A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent ‘the same homotopy theory’. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a ‘ring spectrum with several objects’, i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard’s work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the Eilenberg-Mac Lane spectrum HR and (unbounded) chain complexes of R-modules for a ring R.

To a topological group G, we assign a naive G-spectrum DG, called the dualizing spectrum of G. When the classifying space BG is finitely dominated, we show that DG detects Poincaré duality in the sense that BG is a Poincaré duality space if and only if DG is a homotopy finite spectrum. Secondly, we show that the dualizing spectrum behaves multiplicatively on certain topological group extensions. In proving these results we introduce a new tool: a norm map which is defined for any G and for any naive G-spectrum E. Applications of the dualizing spectrum come in two flavors: (i) applications in the theory of Poincaré duality spaces, and (ii) applications in the theory of group cohomology. On the Poincaré duality space side, we derive a homotopy theoretic solution to a problem posed by Wall which says that in a fibration sequence of finitely dominated spaces, the total space satisfies Poincaré duality if and only if the base and fiber do. The dualizing spectrum can also be used to give an entirely homotopy theoretic construction of the Spivak fibration of a finitely dominated Poincaré duality space. We also include a new proof of Browder’s theorem that every finite H-space satisfies Poincaré duality. In connection with group cohomology, we show how to define a variant of Farrell-Tate cohomology for any topological or discrete group G, with coefficients in any naive equivariant cohomology theory E. When E is connective, and when G admits a subgroup H of finite index such that BH is finitely dominated, we show that this cohomology coincides with the ordinary cohomology of G with coefficients in E in degrees greater than the cohomological dimension of H. In an appendix, we identify the homotopy type of DG for certain kinds of groups. The class includes all compact Lie groups, torsion free arithmetic groups and Bieri-Eckmann duality groups.

These notes are based on lectures given at the Workshop on Structured ring spectra and

products; for example the zeroset of a smooth function on a manifold is not necessarily a manifold, and the non-transverse intersection of submanifolds is

The simplicial objects in an algebraic category admit an abstract homotopy theory via a Quillen model category structure. We show that the associated stable homotopy theory is completely determined by a ring spectrum functorially associated with the algebraic theory. For several familiar algebraic theories we can identify the parameterizing ring spectrum; for other theories we obtain new examples of ring spectra. For the theory of commutative algebras we obtain a ring spectrum which is related to AndreH}Quillen homology via certain spectral sequences. We show that the (co-)homology of an algebraic theory is isomorphic to the topological Hochschild (co-)homology of the parameterizing ring spectrum. # 2000 Elsevier Science Ltd. All rights reserved. MSC: 55U35; 18C10 Keywords: Algebraic theories; Ring spectra; AndreH}Quillen homology; #-spaces The original motivation for this paper came from the attempt to generalize a rational result about the homotopy theory of commutative rings. For...

The quest for an obstruction theory to E ∞ ring structures on a spectrum has led a number of authors to the investigation of homology in the category of E ∞ algebras. In this note we present three, apparently very different, constructions and show that when specialized to commutative rings they all agree. (AMS subject classification 55Nxx. Key words: André-Quillen homology, E∞homology).

Abstract. For a Poincaré duality space X d and a map X → B, consider the homotopy fiber product X × B X. If X is orientable with respect to a multiplicative cohomology theory E, then, after suitably regrading, it is shown that the E-homology of X × B X has the structure of a graded associative algebra. When X → B is the diagonal map of a manifold X, one recovers a result of Chas and Sullivan about the homology of the unbased loop space LX. 1.

Abstract. Let K(n) be the n th Morava K–theory at a prime p, and let T(n) be the telescope of a vn–self map of a finite complex of type n. In this paper we study the K(n)∗–homology of Ω ∞ X, the 0 th space of a spectrum X, and many related matters. We give a sampling of our results. Let PX be the free commutative S–algebra generated by X: it is weakly equivalent to the wedge of all the extended powers of X. We construct a natural map sn(X) : LT(n)P(X) → LT(n)Σ ∞ (Ω ∞ X)+ of commutative algebras over the localized sphere spectrum LT(n)S. The induced map of commutative, cocommutative K(n)∗–Hopf algebras sn(X) ∗ : K(n)∗(PX) → K(n)∗(Ω ∞ X), satistfies the following properties. It is always monic. It is an isomorphism if X is n–connected, πn+1(X) is torsion, and T(i)∗(X) = 0 for 1 ≤ i ≤ n−1. It is an isomorphism only if K(i)∗(X) = 0 for 1 ≤ i ≤ n − 1. It is universal: the domain of sn(X) ∗ preserves K(n)∗–isomorphisms, and if F is any functor preserving K(n)∗–isomorphisms, then any natural transformation F(X) → K(n)∗(Ω ∞ X) factors uniquely through sn(X)∗. The construction of our natural transformation uses the telescopic functors constructed and studied previously by Bousfield and the author, and thus depends heavily on the Nilpotence Theorem of Devanitz, Hopkins, and Smith. Our proof that sn(X) ∗ is always monic uses Topological André–Quillen Homology and Goodwillie Calculus in nonconnective settings.

A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent `the same homotopy theory'. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a `ring spectrum with several objects', i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard's work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the Eilenberg-Mac Lane spectrum HR and (unbounded) chain complexes of R-modules for a ring R. 1.

Abstract. This paper is based on talks I gave in Nagoya and Kinosaki in August of 2003. I survey, from my own perspective, Goodwillie’s work on towers associated to continuous functors between topological model categories, and then include a discussion of applications to periodic homotopy as in my work and the work of Arone–Mahowald. 1.