Abstract not found

I will give a broad survey of the general area of structured ring spectra in modern constructions of the stable homotopy category. I will give some background and history, but my main focus will be a description of work in progress of Mike Mandell, Stefan Schwede, Brooke Shipley, and myself [24, 25]. By “modern ” I mean that there must be a smash product that gives a point-set level symmetric monoidal

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.

Algebraic topology is a young subject, and its foundations are not yet firmly in place. I shall give some history, examples, and modern developments in that part of the subject called stable algebraic topology, or stable homotopy theory. This is by far the most calculationally accessible part of algebraic topology, although it is also the least intuitively grounded in visualizable geometric objects. It has a great many applications to such other subjects as algebraic geometry and geometric topology. Time will not allow me to say as much as I would like about that. Rather I will emphasize some foundational issues that have been central to this part of algebraic topology since the early 1960’s, but that have only been satisfactorily resolved in the last few years. It was only in 1952, with Eilenberg and Steenrod’s book “Foundations of algebraic topology ” [9], that the nature of ordinary homology and cohomology theories was reasonably well understood. Even then, the modern way of thinking about cohomology as represented by Eilenberg-Mac Lane spaces was nowhere mentioned. It may have been known by then, but it certainly was not known to be important.

In this series of lectures we give an exposition of the seminal work of Devinatz, Hopkins, and Smith which is surrounding the classification of the thick subcategories of finite spectra in stable homotopy theory. The lectures are expository and are aimed primarily at non-homotopy theorists. We begin with an introduction to the stable homotopy category of spectra, and then talk about the celebrated thick subcategory theorem and discuss a few applications to the structure of the Bousfield lattice. Most of the results that we discuss below were conjectured by Ravenel [Rav84] and were proved by Devinatz, Hopkins, and Smith [DHS88, HS98]. 1. The stable homotopy category of spectra Recall that in homotopy theory one is interested in studying the homotopy classes of maps between CW complexes (spaces that are built in a systematic way by attaching cells): If f and g are maps (continuous) between CW complexes X and Y, we say that they are homotopic if there is a map from the cylinder X × [0, 1] to Y whose restriction to the two ends (top and bottom) of the cylinder gives f and g respectively. The homotopy classes of maps between X and Y is denoted

WE STUDY bordism of G-manifolds from a new point of view. Our aim is to combine the geometric approach of Conner and Floyd (see [9], [lo], [ll], [13]) and the K-theory approach which is contained in papers by Atiyah, Bott, Segal and Singer ( [ 11, [2], [3]). For simplicity of exposition we restrict to unitary cobordism. We develop cobordism analogue of K-theory integrality theorems and show their relation to the results of Conner and Floyd. We get a systematic and conceptual understanding of various results about (unitary) G-manifolds. We now describe our techniques and results. Tn Section 1 we define equivariant cobordism U,*(X) along the lines of G. W. Whitehead [23], using all representations of the compact Lie group G for suspending. We construct a natural transformation a: U,*(X)-+ U*(EG x,X) of multiplicative equivariant cohomology theories which preserves Thorn classes. Special cases of c ( have been studied by Boardman [6] and Conner [9]. In particular we answer a question of Boardman ([6], p. 138). The computation of a is interesting and very difficult in general. We have only partial results for cyclic groups. It is here that the methods of Atiyah-Segal [2] come into play: the fixed point homomorphism (Section 2) and localization (Section 3). We consider the set S c lJG * of Euler classes of representations (considered as bundles over a point) without trivial direct summand. The first main theorem is the computation of the localization S- ’ U, * in terms of ordinary cobordism of suitable spaces. The Pontrjagin-Thorn construction gives a homomorphism from geometric bordism aeG of unitary G-manifolds to homotopical bordism. The map i is by no means an isomorphism (due to the lack of usual transversality theorems). The elements x E S, x # 1, do not lie in the image of i. One might conjecture that UsG is generated as an algebra by S and the image of i. We prove this for cyclic groups Z, of prime order p (Section