Mod p K-theory of Ω∞Σ∞X revisited

Abstract

In this note we present a new proof of a theorem of McClure on K*(Ω∞Σ∞X, Z/p) [11], in the special case when X is a finite complex with K1(X; Z/p) = 0. Although our method does not work in the full generality covered by his work, our argument requires neither a geometric interpretation of complex k-theory nor all the delicate coherence properties of its multiplication. Since BP-theory is not likely to possess such coherence properties [9], the possibility of generalizing his approach to the case of higher Morava K-theory does not seem feasible. On the contrary, the main ingredient of our approach is the rank formula for the Morava K-theory of the Borel construction [5], which works for any K(n); thus our approach is better adapted to the potential generalization [8]. Throughout the paper we assume that p > 2 so that mod p K-theory possesses a commutative multiplication, and denote by K*(−) the mod p K-theory. Since it is simpler to state our results in terms of CX, the combinatorial model for QX, rather than QX itself, we shall do so. This is sufficient, as when X is connected CX is homotopy equivalent to QX, and when not, K*(QX) can be easily recovered from K*(CX) (see e.g. [11]).