Abstract not found

1. Preliminaries about topological model categories 5 2. Preliminaries about equivalences of model categories 9 3. The level model structure on D-spaces 10 4. Preliminaries about π∗-isomorphisms of prespectra 14

We identify the topological Hochschild homology (THH) of the Thom spectrum associated to an E ∞ classifying map X → BG, for G an appropriate group or monoid (e.g. U, O, and F). We deduce the comparison from the observation of McClure, Schwanzl, and Vogt that THH of a cofibrant commutative S-algebra (E ∞ ring spectrum) R can be described as an indexed colimit together with a verification that the Lewis-May operadic Thom spectrum functor preserves indexed colimits. We prove a splitting result THH(Mf) ≃ Mf ∧BX+ which yields a convenient description of THH(MU). This splitting holds even when the classifying map f: X → BG is only a homotopy commutative A ∞ map, provided that the induced multiplication on Mf extends to an E ∞ ring structure; this permits us to recover Bokstedt’s calculation of THH(HZ).

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.

Abstract. In this paper, we investigate the properties of the category of equivariant diagram spectra indexed on the category WG of based G-spaces homeomorphic to finite G-CW-complexes for a compact Lie group G. Using the machinery of [10], we show that there is a “stable model structure ” on this category of diagram spectra which admits a monoidal Quillen equivalence to the category of orthogonal G-spectra. We construct a second “absolute stable model structure ” which is Quillen equivalent to the “stable model structure”. Our main result is a concrete identification of the fibrant objects in the absolute stable model structure. There is a model-theoretic identification of the fibrant continuous functors in the absolute stable model structure as functors Z such that for A ∈ WG the collection {Z(A ∧ S W)} form an Ω-G-prespectrum as W varies over the universe U. We show that a functor is fibrant if and only if it takes G-homotopy pushouts to G-homotopy pullbacks and is suitably compatible with equivariant Atiyah duality for orbit spaces G/H+ which embed in U. Our motivation for this work is the development of a recognition principle for equivariant infinite loop spaces. The description of fibrant objects in the absolute stable model structure makes it clear that in the equivariant setting we cannot hope for a comparison between the category of equivariant continuous functors and equivariant Γ-spaces, except when G is finite. We provide an explicit analysis of the failure of

Abstract. E ∞ ring spectra were defined in 1972, but the term has since acquired several alternative meanings. The same is true of several related terms. The new formulations are not always known to be equivalent to the old ones and even when they are, the notion of “equivalence ” needs discussion: Quillen equivalent categories can be quite seriously inequivalent. Part of the confusion stems from a gap in the modern resurgence of interest in E ∞ structures. E∞ ring spaces were also defined in 1972 and have never been redefined. They were central to the early applications and they tie in implicitly to modern applications. We summarize the relationships between the old notions and various new ones, explaining what is and is not known. We take the opportunity to rework and modernize many of the early results. New proofs and perspectives are sprinkled throughout.