Preface The classifying spaces of discrete groups have been a subject for extensive study in the last two decades. We mention the remarkable result of H. Miller [48] giving a proof of the Sullivan conjecture. J. Lannes [36] studied mapping spaces out of a classifying space of an elementary abelian p-group and reproved Miller's theorem. W. Dwyer and A. Zabrodsky [23, 62] studied maps between classifying spaces of compact Lie groups. S. Jackowsky and J. McClure [30] have obtained a homotopy decomposition for the p-completed classifying space of any compact Lie group. J. Martino and S. Priddy [43] refined the theory of [30] to obtain a criterion for p-local equivalence of classifying spaces of finite groups. E. Fried- lander [27] has shown that there is a strong relationship between the classifying spaces of complex reductive Lie groups and the corresponding discrete groups pf Lie type. Regarding stable homotopy theory, G. Carlsson [13] has studied the stable homotopy theory of BG and obtained a proof of the Segal conjecture. P. May [47] later generalized Carlsson's result. The Segal conjecture was also used by J. Martino and S. Priddy [40, 41, 42] in the study of stable homotopy theory associated with classifying spaces of finite groups. The list of authors who dealt with the various aspects of the homotopy theory associated with classifying spaces of finite groups and compact Lie groups is indeed too long to be spelled out. In the early 1970's D. Quillen introduced the "plus" construction and used it to define higher algebraic K- groups [54, 55]. The "plus" construction can be thought of as a method of associating with a space X and a perfect normal subgroup P of 7Ti(X), a new space X+ and a map i : X — X+ such that i induces a homology isomorphism with respect to any coefficients system o n I + and such that TTI(X+) ^ TT1{X)/P. A natural question that arises at this point is whether one can go the other way, namely given a space X, does there exist a discrete group 7r such that H*(X) = H*(BTT) with respect to some coefficient system on X. Indeed, D. Kan and W. Thurston have established in [33] that given any connected space X with some reasonable properties, there exists a space TX together with a map

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 1996 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.