K(Z/2) as a Thom Spectrum

Abstract

Dyer-Lashof operations are used to give a simple proof that K(Z/2) is a Thom spectrum. The purpose of this note is to give a simple proof of Mahowald's striking observation that the Eilenberg-Mac Lane spectrum K(Z/2) is a Thom spectrum. Our proof follows closely a proposed (but incorrect) proof by Madsen and Milgram [2]. Although Mahowald has given another proof [3], the idea of [2] is so appealing that it seems useful to record a corrected version. Let 'q: S1 BO represent the generator of T1 BO = Z/2. Since BO is a double loop space there is an induced map 2Q2S I = Q222(s I n Q2222BO-BO Let M(y) denote the Thom spectrum associated with y (localized at 2). THEOREM (MAHOWALD). M(y)K(Z/2). PROOF. We recall that the Dyer-Lashof operation Q1 is defined for double loop spaces and that H*22S3 = Z/2[xl, x2, ... , Xk,...] where xk = Q, * * * Qltl the (k 1) fold iterate of Q1 applied to the fundamental class 1 (Z/2 coefficients are used throughout). Also H*K(Z/2) = Z/2[t1, (2' * * * I (k* ... ] = A* the dual of the mod 2 Steenrod algebra A. Let a: A -* H*MO denote evaluation on the Thom class U, i.e. a(a) = aU, al* is an algebra morphism in homology. Finally, let I': M(y) -> MO denote the map of Thom spectra induced by y; r'* is also an algebra morphism in homology. Since deg xk = 2k 1 = deg (k, H*22S3 H*M(y) and A* have the same rank in each dimension and so the theorem follows from LEMMA. * a H*M(y) H*MO A* is surjective. PROOF. Consider the commutative diagram Received by the editors August 1, 1977. AMS (MOS) subject classifications (1970). Primary 55B20; Secondary 57D75. 'Supported in part by NSF Grant MPS76-07051. ?D American Mathematical Society 1978