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KO-Rings of S

2k+1

/

Z

2

n

Mehmet 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.

(2)

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

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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+

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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

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