DYER-LASHOF OPERATIONS IN if-THEORY

Dyer-Lashof operations were first introduced by Araki and Kudo in [1] in order to calculate ü*(QS+; Z2). These operations were later used by Dyer and Lashof to determine H*(QY;ZP) as a functor of H*(Y;ZP) [5], where QY = | J n H n E n y . This has had many important applications. Hodgkin and Snaith independently defined a single secondary operation in if-homology (for p odd and p = 2 respectively) which was analogous to the sequence of DyerLashof operations in ordinary homology [7, 13], and this operation has been used to calculate K*(QY] Zp) when Y is a sphere or when p = 2 and y is a real projective space [11, 12]. In this note we describe new primary Dyer-Lashof operations in if-theory which completely determine K*(QY;ZP) in general. We shall remove the indeterminacy of the operation by lifting it to higher torsion groups. First we establish notation. X will always denote an Eoospace [9] and Y will denote an arbitrary space, considered as a subspace of QY via the natural inclusion. We write K*(Y; r) for K0{Y; Zpr) 0 ^ ( 7 ; Zpr)\ in particular if-theory is ^-graded and we write |x| for the mod 2 degree of x. There are evident natural maps pt:Ka(Y;r)-+Ka(Y;r + s) i f s> l , *:KaQ ;r)-+KaO',t) i f l < t < r , and ft:lfa(r;r)->lC0_i(y;r). T H E O R E M 1. For each r>2 and a e Z2 there is an operation Q:Ka(X;r)-+Ka(X;r-l)