1.1. Simplicial sets 5 1.2. Symmetric spectra 6

modules over highly structured ring spectra to give new constructions of MUmodules such as BP, K(n) and so on, which makes it much easier to analyse product structures on these spectra. Unfortunately, their construction only works in its simplest form for modules over MU [ 1] ∗ that are concentrated in 2 degrees divisible by 4; this guarantees that various obstruction groups are trivial. We extend these results to the cases where 2 = 0 or the homotopy groups are allowed to be nonzero in all even degrees; in this context the obstruction groups are nontrivial. We shall show that there are never any obstructions to associativity, and that the obstructions to commutativity are given by a certain power operation; this was inspired by parallel results of Mironov in Baas-Sullivan theory. We use formal group theory to derive various formulae for this power operation, and deduce a number of results about realising 2-local MU∗-modules as MU-modules. 1.

Abstract. We study the structure possessed by the Goodwillie derivatives of a pointed homotopy functor of based topological spaces. These derivatives naturally form a bimodule over the operad consisting of the derivatives of the identity functor. We then use these bimodule structures to give a chain rule for higher derivatives in the calculus of functors, extending that of Klein and Rognes. This chain rule expresses the derivatives of FG as a derived composition product of the derivatives of F and G over the derivatives of the identity. There are two main ingredients in our proofs. Firstly, we construct new models for the Goodwillie derivatives of functors of spectra. These models allow for natural composition maps that yield operad and module structures. Then, we use a cosimplicial cobar construction to transfer this structure to functors of topological spaces. A form of Koszul duality for operads of spectra plays a key role in this. In a landmark series of papers, [16], [17] and [18], Goodwillie outlines his ‘calculus of homotopy functors’. Let F: C → D (where C and D are each either Top ∗, the category of pointed topological spaces, or Spec, the category of spectra) be a pointed homotopy functor. One of the things that Goodwillie does is associate with F a sequence of spectra, which are called the derivatives of F.

In the early 1970’s was introduced a notion of homotopy-coherent diagram (Segal [64], Leitch [52], Vogt [78] and Mather [54]). This is a way of using cubical homotopies to deal at once with all of the higher homotopies of coherence involved in a diagram of spaces which “commutes up to homotopy”. This notion was subsequently studied by Vogt [78]

We discuss an analogue of Morita theory for ring spectra, a thickening of the category of rings inspired by stable homotopy theory. This follows work by Rickard and Keller on Morita theory for derived categories. We also discuss two results for derived equivalences of DGAs which show they differ from derived equivalences of rings.

Abstract. We review and extend the theory of Thom spectra and the associated obstruction theory for orientations. Specifically, we show that for an E ∞ ring spectrum A, the classical construction of gl1A, the spectrum of units, is the right adjoint of the functor To a map of spectra Σ ∞ + Ω ∞ : ho(connective spectra) → ho(E ∞ ring spectra). f: b → bgl1A, we associate an E ∞ A-algebra Thom spectrum Mf, which admits an E ∞ A-algebra map to R if and only if the composition b → bgl1A → bgl1R is null; the classical case developed by [MQRT77] arises when A is the sphere spectrum. We develop the analogous theory for A ∞ ring spectra. If A is an A ∞ ring spectrum, then to a map of spaces f: B → BGL1A we associate an A-module Thom spectrum Mf, which admits an R-orientation if and only if

We show that the free E∞-algebra on a zero-cell cannot be modeled by a commutative Γ-ring. The proof shows that Dyer-Lashof operations of positive degree must vanish on the zero’th homology of such an object. 1.

Abstract not found

has a symmetric monoidal smash product which allows the definition of ring spectra ‘up to homotopy’. In recent years there was an increasing interest in more refined notions of ring spectra which are associative (and possibly commutative)

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.