Characterisation of Plus-Constructive Fibrations

Although originally devised to define the higher algebraic K-theory of rings [ 1, 7, 161, the plus-construction has quickly established its usefulness in such diverse areas as stable homotopy theory [ 15 J, bordism of manifolds [lo], and the study of knot complements [ 141. Recall (from, e.g., 11 ] h w ose notation we foHow) that the plusconstruction qx : X --, X+ is a pointed cofibration which induces isomorphisms on homology with Abelian local coeficients (equivalently, has acyclic fibre) and an epimorphism on fundamental groups whose kernel is the maximal perfect subgroup Br,(X) of z,(X). (These perfect radicals play an important role: in particular an epimorphism G ++Q If preserving perfect radicals (&PG = .PH) will be said to be EP*R [ 3 1.) Evidently, the cardinal question is the effect of the construction on homotopy groups. (For instance, when X = BGLA, A a ring, then zjXt = KjA, thejth algebraic K-theory group of A.) Since the principal tool here is the homotopy exact sequence of a (homotopy) fibration, this effectively reduces to asking when a @ration 8: F --) E jp B is plus-constructive (P.-C.), that is when F+ -+ Et -t B + is also a fibration (i.e., F+ N F,,). We assume that all spaces discussed are of the homotopy type of a connected CW-complex which (for (iii) below) admits a finite Postnikov decomposition.) To date, the literature contains numerous suffLzient conditions (e.g., [2, 6, 8, 9, 13, 17, 181) and the necessary condition that zl(p) be EP*R [ 1, (6.8)]. In view of the awesome complication of some of these (sets of) hypotheses, it is remarkable that a simple characterisation of when 8 is P.-C. is after all possible. (The brevity of the proof is a further blessing.)