The Segal Conjecture for Cyclic Groups

where /4(G) 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] as a functor of W, so n+{F{X, Y)) = [_X, Y\. The functional dual DX of X is F(X, S°). If X is finite, DX is its Spanier-Whitehead dual and for any X, n-SPX) — n~\X). For a space X let 'LX+ denote the suspension spectrum of X union a disjoint base point. Our main result is