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14
Stable model categories are categories of modules
 TOPOLOGY
, 2003
"... 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 ..."
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Cited by 78 (16 self)
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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 EilenbergMac Lane spectrum HR and (unbounded) chain complexes of Rmodules for a ring R.
The structure of Mackey functors
 Trans. Amer. Math. Soc
, 1995
"... JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JS ..."
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Cited by 21 (6 self)
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JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to
On The Equivariant Homotopy Of Free Abelian Groups On GSpaces And GSpectra
, 1997
"... this paper is to provide unstable and stable equivariant versions of this result. We obtain results both in the unstable and in the stable equivariant worlds, for a fixed finite group G. In the unstable context, we start with the simple observation that for a Gspace X, AG (X) gives rise to a Mackey ..."
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Cited by 9 (5 self)
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this paper is to provide unstable and stable equivariant versions of this result. We obtain results both in the unstable and in the stable equivariant worlds, for a fixed finite group G. In the unstable context, we start with the simple observation that for a Gspace X, AG (X) gives rise to a Mackey functor FX with values in the category of topological abelian groups. It follows that the composition of any homotopy functor with
On the nonexistence of elements of Kervaire invariant one, arXiv 0908.3724v2
"... Abstract. We show that the Kervaire invariant one elements θj ∈ π 2 j+2 −2 S 0 exist only for j ≤ 6. By Browder’s Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and possibly 126. Except for dimension 126 this resolves a longstand ..."
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Cited by 8 (3 self)
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Abstract. We show that the Kervaire invariant one elements θj ∈ π 2 j+2 −2 S 0 exist only for j ≤ 6. By Browder’s Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and possibly 126. Except for dimension 126 this resolves a longstanding problem in algebraic topology. Contents
Classification of Stable Model Categories
"... 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 ..."
Abstract

Cited by 6 (5 self)
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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 EilenbergMac Lane spectrum HR and (unbounded) chain complexes of Rmodules for a ring R. 1.
Some examples of free actions on products of spheres
 TOPOLOGY
, 2006
"... If G1 and G2 are finite groups with periodic Tate cohomology, then G1 × G2 acts freely and smoothly on some product S n × S n. ..."
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Cited by 6 (5 self)
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If G1 and G2 are finite groups with periodic Tate cohomology, then G1 × G2 acts freely and smoothly on some product S n × S n.
Characteristic Classes of Proalgebraic Varieties and Motivic Measures
, 2006
"... Michael Gromov has recently initiated what he calls “symbolic algebraic geometry”, in which objects are proalgebraic varieties: a proalgebraic variety is by definition the projective limit of a projective system of algebraic varieties. In this paper we introduce characteristic classes of proalgebra ..."
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Cited by 5 (2 self)
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Michael Gromov has recently initiated what he calls “symbolic algebraic geometry”, in which objects are proalgebraic varieties: a proalgebraic variety is by definition the projective limit of a projective system of algebraic varieties. In this paper we introduce characteristic classes of proalgebraic varieties, using Grothendieck transformations of Fulton–MacPherson’s Bivariant Theory, modeled on the construction of MacPherson’s Chern class transformation of proalgebraic varieties. We show that a proalgebraic version of the Euler–Poincaré characteristic with values in the Grothendieck ring is a generalization of the socalled motivic measure.
Simple Mackey Functors
"... In connection with recent developments in group representation theory which make use of the theory of Mackey functors [TW], the natural question of the classification of simple Mackey functors arose. The purpose of the present paper is to give a complete answer to this question. For applications, t ..."
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Cited by 3 (1 self)
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In connection with recent developments in group representation theory which make use of the theory of Mackey functors [TW], the natural question of the classification of simple Mackey functors arose. The purpose of the present paper is to give a complete answer to this question. For applications, the reader can refer to our paper [TW] where our main results are actually used in an essential way. After recalling the definitions and elementary facts about Mackey functors, we prove in Section 2 that every simple Mackey functor S for a finite group G has (up to conjugation) a unique minimal subgroup H with S(H) � = 0, and that the (NG(H)/H)module V = S(H) is simple. Similar methods easily give a criterion for the simplicity of a Mackey functor (Section 3). We attach to S the pair (H, V) just described and it turns out that this provides a parameterisation of the simple Mackey functors. In order to prove this, we describe explicitly a simple Mackey functor corresponding to an arbitrary pair (H, V). This requires a number of constructions which are introduced in Sections 4, 5 and 6, namely restriction, induction, inflation of Mackey functors and fixed point functors. All of these constructions are very natural and have useful adjointness properties. Sections 7 and 8 are devoted to the classification of simple Mackey functors. In the
INDUCTION AND COMPUTATION OF BASS NIL GROUPS FOR FINITE GROUPS
, 2008
"... Let G be a finite group. We show that the Bass Nilgroups NKn(RG), n ∈ Z, are generated from the psubgroups of G by induction, certain twistings maps depending on elements in the centralizers of the psubgroups, and the Verschiebung homomorphisms. As a consequence, the groups NKn(RG) are generate ..."
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Cited by 1 (1 self)
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Let G be a finite group. We show that the Bass Nilgroups NKn(RG), n ∈ Z, are generated from the psubgroups of G by induction, certain twistings maps depending on elements in the centralizers of the psubgroups, and the Verschiebung homomorphisms. As a consequence, the groups NKn(RG) are generated by induction from elementary subgroups. For NK0(ZG) we get an improved estimate of the torsion exponent.
Index of Notation
"... Abstract. Mackey functors are a framework having the common properties of many natural constructions for finite groups, such as group cohomology, representation rings, the Burnside ring, the topological Ktheory of classifying spaces, the algebraic Ktheory of group rings, the Witt rings of Galois e ..."
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Abstract. Mackey functors are a framework having the common properties of many natural constructions for finite groups, such as group cohomology, representation rings, the Burnside ring, the topological Ktheory of classifying spaces, the algebraic Ktheory of group rings, the Witt rings of Galois extensions, etc. In this work we first show that the Mackey functors for a group may be identified with the modules for a certain algebra, called the Mackey algebra. The study of Mackey functors is thus the same thing as the study of the representation theory of this algebra. We develop the properties of Mackey functors in the spirit of representation theory, and it emerges that there are great similarities with the representation theory of finite groups. In previous work we had classified the simple Mackey functors and demonstrated semisimplicity in characteristic zero. Here we consider the projective Mackey functors (in arbitrary characteristic), describing many of their features. We show, for example, that the Cartan matrix of the Mackey algebra may be computed from a decomposition matrix in the same way as for group representations. We determine the vertices, sources and Green correspondents of the projective and simple Mackey functors, as well as providing a way to compute the Ext groups for the simple Mackey functors. We parametrize the blocks of Mackey functors and determine the groups for which the Mackey algebra has finite representation type. It turns out that these Mackey algebras are direct sums of simple algebras and Brauer tree algebras. Throughout this theory there is a close connection between the properties of the Mackey functors, and the representations of the group on which they are defined, and of its subgroups. The relationships between these representations are exactly the information encoded by Mackey functors. This observation suggests the use of Mackey functors in a new way, as tools in group representation theory.