KO-Rings of S
2k+1
/
Z2
nMehmet Kırdar
kirdar@fen.bilkent.edu.tr
March 7, 1997
Abstract
For n ≥ 2, we describe the reduced real K-theory K O˜ (Lk(2 n
)) of the standart lens space Lk(2
n
) in terms of generators and relations. In particular, we show that ˜K O-order of the Hopf bundle is 2n
−1+2[
k 2]. Introduction.
Let Lk(2n) = S2k+1/Z2n be the (2k +1)th-dimensional standard lens space
mod 2n. For n = 1, L
k(2) is the real projective space and the structure of
KO-rings are well-known, see e.g., [Ka] Chapter IV, 6.46. For KO(Lk(4)),
see [KS]. We will deal with the problem for all n ≥ 2.
We have the following set-up for the problem, see e.g. [KS]. Let ξ and η be the non-trivial real line bundle and the canonical complex line bundle (the Hopf bundle) over Lk(2n) and λ = ξ − 1 and µ = η − 1
be their reductions respectively. Bundle complexification and realification yield the corresponding homomorphisims c : ˜KO(Lk(2n)) → ˜K(Lk(2n)) and
r : ˜K(Lk(2n)) → ˜KO(Lk(2n)). Set w = r(µ). The following relations hold :
c(ξ) = η2n−1
and η2n
= 1. Let π : Lk(2n) → Pn(C) be the standard
projec-tion, then we have π!(µ) = µ and π!(w) = w.
The results are consequences of Theorem 2.2 from [AW] and some elemen-tary numerical computations using the relations and the Atiyah-Hirzebruch spectral squence for K-theory.
K-rings.
We consider the Atiyah-Hirzebruch spectral sequence for K(Lk(2n)). It
collapses and we have
Theorem 1. K((Lk(2n)) = Z[µ]/ < µk+1, (1 + µ)2
n
− 1 > KO-rings.
We will recall some facts about KO(P˜ n(C)) from [AW]. Due to
Theo-rem 2.2(ii), Ψk
R(w) = Tk(w). The polynomials Tk are given by Tk(w) =
Pk j=1αk,jwj, where αk,j = (k j)( k+j−1 j ) (2j−1 j ) .
The following lemma will be the key to the problem. Lemma 2. r(µk) =P[ k 2] r=0βk,rwk−r, where βk,r = ( 2r r)( k 2r) (k−1 r ) .
Proof : We prove by induction. It is true for k = 1. Assume that it is true for < k. Due to Lemma A 2 [AW], r commutes with Ψk
Λ. Thus we have rΨk C(µ) = ΨkR(w), i.e, r((1 + µ)k − 1) = ΨkR(w). By induction, r(µk) =Pk j=1αk,jwj−Pk−1i=1 k i P[ i 2] r=0βi,rwi−r. The coefficient of wj in r(µk) is α k,j −Pk−j−1r=0 k j+r βj+r,r. If we equate it
to βk,k−j and simplify, we find
k+j−1 j =Pk−j r=0 k−j r 2j−1 j−r . By substituting m = k + j − 1, p = k − j we have mj = Pp r=0 p r m−p j−r
. This verifies the formula.
Theorem 3. KO(L˜ k(2n)), n ≥ 2, is generated by λ and w with the
following relations: (i)Pk−2j p=0 Pp+jr=p−[p2] 2n p+1 β2j+p,j+p−rwj+r = 0, 1 ≤ j ≤ [k2]. If k 6≡ 1 (mod 4) then w[k2]+1 = 0. If k ≡ 1 (mod 4) then 2w[k 2]+1 = 0, w[ k 2]+2= 0. (ii) 2λ = Ψ2n−1 R (w) (iii) λw = (Ψ2Rn−1+1− Ψ2n−1 R − 1)(w).
Proof : Let k = 4t + 3. Consider c : KO(L˜ k(2n)) → ˜K(Lk(2n)). We
have c(λ) = (1 + µ)2n−1
− 1 = 2n−1µ + ..., c(w) = µ + (1 + µ)2n−1
− 1 = µ2+ ...
Let F be the subgroup of ˜K(Lk(2n)) generated by c(λwi), i ≥ 0 and c(wi),
i > 0. Then using Theorem 1, we have |F | = 2[k 2]n+[
k
2]+1. Consider the E
2
-level of the Atiyah-Hirzebruch spectral sequence of ˜KO(Lk(2n)). It is clear
that ˜KO(Lk(2n)) has at most 2[
k 2]n+[
k
2]+1 elements. Thus c is monomorphism
and the differentials dr, r ≥ 2, are zero on the groups of total degree zero.
For other values of k, they vanish by naturality of the spectral squences un-der the inclusions jk,k0 : Lk0(2n) → Lk(2n), k
0
< k. This together with the relation (iii) show that the additive relations (i) are the smallest.
(i) These are simply the trivial relations r(µrη2n
− µr) = 0. From η2n
= 1, after the binomial expansion and multiplication by µ2j−1, we have
Pk−2j
p=0
2n
p+1
µ2j+p = 0. Using Lemma 2, we realify both sides, then we have
Pk−2j
p=0 P j+[p2]
q=0 β2j+p,qw2j+p−q = 0. Put r = j + p − q. This gives (i). It is clear
from the expression that j ≤ [k
2]. The other statements follow from Theorem
2.2 (i) in [AW].
(ii) follows from the relation c(ξ) = η2n−1
. (iii) It is clear that c(λw) = c((Ψ2Rn−1+1− Ψ2n−1
R − 1)(w)). For k = 4t + 3, c is monomorphism and the relation holds. For other values of k, it holds by naturality.
Remark. The relations (i) are the periodicity relations ψ2Rn+i(w) = ψi
R(w) .
Remark. Consider the Atiyah-Hirzebruch spectral sequence for ˜KO(Pk(C)).
Due to Lemma 2.4 in [AW], E8i+2j,−8i−2j
∞ are generated by w
2i+1, 2w2i+1, w2i+2
for j = 1, 2, 4 respectively. This is the same for the spectral sequence of ˜
KO(Lk(2n)). Due to relation (iii), E∞8i+1,−8i−1is generated by λw
2i(or 2nw2i).
Corollary 4. KO-order of w˜ i is 2n+2[k
2]+1−2i, 1 ≤ i ≤ [k
2].
Proof : The coefficient of wj+r of the relation in Theorem 3 (i) is
P2r
p=0
2n
p+1
β2j+p,j+p−r. In particular, for r = 0, 2n+1 and for r = 1, −2n−1+
2n(...). For r ≥ 2, since v 2( 2n 2r ) ≥ n + 3 − 2r, it has at least 2n−2(r−1) as
a factor. Thus, we have 2n+1wj = 2n−1wj+1+ higher order terms which are
annihilated faster than 2n−1wj+1 by multiplication by 2. Now the result
fol-lows by induction and from the fact that 2w[k
2]+1= 0 when k ≡ 1 (mod 4).
References.
[AW] J. F. Adams-G. Walker, On complex stiefel manifolds, Proc. Camb. Phil. Soc. (1965), 61, 81.
[Ka] M. Karoubi, K-Theory. An Introduction, Grundlehren der mathe-matischen Wissenschaften 226, Springer-Verlag, 1978.
[KS] T. Kobayashi-M. Sugawara, KΛ-rings of Lens spaces Ln(4),
Hi-roshima Math. J. (1971), 253-271.
BILKENT UNIVERSITY DEPARTMENT OF MATHEMATICS 06533 BILKENT, ANKARA, TURKEY
kirdar@fen.bilkent.edu.tr