 and = 1 in the other cases, *
*this
proves part (i).
For part (iii), we note that first by a standard argument (cf. [3]) the p-adic*
* part
of K*(BDqpr) is isomorphic to K*(BDpr); so, in considering the p-local theory, *
*we
need only consider the latter. If we consider the group extension spectral seq*
*uence
H*(BC(2); K*(BC(pr); Zp) ) K*(BDpr; Zp), we see that K*(BDpr; Zp) is naturally
identified with the invariants of the involution on K*(BC(pr); Zp). The latter *
*is gener-
ated by an element "OE= OE -r1 where OE is a character of order pr, and can be *
*described
[3] as Zp[[O"E]]=((1 + "OE)p - 1); while the involution takes OE to OE-1.
Let = OE + OE-1, so " = - 2 2 K0(BDpr; Zp). It is now easy to see that ge*
*nerates
the relevant part of the representation ring of Dpr, so that K*(BDpr; Zp) is a *
*quotient
of Zp[[ "]]. The sum t = OEt + OE-t for t 1 can be expressed as a polynomial
of degree t in 1, which is related to the usual expression for cos(tx). Howev*
*er, by
considering their expressions in terms of OE; OE-1, we can see that t = pr-t.*
* Hence,
while 1; 1; : :;: pr-1_2are linearly independent, pr+1_2= pr-1_2; and when t*
*ranslated
as a function of or ", this is a polynomial relation of minimum degree.
We can express this more explicitly as a function of "(which is what we want) *
*as
follows. Put = 2 cos(x), then (a) t = 2 cos(tx) and (b) "= -4 sin2(x=2). *
*Our
relation becomes
r + 1 pr - 1 x prx
cos(p_____2x) - cos(______2x) = -2 sin(__2) sin(____2) = 0
r+1 2 x
which can be written as a polynomial of degree p___2in " = -4 sin(_2), since pr*
* is odd.
Since the relation is satisfied when x = 0, the polynomial has no constant term*
*. It is
also clear (e. g. by differentiating) that the term of lowest degree is pr "(up*
* to units).
Finally, using sin(pu) = 1_2i(epu- e-pu), we see that sin(pu) (2i)p-1(sinp(u))*
* (mod p),
so the polynomial is monic up to Zp-units (and all the coefficients apart from *
*the
leading one are zero mod p). Since the restriction from R(SO(3)) to R(Dpr)takes*
* to
, this proves part (iii). Part (iv) is now an immediate consequence.
It should be noted that the result above is slightly unexpected. There is no re*
*ason to
expect that the map Bk ! BSO(3) factors through BC(q(k) to give a splitting of
the inclusion j, even when p-completed. However, from the viewpoint of K-theory*
* it
does.
We next consider the slightly different case where ff = 1; and here it is best *
*to modify
the approach, principally because R(SO(3)p^no longer maps onto R(C(q(k)))p^, as*
* we
26
19/1/1998
shall see. Instead, when k = qpr+ 1, we consider directly a map from Bk to BSO(*
*2).
This will involve something of a detour.
Proposition 6.2. (i) The map u : Bqpr+1!rBSO(3)p factors through the inclusion
Bi : BSO(2)p ! BSO(3)p via a map v : Bqp +1! BSO(2)p: u = Bi O v;
(ii) If "ois the class of the inclusion BSO(2)p = BU(1)p ! BUp 2 K0(BSO(2)p; Zp*
*),
v*("o) restricts to the class qpr. 2 K0(Fqpr+1(S2); Zp).
Proof. Let qpr;1 qpr+1 be the subgroup of diffeomorphisms leavingrone point
fixed (which we mayrtake to be thernorth pole).r Clearly (a) qp ;1is a subgroup*
* of
index qpr+ 1 in qp +1, so that Bqpr;1ss!Bqp +1is a (qpr+ 1)-fold covering; (b) *
*the
map u O ss factors through v0: Bqp ;1!rBSO(2)p. (This is because we can interpr*
*et
u as coming from the `inclusion' of qp +1 in Diff+ (S2) ' SO(3).) We thus have a
commutative diagram:
0
Bqpr;1 -v! BSO(2)p
(34) ss# Bi #
Bqpr+1 -v! BSO(3)p
Since BSO(2)prclassifies H2( ; Zp), v0 can be regarded as an element of the gr*
*oup
H2(Bqp ;1; Zp), and part (i) of the proposition will follow from:
Lemma 6.3. The map ss in the diagram (34) induces an isomorphism on H2( ; Zp).
Proof. The transfer of the covering on H2, ss*, satisfies ss* O ss* = qpr + 1 (*
*degree of
the covering). Sincerthis is a unit in Zp, ss* is an injection onto a direct s*
*ummand.
We knowrthat H2(Bqp +1; Zp) = Z=pr; so if we can show that the same is true of
H2(Bqp ;1; Zp), we are done.
For this, consider the fibration
r;1
(35) Fqpr(R2) ! Bqp ! BSO(2)
which follows from the preceding descriptions. The spaces Fn(R2) stand in the s*
*ame
relation to 2S2N+2 as the Fn(S2)'s do to 2S2N+2 (they are the braid group cla*
*s-
sifying spaces, cf. [9]). Now H*(2S2N+2 ; Zp) contains a divided polynomial alg*
*ebra
on generators zi of weight i corresponding to the i's in K-theory; and also a g*
*ener-
ator y of degree 1 and weight 2 corresponding to j0. Other generators are of de*
*gree
2p - 1. It is easy to see that d2(y) = z2 x in the spectral sequence of (34),*
* where
x generates H2(BSO(2); Z). Hence, the terms of weight n which affect the calcul*
*ation
of H2 come from zn-2y; zn x, and d2(zn-2y) = zn-2z2 x = (n - 2; 2)zn x. If we
substitute qpr + 1 for n, and evaluate in p-local cohomology, we obtain the req*
*uired
result, since (qpr - 1; 2) = pr up to units. If we define v = __1__qpr+1ss*[v0]*
*, we have the
required factoring exactly.
But now, from the cohomologyrspectral sequence of Fqpr+1(S2) ! Bqpr+1! BSO(3),
the restriction from H2(Bqp +1; Zp) to H2(Fqpr+1(S2); Zp) is an isomorphism. So
v*("o) restricts to the image of this generator under the map H2(Fqpr+1(S2); Zp*
*) !
27
19/1/1998
K0(Fqpr+1(S2); Zp). By the remarks following proposition 2.4, this is adequate*
* as a
characterization of qpr. , which proves part (ii).
To obtainrthe more delicate result which we want parallel to proposition 6.1, w*
*e consider
K*(Bqp +1; Zp) as a module over S0 = K*(BSO(2); Zp) via v*.
Proposition 6.3. (i) The S0-module S0:qpr+1 is isomorphic to K*(BCqpr; Zp).r
(ii) The elements qpr. are infinite cycles in the spectral sequence of Bqp +1,*
* and no
non-zero element ps:xi qpr+1 or ps:xi qpr. is a boundary.
(iii)rThe annihilator of qpr+1 in S0 is generated by the polynomial 0Fr("o) = *
*(1 +
"o)p - 1; this has degree pr,rand its lowest non-vanishing term is pr"o. And "*
*o. qpr+1
restricts torqpr. in K0(F qp +1(S2); Zp).
(iv) K*(Bqp +1; Zp) is an algebra over the ring S0:qpr+1; in particular, the po*
*lynomial
0Fr annihilates every element of K*(Bqpr+1; Zp)
Proof. This is a matter of sorting out how much needs to be added to the proof *
*of
proposition 6.1. Here we note first that the spectral sequence of the fibration*
* (33) in
this case is different, but no harder to evaluate; K0(SO(3)=C(q(k)); Zp) is in *
*this case
generated by 1 and a class z say, which is the restriction of q:pr. , and which*
* is clearly
an infinite cycle. Furthermore, qpr. is the restriction of the class defined b*
*y [v], and
so is in turn an infinite cycle which restricts to z. Next, K1(SO(3)=C(q(k)); *
*Zp) is
again killed by d4. The same argument as before therefore shows that it is impo*
*ssible
for the elements ps:xi qpr+1, ps:xi qpr. to be boundaries, which proves part (*
*ii).
For part (i), note that v 2 H2(Bqpr+1;rZp) is of orderrpr, so is the image unde*
*r the
Bockstein of anrelement of H1(Bqp +1; Z=pr) = [Bqp +1; BCpr], say v = u O v1,
where v1 : Bqp +1 ! BCprrand u : BCpr ! BSO(2) classifies the inclusion. Also,
since j : BCqpr! Bqp +1is an isomorphism on H1( ; Z=pr), v1O j : BCqpr! BCpr
is a p-local equivalence. From the second statement we can deduce that j*v*1is*
* an
isomorphism, so v*1is injective; from the first, v* = v*1u*. So Ker(v*) = Ker(u*
**), which
proves (i). The first part of (iii) follows from this, together with (iv), by t*
*he methods
used in proposition 6.1;rwe need only note that K*(BCpr; Zp) = R(Cpr)^ Zp is we*
*ll
known to be Zp[o]=(op - 1) completed with repect to the topology generated by *
*"o. As
for the second part of part (ii), this is proposition 6.2 (ii).
Our major step is to prove that the elements are also cycles.
Proposition 6.4. (i) The element qpr is a permanent cycle in the spectral se-
quence {En(Zp)}rfor all q; r, and so defines an element which we shall call "qp*
*r 2
K1(B2qp ; Zp).
(ii) Let Gr(") = Fr(")=". Then the module S:"qpr is a quotient of S=(Gr(")).
(iii) The H*(BSO(3))-submodule of E1 generated by qpr gives rise to a submodule
S:"qpr of K1 which is profinite as an abelian group, and whose rank over Zp^is *
*at most
pr-1_
2 .
Note. Once again we have a problem arising from the possible existence of bound*
*aries
among the elements psxi qpr; and again this will be eliminated at a later stage.
28
19/1/1998
The method of proof will, as usual, be by induction on the integer qpr; we shal*
*l use the
induction hypothesis to derive information aboutrthe structure of the spectral *
*sequence,
and its consequences for the K-theory of B2qp , and deduce that any non-vanishi*
*ng
differential on q:prwould give us a wrong p-adic rank.
First, let us show that (i) implies (ii) and (iii). Suppose, then, that we hav*
*e shown
that qpr is a permanent cycle. Then so is xi qpr for any k. Hence, in E1 , qpr
generates a graded module over H*(BSO(3); Zp) equal to the quotient by boundari*
*es
of
Mqpr= (1 qpr):H*(BSO(3); Zp)=(pr)
corresponding to at most Z=pr in each degree 4k 0. This implies in particular *
*that
pr:"qpr 0 (mod "). Hence, there is a polynomial G(") (of minimum degree) with
constant term pr such that G("):"qpr = 0. Since Fr("):"qpr = 0 by proposition 2*
*.1(iii),
G(") divides Fr("), and since its term of lowest degree is pr, it must in fact *
*divide
Fr(")=" = Gr(").
We therefore have an epimorphism from S=(Gr(")) to S:"q:pr, (which will induce *
*an
epimorphism of the corresponding graded modules over H*(BSO(3); Zp)), as stated
in (ii).Now (iii) follows from what we know of the structure of the augmentatio*
*n ideal
I(Dqpr)^.
Part (iii) will be crucial in the inductive proof of part (i) which follows. I*
*n fact,
the upper bound for the module generated by " as measured byrpart (ii) is exact*
*ly
right in terms of what we know of the p-adic part of K*(B2qp ). We shall show (*
*in
consequence) that if there are any non-zero differentials on qpr, then the corr*
*esponding
number of p-adic summands must be strictly less than it should be.
We next deduce a simple consequence of the inductive hypothesis, the first stag*
*e in an
eventual computation of the whole spectral sequence:
Lemma 6.4. Assume that qpris a permanent cycle in the spectral sequence. Then,
(i) the product q0ps+ff. q:pris also a permanent cycle, for all q0; s;
(ii) if s r and ff = 0; 2, then d4(q0ps+ff-3. fl3 . qpr) = 0;
(iii) if s < r, and ff = 0; 2, then d4(q0:ps+ff-3. fl3:qpr) = ps:x q0ps+ff. qp*
*r.
(iv) if 2 < ff < p, then d4(q0ps+ff-3. fl3 . qpr) = x q0ps+ff. qpr.
Proof. These relations are all easy consequences of lemma 6.2 and of the Leibni*
*z rule.
Part (i) is immediate; parts (iii) and (iv) follow using lemma 6.2; while part *
*(ii) is the
same result as part (iii) once we have taken account of the fact that ps annihi*
*lates qpr
under the given conditions.
Lemma 6.4 enables us now to take care of the cases where there are no p-adic su*
*mmands
_ which is important, if irrelevant to the induction argument.
Lemma 6.5. Suppose that j is a permanent cycle in the spectral sequence for a*
*ll
j qpr. Then the group K"*(Bmp+ff; Zp) where 2 < ff < p and m 2qpr-1 + 1
is (a) entirely torsion and (b) entirely in K1. Its generators are all the ele*
*ments
"(m-2s)p+ff:"sp; s = 1 : :[:m=2]; and "(m-2s)p+ff:"sp generates a torsion summa*
*nd of
order pvp(s)+1, where as usual vp denotes the p-value. There are accordingly [m*
*=2] such
summands.
29
19/1/1998
Proof. From lemma 6.1 we have a description of the generators for K*(Fmp+ff(S2)*
*; Zp)
and of their torsion orders under the given conditions. From lemma 6.4, the dif*
*ferential
d4 in the spectral sequence kills almost everything, leaving only the identity *
*element
mp+ff and the classes (m-2s)p+ff. sp, which are of the order stated and which c*
*orre-
spond to the elements "(m-2s)p+ffcdot"sp. The description of their order in E1 *
* prima
facie only implies that for each s, the product pvp(s)+1:"(m-2s)p+ff. "spis zer*
*o modulo
powers of "2 S. However, by proposition 6.1 (iv), this implies that it is actua*
*lly zero,
and the lemma follows.
We now embark on the inductive proof of Propositionr6.4 (i). Accordingly, we a*
*re
interested in the spectral sequence for B2qp . Let us suppose from now on (unt*
*il
the proof has been completed) that the proposition has been proved for mp ; m =
1; : :;:qpr-1 - 1. Note first that by lemma 6.4 (iii) and the induction hypoth*
*esis,
if s < t and aps; bpt are less than qpr, ps:x aps . bpt is a d4-boundary, while
pt-s:aps-3 . fl3 . bptgenerates a summand of the d4-cycles, of order ps. These*
* two
elements contribute to E5 as follows:
1) aps. bptgenerates an H*(BSO(3); Zp)-module which is isomorphic to Z=pt:1
Z=ps:{x; x2; : :}:;
2) pt-saps-3. fl3 . bptgenerates an H*(BSO(3); Zp)-module which is isomorphic to
Z=ps:{1; x; x2; : :}:.
We now can deduce two important upper bounds:
Lemma 6.6. Suppose that s < t and aps; bpt are less than qpr. Then:
(i) the generator "aps. "bptcorresponding to aps. bptcontributes a submodule to
K1(Baps+2bpt; Zp) which is isomorphic to a quotientsof S=(Fs("); Gt(")). This r*
*ing is
(additively) the direct sum of Z=pt-s and p_-1_2copies of Zp^.
(ii) If the generator pt-saps-3cdotfl3. b:ptsurvives to E1 , it contributes a s*
*ubmodule
to "K0(Baps+2bpt;sZp) which is isomorphic to a quotient of S=(Gs(")), i.e. of t*
*he sum
of p_-1_2copies of Zp^.
Proof. Clearly both Fs and Gt annihilate the product u = "aps.b"pt. It is clear*
* (cf the
description in terms of sin functions above) that Gs divides Gt, so that Gt = H*
*(t; s)Gs;
and H(t; s) is a monic polynomial whose constant term is pt-s. Hence, (since F*
*s =
x:Gs), Gt pt-sGs (mod Fs). So u is annihilated by Fs = ":Gs and by pt-s:Gs.
Now, from the exact sequence
0 ! Zp . Gs ! S=(Fs) ! S=(Gs) ! 0
we can deduce that Gs generates a Zp-summand in S=(Fs), and from this that addi-
tively S=(Fs; pt-sGs)sis isomorphic to the sum of Z=pt-s and S=(Gs). The latte*
*r is
exactly the sum of p_-1_2copies of Zp^as claimed. Hence, the contribution of u *
*to K1
is at most a quotient of this.
In the case of pt-saps-3. fl3 . bpt, we know that the corresponding generator o*
*f K1
generates a module over S=(Fs), since ps is the highest power of p dividing aps*
* + 2bpt
(proposition 6.1 (iii)). However, from the fact that its contribution to E0;05i*
*s a Z=ps
30
19/1/1998
(see (2) above), we can use the argument used above for proposition 6.1 (iii) a*
*nd (iv) to
show that this module is in fact a quotient of S=(Gs), which is a p-adic sum as*
* claimed.
The corresponding case to lemma 6.6 when s t is different. In fact, by lemma 6*
*.4 (ii),
the differentials d4 in this case are always zero, and the upper bounds are as *
*follows:
Lemma 6.7. (i) If s t and aps; bpt are less than qpr, then the generator "aps*
*. "bpt
which corresponds to aps. bptcontributes a submodule to K1(Baps+2bpt; Zp) which
*
* t-1
is isomorphic to a quotient of S=(Gt(")). This is (additively) the direct sum o*
*f p___2
copies of Zp^.
(ii) If s > t, and the generator aps-3 . fl3cdotbptsurvives to E1 , it contribu*
*tes a
submodule to K"0(Baps+2bpt; Zp) which is isomorphic to a quotient of S=(Gt(")),*
* i.e.
t-1
of the sum of p___2copies of Zp^.
The proof for part (i) is similar to that of lemma 6.6 with the appropriate cha*
*nges, and
without the problems caused by the need to compute a complicated quotient of S.*
* For
part (ii), suppose that s > t; then the p-value of aps + 2bpt is t, so that all*
* elements
are annihilated by Gt(") as required. Note that this argument does not work if *
*t = s,
since it is possible that the p-value of (a + 2b)ps is greater than s. This cas*
*e will be
dealt with later (see proposition 6.7).
We should now look at the spectral sequence for B2qpr. However, for greater gen*
*eral-r
ity, we extend the scope of the inquiry by taking together the two cases of {En*
*(Bkp )}
where k = 2q; 2q + 1.
For k = 2q, the E2-term is the module over H*(BSO(3); Zp) generated by:
(A) qpr, of order pr;
(B) All products 2mp . qpr-mp for m = 1; : :;:qpr-1 - 1, of order ps, where s i*
*s the
p-value of qpr - mp;
(C) All products 2mp-3 . fl3 . qpr-mp for m = 1; : :;:qpr-1 - 1, of order as in*
* (B);
(D) 2qpr, of infinite order;
(E) 2qpr-3. fl3, of infinite order.
For k = 2q + 1, it is the module over H*(BSO(3); Zp) generated by:
(B) All products (2m+1)p. qpr-mp for m = 1; : :;:qpr-1 - 1, of order ps, where *
*s is
the p-value of qpr - mp;
(C) All products (2m+1)p-3. fl3 . qpr-mp for m = 1; : :;:qpr-1 - 1, of order as*
* in (B);
(D) (2q+1)pr, of infinite order;
(E) (2q+1)pr-3. fl3, of infinite order.
We have calculated the E4 term and found various implications about the S-module
structure;rwe shall now use the upper bounds above to find upper bounds for the*
* size
of K1(Bkp ; Zp) (odd part of the spectral sequence).
Lemma 6.8. Suppose that, in the spectral sequence
r
{En(Zp)} : H*(BSO(3); K*(Fkpr(S2); Zp) ) K*(Bkp ; Zp))
all differentials vanish after d4. Then the rank of "K1(Bkpr; Zp) as a module o*
*ver Zp^
31
19/1/1998
is at most equal to
X r
(36) 1_2 OE(pi)(qpr-i)
i=1
if k = 2q, and at most equal to
Xr h(2q + 1)pr-ii
(37) 1_2 OE(pi) ___________
i=1 2
if k = 2q + 1.
Proof. Firstrconsider the even case k = 2q. The terms (A) give a summand of rank
at most p_-1_2, by proposition 6.2 (ii); while the terms (E) are killed by d4.
The remaining odd terms are those of type (B). We count them according to the p*
*-value
of 2mp (i.e. of the part). We have the following estimates:
(i) If the p-value of 2mp is s < r, then so is that of qpr - mp and they are eq*
*ual;
so lemma 6.7 applies.s Hence, the corresponding (B) term contributes a module of
rank at most p_-1_2. The number of contributions for this value of s is the num*
*ber of
integers nps < qprswhich are not divisible by ps+1. This is q:OE(pr-s). The t*
*otal is
ns = q:OE(pr-s)p_-1_2.
(ii) If the p-valuerof 2mp is s r, then I claim that the (B) term contributes *
*a module
of rank at most p_-1_2. First, if s > r, the p-value of qpr - mp is exactly r,*
* and the
result follows from lemma 6.7. If s = r, then the p-value of qpr- mp is r, and*
* either
lemma 6.6 or lemma 6.7 implies the same bound. The number of such contributions
(i.e. of integers 2mp < 2qpr of p-value r) is thernumber of multiples of pr wh*
*ich are
< qpr, i.e. q - 1, and the total is nr = (q - 1)p_-1_2.
We find for the upper bound on the rank:
pr_-_1_+ Xr q:OE(pi)pr-i-_1_; +(q - 1):pr_-_1_
2 i=1 2 2
(setting i = r - s). This is not exactly the formula of equation (36), but it i*
*s not far
different. In fact, the difference is equal to
1_q: pr - 1 - Xr OE(pi)
2 i=1
= 1_2q:(pr - 1 - (p - 1)(1 + p + : :+:pr-1))
= 0
The case where k is odd is slightly more complicated. The point to notice is t*
*hat
the number of integers nps < (2q + 1)pr which are odd and not divisible by ps+1*
* is
32
19/1/1998
(q + 1_2):OE(pr-s) for s < r, while the number of odd integers npr < (2q + 1)pr*
* is exactly
q. We deduce the bound
Xr pr-i- 1 pr - 1
(38) (q + 1_2) OE(pi)________ + q ______
i=1 2 2
as before. Using the fact that
h (2q + 1)pr-ii 1 1
___________ = (q + __)pr-i- __
2 2 2
the difference between (37) and (38) is
r - 1 1 Xr 1 1 Xr
q p_____2+ __4 OE(pi) - __(q + __) OE(pi)
i=1 2 2 i=1
which is zero as before. This completes the proof of lemma 6.8.
The point of the lemma is that thercount of p-rank which it provides is exactly*
* the
rank of the p-adic part of "K*(Bkp ; Zp) as computed in [7]. The key remaining *
*result
for Proposition 6.4 is now:
Lemma 6.9. If there is a non-zero differentialrdion the generator qprin the sp*
*ectral
sequence for i > 4, then the rank of "K1(B2qp ; Zp) must be strictly less than *
*the upper
bound computed by lemma 6.8.
From this, Proposition 6.4 follows since if the rank is less than that given by*
* formula
(36), we have a result in contradiction with [7].
It remains to prove lemma 6.9. Suppose then that for some v 2 K0(F2qpr(S2); Zp)*
* and
n > 1, we have d4n(qpr) = xn v. The only possibilities for v are elements of t*
*ype
(C), (D) in the list above. Now, if v is of order ps (s r), psqpr is a d4n-cyc*
*le; and
for k > 0, d4n(xk qpr) = xn+k v; d4n(xk psqpr) = 0.
It is possible that yet other differentials will fail to vanish on multiples of*
* qpr; however,
for some smallest s we will have a module (over H*(BSO(3); Zp)) of infinite cyc*
*les
which is generated by ps:qpr and its products with x; x2; : :,:modulo boundarie*
*s if
any. (If s = r, this is zero.) Call this module M. Note that since we know b*
*y the
inductive hypothesis that the other0odd generators are infinite cycles, it is i*
*mpossible
that any element of form xk ps :qprfor s0< s should become a cycle `accidental*
*ly',
i.e. because its boundary is the boundary of something else.
Now I claimrthat M E1 gives rise to a module in K1 whose rank over Zp^is stri*
*ctly
less than p_-1_2. From this, lemma 6.9 clearly follows. To justify the claim, n*
*ote that
the cycle psqpr which survives to E1 gives rise to an element in K1, say "u; a*
*nd "u
generates a filtered S-modulerN whose associated graded module is just M. Next,*
* N
is the entire part of K1(B2qp ; Zp) which comes from the (A) terms above. To fi*
*nd
the size of N, we note that pr-s:u 0 (mod ") by considering the E1 term, so *
*"uis
annihilated by some polynomial f(") whose constant term is pr-s. However, "uis *
*also
33
19/1/1998
annihilated by Gr, a monic polynomial whose constant term is pr. We can divide *
*frby
Gr if necessary and get a remainder, a polynomial whose degree d is less than p*
*_-1_2,
and which also annihilates "u. We deduce that N is generated over Zp^by "uand *
*its
products with "; : :;:"d-1. This completes the proof.
We have now completed the induction, and can deduce that all products of 's with
's (elements of type (B) or (D) in our list) are infinite cycles. However, we *
*can do
better. It does not follow from proposition 6.4 that the even generators of typ*
*e (C) are
infinite cycles; in fact, they cannot easily be decomposed into products, if at*
* all. We
prove now that this too is the case _ i.e. that the spectral sequence is trivia*
*l from E5
on.
Lemma 6.10. If any differential di is non-vanishingron one of the generators *
*(C)
above, for i > 4, then the rank of K"1(Bsp ; Zp) must be strictly less than the*
* upper
bound computed by lemma 6.8.
Proof. Similar to lemma 6.9. Suppose that d4n(u) 6= 0, where u is a generator o*
*f type
(C). Then d4n(u) = ptxn v, where v is of type (A) or (B). Furthermore, if 1 v*
* is of
order ps in E5, (so that the generator "vcorresponding to v in K1 is annihilate*
*d by Fs),
then t < s for the relation to be non-trivial. From the boundary relation on xn*
* v, we
can deduce that "vsatisfies a relation of form
pt"n:"v= a1"n+1:"v+ : : :
i.e., "vis annihilated by a polynomial f(") whose term ofslowest degree is pt"n.
I now claim that we can choose f to have degree < n + p_-1_2. In fact, from Fs(*
*"):"v= 0,
ps+1_ ps-1_
we deduce that " 2:"vis a linear combinationsof ":"v; : :;:" 2:"v. Hence we*
* can divide
f by Fs, leaving a remainder of degree < n + p_-1_2which annihilates "u. Since *
*Fs does
not divide f, whose lowest degree term is pt < ps, this remainder is non-trivia*
*l. This
ps-3_
gives a linear relation between "n:"v; : :;:"n+ 2 :"v.
ps-3_
However, the estimates in lemmas 6.6, 6.7 require that "v; ":"v; : :;:" 2:"vs*
*hould be
linearly independent over Zp^. From this it follows necessarily that the same *
*is true
of their products with "n. Since we have shown that if ptxn v is a boundary, t*
*hese
products are linearly dependent, the existence of such a boundary must make the*
* rank
of K1 strictly less than the upper bound.
We can now apply the previous argument. If there is a non-zero boundary, the ra*
*nk
of K1 is less than the upper bound, which contradicts [7]. Hence, there are no*
* such
boundaries. Putting the previous results together (and noting that the case ff *
*= 2 is
obtained from that for ff = 0 simply by multiplying by 2) we have:
Proposition 6.5. If ff = 0; 2, all differentials after d4 vanish in the spectr*
*al sequence
for K*(Bmp+ff; Zp).
Proof. For the generators of type (A), (B), this follows from proposition 6.4; *
*for those
of type (C) from lemma 6.10; while those of type (D) are taken care of by lemma*
* 6.2.
The generators (E) have, of course, already been killed by d4.
34
19/1/1998
Corollary 6.1. The S-module generated by any element mp . np is exactly that
given by lemmas 6.6, 6.7.
In fact, we have seen that there can be no further relations on the odd generat*
*ors
beyond the ones we have already specified as minimal.
The structure of the spectral sequence is now determined. From lemma 6.6 we can
also deduce the location and order of the torsion. In fact, we see that for eac*
*h integer
2qps < kpr such that s > r, we have a torsion summand of order ps-r in K1 coming
from a relation onsthe generator "kpr-2qps. "qps. Hence, the total number of to*
*rsion
summands in K*(Bqp ; Zp) is at least equal to the number of multiples of 2ps+1 *
*which
are less than qps, i.e. to q_2p. However, this is exactly the number of torsio*
*n summands
given by theorem 5.1, and so there can be no further torsion. We have:
Proposition 6.6. (i) The torsion in K*(Bkpr; Zp) is all in K1.
(ii) For each integer 2qps < kpr, where s > r, there is a torsion summand of or*
*der
ps-r, and these are all the summands.
This result, together with lemma 6.5 (which is now true `unconditionally', sinc*
*e we
know that the 's are permanent cycles), establishes theorem 1.1 except in the c*
*ase
ff = 1. Before proceeding to this case, we clear up the remaining questions abo*
*ut the
S-module structure.
We know (since there is no torsion) the additive structure of K"0; it is a dire*
*ct sum
of copies of Zp^, the number of copies being given by the formula of [7]. It s*
*eems
reasonable to deduce (since they add to the right number) that the generator wh*
*ich
comes from (a multiple of) qps-3fl3qpr gives rise to an S-module of the right s*
*ize.
This is in fact true, but there are a few details to be taken care of.
Proposition 6.7. Let "u(q0pr; qps) 2 "K0(Bq0pr+2qps; Zp) be a generator corres*
*pond-
ing to (a multiple of) q0pr-3. fl3 . qps. Then the annihilator of "u(q0pr; qps*
*) in S is
precisely the ideal (Gs) if s > r and (Gr) if s r.
Proof. What, essentially, we need to prove is that for all values of q0pr, qps,*
* "u(q0pr; qps)
is annihilated by the polynomial in question. Then as before, having deduced an*
* upper
bound on the size0ofrthesS-module generated by "u(q0pr; qps), the computations *
*of the
rank of "K0(Bq p +2qp; Zp) ensure that this is a lower bound also, so that ther*
*e is no
larger annihilator.
In most cases, this is straightforward. Specifically, if r > s resp. r < s, a*
*s we have
seen in lemmas 6.6 and 6.7, the element is annihilated by Gs resp. Gr, simply u*
*sing
the p-value of the sum q0pr + 2qps. However, if r = s and vp(q0+ 2q) > 0, we ca*
*nnot
immediately assert that "u(q0pr; qpr) is annihilated by Gr. We shall prove this*
* for all
q0; q; r by an induction on r. Specifically, we assert:
Lemma 6.10. (i) If "u(aps; bps) is annihilatedrby Gs for all q0; q and all s *
*< r, then
every generator "u(aps; bpt) in every "K0(Bqp ; Zp) is annihilated by precisely*
* the right
polynomial;
(ii) Under the hypotheses of (i), every "u(apr; bpr) is annihilated by Gr.
35
19/1/1998
To prove part (i), note that if qpr = m + n, then either vp(m); vp(n) are equal*
* and
less than r (in which case the induction hypothesis applies); or at least one o*
*f the
two p-values is equal to r (and then we know that Gr annihilates the elements of
K"0(Bqpr; Zp)).
To prove (ii), consider "u(apr; bpr) 2 K"0(B(a+2b)pr; Zp). If t = vp((a + 2b)p*
*r), then
certainly Gt("):"u(apr; bpr) = 0. If t = r (i.e. a + 2b is prime to p), we ar*
*e done, so
suppose t > r. Then we still know that the order of apr-3 . fl3 . bpr in E1 i*
*s pr;
whence, "u(apr; bpr) is annihilated by some polynomial f(") whose constant term*
* is pr.
However, if we now consider "pr. "u(apr; bpr) = "u((a + 1)pr; bpr), this has we*
*ight (a +
2b + 1)pr, whose p-value is exactly r. Hence, it is annihilated both by Gr and *
*by f. If
Gr is not a multiple of f, we can divide to find that "pr. "u(apr; bpr) is anni*
*hilated by
a polynomial f0 of lower degree than Gr. This contradicts the conclusion of par*
*t (i);
so f is a factor of Gr. Furthermore, since the constant terms of f and of Gr ar*
*e both
pr, Gr = f:h where the constant term of h is 1.
We now use the fact that the irreducible factors of Gr are known; they are in f*
*act
just G1, G2=G1,: :,:Gr=Gr-1. This is an easy consequence of Galois theory. In f*
*act,
the roots of Gr=Gr-1 correspond to the angles 2ssk_prwhere k is prime to p (in *
*our
description above, they are cos2ssk_pr- 1 = -2 sin2ssk_pr). Let fi be a generat*
*or of (Z=pr)*.
Then the automorphism of the cyclotomic field which takes z to zfipermutes the *
*roots
of Gr=Gr-1 cyclically. Hence, each factor is irreducible. The constant term o*
*f each
factor is p, so Gr has no non-trivial factors with constant term 1. We can con*
*clude
that f = Gr. This implies lemma 6.10 and hence proposition 6.7.
Finally, with waning enthusiasm (are you still there, reader?), we must discuss*
* the case
ff = 1. Accordingly, we consider the spectral sequence
r+1
{Ekpr+1(Zp)} : Hi(BSO(3); Kj(Fkpr+1(S2); Zp)) ) K*(Bsp ; Zp)
We have a description of K*(Fkpr+1(S2); Zp)) from proposition 2.6, and we also *
*can
use the substantial information already derived on the spectral sequence when f*
*f =
0. Finally, we can use the S0-algebra structure derived from proposition 6.3 f*
*or our
description as we used the S-algebra structure in the other cases. Parallel to*
* our
description of generators for the K-theory of Fkpr(S2), we have:
For k = 2q, the E2-term is the module over H*(BSO(3); Zp) generated by:
(A) 1 . qpr, of order pr;
(B) All products 2mp+1 . qpr-mp for m = 1; : :;:qpr-1 - 1, of order ps, where s*
* is the
p-value of qpr - mp;
(C) All products 2mp-2 . fl3 . qpr-mp for m = 1; : :;:qpr-1 - 1, of order as in*
* (B);
(D) 2qpr+1, of infinite order;
(E) 2qpr-2. fl3, of infinite order;
(F) . 2qpr, of order pr;
(G) . qpr, of order pr;
(H) All products pm(r;s)j0 . 2mp-1 . qpr-mp for m as in (B), where m(r; s) = ma*
*x (r -
s; 0), and s is the p-value of qpr - mp;
36
19/1/1998
(I) All products . 2mp . qpr-mp for m as in (B), where s is the p-value of qpr*
* - mp;
the products in (H), (I) having order pmin(r;s).
For k = 2q + 1, it is the module over H*(BSO(3); Zp) generated by:
(B) All products (2m+pr-1)p+1. qpr-mp for m = 1; : :;:qpr-1 - 1, of order ps, w*
*here
s is the p-value of qpr - mp;
(C) All products (2m+pr-1)p-2. fl3 . qpr-mp for m = 1; : :;:qpr-1 - 1, of order*
* as in
(B);
(D) (2q+1)pr+1, of infinite order;
(E) (2q+1)pr-2. fl3, of infinite order.
(F) . (2q+1)pr, of order pr;
(H) All products pm(r;s)j0 . (2m+pr-1)p-1. qpr-mp for m as in (B), where m(r; s*
*) =
max (r - s; 0), and s is the p-value of qpr - mp;
(I) All products . (2m+pr-1)p. qpr-mp for m as in (B), where s is the p-value *
*of
qpr - mp; the products in (H), (I) having order pmin(r;s).
The prospect of dealing with the associated spectral sequence is not so forbidd*
*ing (given
all these generators) as it at first appears. In fact, from lemma 6.1 and propo*
*sitions
6.4 and 6.5, all differentials after d4 vanish on all generators except (perhap*
*s) the
generators (H). In fact, even d4 is only non-vanishing on the (C)'s and (E)'s. *
*We can
therefore use the same strategy as before; knowing that the (H)'s have even deg*
*ree _
so that theirrboundaries, if any, are odd _ we can (a) find an upper bound for *
*the rank
of K1(Bkp +1; Zp), assuming all differentials are zero; (b) show that this is e*
*xactly
the rank we should have; and (c) deduce that if any differentials are non-vanis*
*hing on
the (H)'s, the rank must be strictly less.
To combine as far as possible brevity with honesty, this is carried out as foll*
*ows. First,
we note that the odd generators in case k = 2q are those of type (A), (B), (E),*
* (G),
(I); and as before, the (E)'s are knocked out by d4. Next, in terms of the S0-m*
*odule
structure, the term (A) is paired with (G), giving a single summand in K"1gener*
*ated
by "1. "qpr. By the argument used in proposition 6.4(ii) and (iii), this has r*
*ank at
most pr - 1 over Zp^(and the same applies to the module generated by the product
of "1. "qprwith anything else). Now the important point to note is that althoug*
*h the
orders of the terms (B), (I) which correspond are different in E2, they become *
*the same
in E4, i.e. pmin(r;s). We can deduce that the S0-modules which arise as summand*
*s have
rank at most pmin(r;s)- 1 in each case. By a calculation which exactly parallel*
*s that
of lemma 6.8, we deduce:
Lemma 6.11. Suppose that, in the spectral sequence
r+1
{En(Zp)} : H*(BSO(3); K*(Fkpr+1(S2); Zp) ) K*(Bkp ; Zp))
all differentials vanish after d4. Then the rank of K"1(Bkpr+1; Zp) as a module*
* over
Zp^is at most equal to
Xr
(39) OE(pi)(qpr-i)
i=1
37
19/1/1998
if k = 2q, and at most equal to
Xr h(2q + 1)pr-ii
(40) OE(pi) ___________
i=1 2
if k = 2q + 1.
That is, it is twice the sum (in each case) computed by lemma 6.8. This is agai*
*n the
count for the rank given by [7].
We don't need lemma 6.11 to deduce that the odd generators are cycles _ as has *
*been
pointed out, we already know that. However, we do need it to show that none of *
*them
are boundaries, in particular to show that the unusual generators of type (H) a*
*re also
infinite cycles. This result, parallel to lemma 6.10, is proved in exactly the *
*same way.
Accordingly, we have:
Proposition 6.8. All differentials after d4 vanish in the spectral sequence*
* for
K*(Bmp+1 ; Zp).
Corollary 6.2. The S0-module generated by an element "aps+1."bpr has rank exac*
*tly
pmin(r;s)- 1 over Zp^.
(Compare proposition 6.5 and corollary 6.1.) We therefore once again know the a*
*dditive
structure of K1(Bmp+1 ; Zp), in particular the torsion. Here we have something *
*of a
surprise. The previous arguments show that there is torsion coming from generat*
*ors
of type (B) _ a torsion summand of order ps-r in K1 resulting from a relation o*
*n the
generator "kpr-2qps+1. "qpswhenever s > r; these have the same order and number*
* as
before. However, the same argument does not apply to . "kpr-2qps. "qps, since*
* this
already has order pr, and so does not contribute torsion in the same way. Thank*
*fully,
we observe that the results of x5 imply that this is the correct count for the *
*number
of torsion summands (i.e. the same as for ff = 0); there can therefore be no mo*
*re, and
we can state:
Proposition 6.9. (i) The torsion in K*(Bkpr+1; Zp) is all in K1.
(ii) For each integer 2qps < kpr, where s > r, there is a torsion summand of or*
*der
ps-r, and these are all the summands.
This completes the remaining case of the proof of theorem 1.1.
Finally, what about the S0-module structure of K0? Here the same argument appli*
*es,
but with a significant difference. In fact, the generators of the modules do no*
*t come
from the terms (C) (as they did in the previous case), but from the (H)'s; this*
* isr
because of the relation fl3 = j0 . , properly interpreted in the K-theory of Bk*
*p +1.
We can all the same pursue the argument on the same lines, noting that by the t*
*ime
we reach E4, the (H) term and the corresponding (B) term have the same order. I*
*f we
write "v(apr; bps) for the p-local generator corresponding to an appropriate mu*
*ltiple of
j0 . aps-1. bpr, it is again easy to show that "v(apr; bps) is annihilated by t*
*he right
polynomial, and so generates the right module, if r 6= s; while if they are equ*
*al, a
special argument on the lines of lemma 6.10 will prove:
38
Proposition 6.9. The annihilator of "v(apr; bps) in S' is precisely the ideal *
* (0Gr) if
s > r and (0Gr) if s r where 0Gr("o) = 0Fr("o)="o.
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