Infinite Loop Space Theory Revisited

Just over two years ago I wrote a summary of infinite loop space theory [37]. At the time, there seemed to be a lull in activity, with little immediately promising work in progress. As it turns out, there has been so much done in the interim that an update of the summary may be useful. The initial survey was divided into four chapters, dealing with additive infinite loop space theory, multiplicative infinite loop space theory, descriptive analysis of infinite loop spaces, and homol-ogical analysis of infinite loop spaces. We shall devote a section to developments in each of these general areas and shall also devote a section to the newly evolving equivariant infinite loop space theory. Two of the biggest developments will hardly be touched on here however. I ended the old survey with the hope that "much new information will come when we learn how the rich space level structures described here can effectively be exploited for calculations in stable homotopy theory." This hope is being realized by work in two quite different directions. As discussed in [37, §4], the approximation theorem to the effect that ~nznx is a group completion of the simple combinatorial space CnX plays a central role in the general theory. I stated there that "homotopical exploitation of the approximation theorem has barely begun." This is no longer the case. Such exploitation is now one of the more active areas of homotopy theory, recent contributions having been made to summarize the present state of the art in [42], and will content myself here with a remark in section two and a brief discussion of the equivariant approximation theorem in section five. Second, the notion of E ring spectrum discussed in [37, §ii] led to a simpler homotopical notion of H ring spectrum. This concept is really part of stable homotopy theory as understood classically, rather than part of infinite loop space theory, and seems to be basic to that subject. An introduction and partial summary of results based on this concept are given in [39]. A complete treatment will appear in the not too distant future [5]; meanwhile, the main results are avail