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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 83 (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
"... 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, ..."
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Cited by 30 (6 self)
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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.
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 10 (6 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 9 (4 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 ..."
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Cited by 9 (6 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.
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 7 (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
When projective does not imply flat, and other homological anomalies
 Theory Appl. Categ
, 1999
"... Abstract. The category MG of Mackey functors for a group G carries a symmetric monoidal closed structure. The product providing this structure encodes the Frobenius axiom, which describes the interaction of induction and multiplication in Mackey functor rings. Mackey functors are of interest in equ ..."
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Cited by 6 (1 self)
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Abstract. The category MG of Mackey functors for a group G carries a symmetric monoidal closed structure. The product providing this structure encodes the Frobenius axiom, which describes the interaction of induction and multiplication in Mackey functor rings. Mackey functors are of interest in equivariant homotopy theory since good equivariant cohomology theories are Mackey functor valued. In this context, the product is useful not only because it encodes the interaction between induction and the cup product, but also because of the role it plays in the not yet fully understood universal coefcient and Künneth formulae. This role makes it important to know whether projective objects in MG are flat, and whether the product of projective objects inMG is projective. In the most familiar symmetric monoidal abelian categories, the tensor product obviously interacts appropriately with projective objects. However, the product forMG need not be so well behaved. For example, if G is O(n), projectives need not be flat in MG and the product of projective objects need not be projective. This misbehavior complicates the search for full strength equivariant universal coecient and Künneth formulae. These questions about the interaction of the tensor product with projective objects can be regarded as compatibility conditions which may be satised by a symmetric monoidal closed category M. The primary purpose of this article is to investigate these, and related, conditions. Our focus is on functor categories whose monoidal structures arise in a fashion described by Day. Conditions are given under which such a structure interacts appropriately with projective objects. Further, examples are given to show that, when these conditions aren’t met, this interaction can be quite bad. These examples were not fabricated to illustrate the abstract possibility of misbehavior. Rather, they are drawn from the literature. In particular, MG is badly behaved not only for the
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.
DRESS INDUCTION AND THE BURNSIDE QUOTIENT Green Ring
, 2009
"... We define and study the Burnside quotient Green ring of a Mackey functor, introduced in our MSRI preprint [18]. Some refinements of Dress induction theory are presented, together with applications to computation results for Ktheory and Ltheory of finite and infinite groups. ..."
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Cited by 1 (1 self)
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We define and study the Burnside quotient Green ring of a Mackey functor, introduced in our MSRI preprint [18]. Some refinements of Dress induction theory are presented, together with applications to computation results for Ktheory and Ltheory of finite and infinite groups.