The uniqueness of infinite loop space machines

Abstract

AN INFINITE loop space machine is a functor which constructs spectra out of simpler space level data. There are many such machines known[ 14,8,151. They differ somewhat in the data they accept. Worse, they are given by such widely disparate topological constructions that it is far from obvious that they turn our equivalent spectra when fed the same data. The purpose of this paper is to prove that all machines which satisfy certain reasonable properties do in fact turn out equivalent spectra. The properties are satisfied by Segal’s machine[l5], but require use of somewhat more general input data than the other machines in the literature are geared to accept. We generalize May’s machine [8,9] so that it acts in the requisite generality and satisfies the requisite properties. Thus the May and Segal machines are equivalent. This proof will illustrate what would be involved in the corresponding generalization of other machines, and we are quite confident that an exhaustive case-by-case verification would lead to the conclusion that there is really only one infinite loop space machine. To avoid leaving a wrong impression, we hasten to add that this does not mean we can now discard all but one of the explicit constructions. The purpose of the constructions is to prove theorems and make calculations, of the sort sketched in [ll], and such applications may only be accessible to one or another of the machines. For example, the passage from E, ring spaces to E, ring spectra, the construction of classifying spectra for bundle and fibration theories oriented with respect to an E, ring spectrum, and the passage from E, ring spectra to H, ring spectra[l2,131 are part of a calculationally powerful circle of ideas which depends on use of the particular geometry of May’s machine. The point here is that while there is now a uniqueness theorem for infinite loop space machines, there is no uniqueness theorem for the assembly lines of multiplicative infinite loop space factories. On the other hand, Segal’s machine has the distinct advantage of being very much simpler to construct than the others. Moreover, it will play a canonical role in our theory. Rather than compare two machines directly, we compare each of them to Segal’s machine. We give a general discussion of the input data of infinite loop space machines in 6 I and give a way to construct examples in 04. We prove the uniqueness theorem in $02 and 3, except that we relegate the proof of a key result about spectra to the first appendix. We give the promised generalization of May’s machine in §§S and 6. As is traditional in this subject, there is also an appendix about cofibrations. In the course of proving our new results, we have had to redevelop and systematize the foundations of infinite loop space theory, and it is our hope that the present paper can serve as a readable source for its main ideas and techniques. The first author wishes to acknowledge that the key new idea is entirely due to the second author and the latter wants to thank Waldhausen for a very helpful conversation. Both authors wish to acknowledge that the basic insight comes from Fiedorowicz’ paper [6].