The Segal conjecture for cyclic groups

homotopy type of BG, the classifying space of the finite group G. In one form it partly describes the stable cohomotopy of BG, asserting that 0 if i> 0. A{GY ifi = 0, where /4(G)A denotes the completion of the Burnside ring of G with respect to the ideal of virtual G-sets of degree 0. The conjecture was proved for G = Z/(2) by W. H. Lin [5], [3] and for G = Z/(p), where p is an odd prime, by J. H. C. Gunawardena [4]. In this note we outline a proof for G cyclic. We will assume that G has prime power order as the general case follows easily. The proofs cited above make essential use of the Adams spectral sequence [2] and include a hard calculation of a certain Ext group. Our proof uses their results to start an inductive argument. While it does not require any more hard calculations, it does use a certain generalization of the Adams spectral sequence to be described below. Recall that the function spectrum F{X, Y) by definition represents \_W A X, V] a