Abstract

I presented a cal-culation of H,(F;Zp) as an algebra, for odd primes p, where F = lim F(n) and F(n) is the topological monoid of homotopy equivalences of an n-sphere. This computation was meant as a preliminary step towards the computation of H*(BF;Zp). Since then, I have calculated H*(BF;Zp), for all primes p, as a Hopf algebra over the Steenrod and Dyer-Lashof algebras. The calculation, while not difficult, is somewhat lengthy, and I was not able to write up a co-herent presentation in time for inclusion in these proceed-ings. The computation required a systematic study of homology operations on n-fold and infinite loop spaces. As a result of this study, I have also been able to compute H,(2nsnx;Zp), as a Hopf algebra over the Steenrod algebra, for all connected spaces X and prime numbers p. This result, which generalizes those of Dyer and Lashof [3] and Milgram [8], yields explicit descriptions of both H,(~nsnx;Zp) and H,(QX;Zp), QX = li~> 2nsnx, as functors of H,(X;Zp). An essential first step towards these results was a systematic categorical analysis of the notions of n-fold and infinite loop spaces. The results of this analysis will- 449 be presented here. These include certain adjoint functor relationships that provide the conceptual reason that H.(~nsnx;zp) and H,(QX;Zp) are functors of H,(X;Zp) and that precisely relate maps between spaces to maps between spectra. These categorical considerations moti-vate the introduction of certain non-standard categories, I and i, of (bounded) spectra and ~-spectra, and the main purpose of this paper is to propagandize these cate-gories. It is clear from their definitions that these categories are considerably easier to work with topolog-ically than are the usual ones, but it is not clear that they are sufficiently large to be of interest. We shall remedy this by showing that, in a sense to be made precise, these categories are equivalent for the purposes of homotopy theory to the standard categories of (bounded) spectra and ~-spectra. We extend the theory to unbounded spectra in the last section.

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