On the Connective $K$-Homology Groups of the Classifying Spaces $\textit{BZ}/p^r$

Abstract

Let p be a prime number and Z/p be a cyclic group of order p '. We choose BZ/p, the classifying space of Z/p, as the colimit of the lens spaces L(p), It is well known that K(BZ/p) =0Z^ by Atiyah [1], Atiyah-Segal [2], and K^-^BZ/p*} =®Z/p°° by Vick [9] (and the groups of other degrees are trivial) . We consider the connective X-theory k. Using the Atiyah-Hirzebruch spectral sequence, connective K(co) homology group of BZ/p is a subgroup of periodic K(co) homology group of BZ/p. And the Atiyah Hirzebruch spectral sequence also shows that k2n-i (BZ/p ) is a finite group for any n. Moreover, the Atiyah-Hirzebruch spectral sequence determines its order. The purpose of this paper is to calculate the additive structure of