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HZalgebra spectra are differential graded algebras
 Amer. Jour. Math
, 2004
"... Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZalgebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZalgebra spectra. We also construct Qu ..."
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Cited by 67 (17 self)
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Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZalgebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZalgebra spectra. We also construct Quillen equivalences between the differential graded modules and module spectra over these algebras. We use these equivalences in turn to produce algebraic models for rational stable model categories. We show that bascially any rational stable model category is Quillen equivalent to modules over a differential graded Qalgebra (with many objects). 1.
Complete modules and torsion modules
 Amer. J. Math
"... Abstract. Suppose that R is a ring and that A is a chain complex over R. Inside the derived category of differential graded Rmodules there are naturally defined subcategories of Atorsion objects and of Acomplete objects. Under a finiteness condition on A, we develop a Morita theory for these subc ..."
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Cited by 57 (6 self)
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Abstract. Suppose that R is a ring and that A is a chain complex over R. Inside the derived category of differential graded Rmodules there are naturally defined subcategories of Atorsion objects and of Acomplete objects. Under a finiteness condition on A, we develop a Morita theory for these subcategories, find conceptual interpretations for some associated algebraic functors, and, in appropriate commutative situations, identify the associated functors as local homology or local cohomology. Some of the results are suprising even in the case R = Z and A = Z/p. 1.
Separable extensions in ttgeometry and generalized Quillen stratification
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An Algebraic Model For Rational ...Equivariant Stable Homotopy Theory
 of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK. Email address: j.greenlees@sheffield.ac.uk
"... Greenlees defined an abelian category A whose derived category is equivalent to the rational S 1 equivariant stable homotopy category whose objects represent rational S 1  equivariant cohomology theories. We show that in fact the model category of di#erential graded objects in A models the ..."
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Cited by 3 (3 self)
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Greenlees defined an abelian category A whose derived category is equivalent to the rational S 1 equivariant stable homotopy category whose objects represent rational S 1  equivariant cohomology theories. We show that in fact the model category of di#erential graded objects in A models the whole rational S 1 equivariant stable homotopy theory. That is, we show that there is a Quillen equivalence between dgA and the model category of rational S 1 equivariant spectra, before the quasiisomorphisms or stable equivalences have been inverted. This implies that all of the higher order structures such as mapping spaces, function spectra and homotopy (co)limits are reflected in the algebraic model. The construction of this equivalence involves calculations with Massey products. In an appendix we show that Toda brackets, and hence also Massey products, are determined by the derived category. 1.
The ArfKervaire invariant problem in algebraic topology: introduction
 CURRENT DEVELOPMENTS IN MATHEMATICS, 2009, INT
, 2010
"... This paper gives the history and background of one of the oldest problems in algebraic topology, along with a short summary of our solution to it and a description of some of the tools we use. More details of the proof are provided in our second paper ..."
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This paper gives the history and background of one of the oldest problems in algebraic topology, along with a short summary of our solution to it and a description of some of the tools we use. More details of the proof are provided in our second paper
The Burnside ring and equivariant stable cohomotopy for infinite groups
 Pure Appl. Math. Q
"... After we have given a survey on the Burnside ring of a finite group, we discuss and analyze various extensions of this notion to infinite (discrete) groups. The first three are the finiteGsetversion, the inverselimitversion and the covariant Burnside group. The most sophisticated one is the four ..."
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After we have given a survey on the Burnside ring of a finite group, we discuss and analyze various extensions of this notion to infinite (discrete) groups. The first three are the finiteGsetversion, the inverselimitversion and the covariant Burnside group. The most sophisticated one is the fourth definition as the zeroth equivariant stable cohomotopy of the classifying space for proper actions. In order to make sense of this definition we define equivariant stable cohomotopy groups of finite proper equivariant CWcomplexes in terms of maps between the sphere bundles associated to equivariant vector bundles. We show that this yields an equivariant cohomology theory with a multiplicative structure. We formulate a version of the Segal Conjecture for infinite groups. All this is analogous and related to the question what are the possible extensions of the notion of the representation ring of a finite group to an infinite group. Here possible candidates are projective class groups, Swan groups and the equivariant topological Ktheory of the classifying space for proper actions. Key words: Burnside ring, equivariant stable cohomotopy, infinite groups.
The Burnside Ring and Equivariant Cohomotopy for Infinite Groups
, 2008
"... After we have given a survey on the Burnside ring of a finite group, we discuss and analyze various extensions of this notion to infinite (discrete) groups. The first three are the finiteGsetversion, the inverselimitversion and the covariant Burnside group. The most sophisticated one is the four ..."
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After we have given a survey on the Burnside ring of a finite group, we discuss and analyze various extensions of this notion to infinite (discrete) groups. The first three are the finiteGsetversion, the inverselimitversion and the covariant Burnside group. The most sophisticated one is the fourth definition as the equivariant zeroth cohomotopy of the classifying space for proper actions. In order to make sense of this definition we define equivariant cohomotopy groups of finite proper equivariant CWcomplexes in terms of maps between the sphere bundles associated to equivariant vector bundles. We show that this yields an equivariant cohomology theory with a multiplicative structure. We formulate a version of the Segal Conjecture for infinite groups. All this is analogous and related to the question what are the possible extensions of the notion of the representation ring of a finite group to an infinite group. Here possible candidates are projective class groups, Swan groups and the equivariant topological Ktheory of the classifying space for proper actions. Key words: Burnside ring, equivariant cohomotopy, infinite groups.
4 HEIGHT h DETECTION AND CONNECTIVE REAL KTHEORY OF ELEMENTARY ABELIAN 2GROUPS
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