On the K-theory of the extended power construction

Abstract

The construction of Dyer-Lashof operations in K-theory outlined in (6) and refined in (12) depends in an essential way on the descriptions of the mod-p K-theory of EZp, ×ZpXp and EΣ ×σ p Xp given there. Unfortunately, these descriptions are incorrect when p is odd except in the case where the Bockstein β is identically zero in K*(X; Zp), and even in this case the methods of proof used in (6) and (12) are not strong enough to show that the answer given there is correct. In this paper we repair this difficulty, obtaining a complete corrected description of K*(EZp ×ZpXp; Zp) and K*(EΣp) (theorem 3·1 below, which should be compared with ((12); theorems 3·8 and 3·9) and ((6); theorem 3)). Because of the error, the method used in (6) and (12) to construct Dyer-Lashof operations fails to go through for odd primes when non-zero Bocksteins occur, and it is not clear that this method can be repaired. We shall not deal with the construction of Dyer-Lashof operations in this paper. Instead, the first author will give a complete treatment of these operations in (5), using our present results and the theory of H∞-ring spectra to obtain strengthened versions of the results originally claimed in ((12); theorem 5·1). There is also a minor error in the mod-2 results of (12) (namely, the second formula in (12), theorem 3·8 (a) (ii)) should readwhere B2 is the second mod-2 Bockstein, and a similar change is necessary in the second formula of ((12), theorem 3·8(b) (ii)). The correction of this error requires the methods of (5) and will not be dealt with here; fortunately, the mod-2 calculations of ((12), §6–9), (10) and (11) are unaffected and remain true as stated.