bu*(BZ/p)^n as a Graded Group

Abstract

Let p be a prime. We calculate bu*(BZ/p)^n, the connective unitary K-theory of the n-fold smash product of the classifying space for the cyclic group of order p, as a graded group using a K\"{u}nneth formula short exact sequence for n=2 and inductively for any n>= 2.
While this smashing is in progress some other spectra appear, for instance, the spectrum HZ/p\wedge (B\Z/p)^r for r<n. In order to producing a new homotopy equivalent to bu\wedge (B\Z/p)^n, we need to find a homotopy equivalence which simplifies the spectrum H\Z/p\wedge (B\Z/p)^r.