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Complete Reduction in K-Theory

We define a reduction, called complete reduction, for the K and KO relations of the Hopf bundle over lens spaces introducing some numbers of interest to various theories of mathematics. Along the way, we make an interesting conjecture in number theory related to the cyclotomic fields.


Introduction
Let BZ n denote the classifying space of the cyclic group Z n . The K-ring of BZ n is given by K(BZ n ) = Z [µ] ((1 + µ) n − 1) where µ = η − 1 is the reduction of the tautological complex line bundle (Hopf bundle) over BZ n . When n is odd, the KO-ring of BZ n is described in the following way: Let w = r(µ) be the realification of µ. Then, KO(BZ n ) = Z [ω] (wf n (w)) where f n (·) is the polynomial The topological K-theory over the real numbers is a little more involved when n is even. See [2] for details. From now on, for simplicity, let n = p be an odd prime number, although the very simple idea of the paper can be extended for all natural numbers. In this case, the topological K-theory of the lens spaces is very well-studied. See [1].
In this note, we will define a reduction called "complete reduction" for the relations of µ and ω coming from the generators of the principal ideals of the above rings. Complete reduction is the smallest way of writing these relations geometrically, respecting the Atiyah-Hirzebruch spectral sequence and detects the group cohomology of Z p by means of the filtrations of that spectral squence.
In order to obtain first few terms of the complete reduction, we make a division trick and this gives some invariants -numbers-which we named K n for the complex case and M n for the real case, interesting not only for KΛ-rings of lens spaces in topological K-theory but for RΛ-rings of cyclic groups in representation theory and for cyclotomic rings of integers in number theory, due to the equivalence of theories K(BZ p ), R(Z p ) and Z exp 2πi p .
Obviously the complete reduction is unique. Example 1.2. For p = 3, the complete reduction is periodic with period 2 and repeating coefficients −1, 1. For p = 5, the complete reduction is periodic with period 6 and repeating coefficients −1, 2, −2, 1, 0, 0. For p = 7, the first 28 coefficients of the complete reduction is as below: 7µ = −µ 7 + 3µ 8 + 3µ 9 + 2µ 10 + 2µ 11 + 3µ 12 + µ 13 − 2µ 14 + 0.µ 15 + µ 16 Quite surprisingly, we couldn't observe periodicity of the coefficients of the complete reduction for p = 7. One should do further reduction to decide if it is periodic or not. Although we had the opposite belief after this experimentation, as an interesting open problem in number theory, we make the following probably false conjecture. Conjecture 1.3. The complete reduction is periodic for all odd prime numbers. Next, we want to express first few coefficients of the complete reduction in terms of p. We introduce a very simple idea, probably done before, many times in history. We will do the division trick used in the example above. Definition 1.4. Define integers K p,n by ∞ n=0 K p,n µ n = −pµ Let us denote K p,n simply by K n when p is understood. Then pµ = ∞ n=0 K n µ p+n is a reduction of the relation of µ. But, of course, it is not the complete reduction except for the primes 3 and 5. On the other hand, the first p + 1 coefficients of the complete reduction of the relation of µ are K n (mod p), 0 ≤ n ≤ p.
The numbers K n satisfy a recursive formula. By using this recursive formula or by direct division, which is the same process, we can compute K n for all n ≤ p − 2 as a polynomial of p. We computed upto K 6 as below: The author doesn't know whether these, probably very well-known, polynomials are used somewhere in number theory. For large primes, we can use these tabulated formulas, to find at least first p − 1 terms of the complete reduction for the prime number p. Example 1.5. For p = 23, K 1 = 11, K 2 = −44 ≡ 2, K 3 = 22 ≡ −1, K 4 = 374 ≡ 6, K 5 = −572 ≡ 3, K 6 = −10494 ≡ −6, and hence the first seven terms of the complete reduction are 23µ = −µ 23 + 11µ 24 + 2µ 25 − µ 26 + 6µ 27 + 3µ 28 − 6µ 29 + ......
Here, we recall the famous Bernoulli number B n . It immediately follows from the definitions that B n = lim p→∞ −n!Kn p n .

KO-Reduction
By iteration, the relation wf p (w) = 0, explicitly, can be written in the form Similar to the complex case, we call the relation above completely reduced if |b n | ≤ p−1 2 for all n ≥ 1. Complete reduction is clearly unique. Example 2.1. For p = 3, the complete reduction is 3ω = −ω 2 . It is periodic of period 1, all coefficients, after a 0 = −1, being 0. For p = 5, 5ω = −ω 3 + ω 4 + 5ω 3 is the complete reduction. It is periodic of period 2 with repeating coefficients −1, +1. For p = 7, we did some reduction and found first 16 terms of the complete reduction as below: Again, similar to the complex case, we couldn't observe a periodicty for the prime number 7. On the other hand, we conjecture that the complete reduction is periodic in the real case too. Next we define some numbers for the computation of the coefficients of the complete reduction in terms of p. Let us denote M p,n simply by M n when p is understood. Then pω = ∞ n=0 M n ω n+ p+1 2 is a reduction for ω. Of course, it is not complete reduction except for the primes 3 and 5. On the other hand, the first p+1 2 coefficients of the complete reduction of the relation of ω are M n (mod p), 0 ≤ n ≤ p−1 2 . Clearly M 0 = −1 for all p. We can calculate M n by writing a recursive formula like we did in complex case, or by direct division. We obtain formulas for M n in terms of p which are valid for p ≥ 2n + 3. The next three are Note also that by means of the realification map, M n should be expressed in terms of K n . So, if the conjecture is true for the complex case, it is also true for the real case.