The Geometry of Iterated Loop Spaces

i Preface This it the first of a series of papers devoted to the study of iterated loop spaces. Our goal is to develop a simple coherent theory which encompasses most of the known results about such spaces. We begin with some history and a description of the desiderata of such a theory. First of all, we require a recognition principle for n-fold loop spaces. That is, we wish to specify appropriate internal structure such that a space X possesses such structure if and only if X is of the (weak) homotopy type of an n-fold loop space. For the case n = 1, Stasheff's notion [28] of an A ∞ space is such a recognition principle. Beck [5] has given an elegant proof of a recognition principle, but, in practice, his recognition principle appears to be unverifiable for a space that is not given a priori as an n-fold loop space. In the case n = ∞, a very convenient recognition principle is given by Boardman and Vogt's notion [8] of a homotopy everything space, and, in [7], Boardman has stated a similar recognition principle for n < ∞. We shall prove a recognition principle for n < ∞ in section 13 (it will first be stated in section 1) and for n = ∞ in section 14; the latter result agrees (up to language) with that of Boardman and Vogt, but our proof is completely different. By generalizing the methods of Beck, we are able to obtain immediate non-iterative constructions of classifying spaces of all orders. Our proof also yields very precise consistency and naturality statements. In particular, a connected space X which satisfies our recognition principle (say for n = ∞) is not only weakly homotopy equivalent to an infinite loop space B 0 X, where spaces B i X with B i X = ΩB i+1 X are explicitly constructed, but also the given internal structure on X agrees under this equivalence with the internal structure on B 0 X derived from the existence of the spaces B i X. We shall have various other consistency statements and our subsequent papers will show that these statements help to make the recognition principle not merely a statement as to the existence of certain cohomology theories, but, far more important, an extremely effective tool for the calculation of the homology of the representing spaces. An alternative recognition principle …