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Topological equivalences for differential graded algebras
 Adv. Math
, 2006
"... Abstract. We investigate the relationship between differential graded algebras (dgas) and topological ring spectra. Every dga C gives rise to an EilenbergMac Lane ring spectrum denoted HC. If HC and HD are weakly equivalent, then we say C and D are topologically equivalent. Quasiisomorphic dgas are ..."
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Cited by 14 (6 self)
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Abstract. We investigate the relationship between differential graded algebras (dgas) and topological ring spectra. Every dga C gives rise to an EilenbergMac Lane ring spectrum denoted HC. If HC and HD are weakly equivalent, then we say C and D are topologically equivalent. Quasiisomorphic dgas are topologically equivalent, but we produce explicit counterexamples of the converse. We also develop an associated notion of topological Morita equivalence using a homotopical version of tilting. Contents
Postnikov extensions for ring spectra
, 2006
"... We give a functorial construction of kinvariants for ring spectra, and use these to classify extensions in the Postnikov tower of a ring spectrum. ..."
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Cited by 9 (3 self)
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We give a functorial construction of kinvariants for ring spectra, and use these to classify extensions in the Postnikov tower of a ring spectrum.
Homotopy theory of modules over operads in symmetric spectra
, 2009
"... We establish model category structures on algebras and modules over operads in symmetric spectra, and study when a morphism of operads induces a Quillen equivalence between corresponding categories of algebras (resp. modules) over operads. ..."
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Cited by 4 (2 self)
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We establish model category structures on algebras and modules over operads in symmetric spectra, and study when a morphism of operads induces a Quillen equivalence between corresponding categories of algebras (resp. modules) over operads.
Homotopy theory of modules over operads and nonΣ operads in monoidal model categories
 J. Pure Appl. Algebra
"... There are many interesting situations in which algebraic structure can be described ..."
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Cited by 4 (2 self)
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There are many interesting situations in which algebraic structure can be described
MOONSHINE ELEMENTS IN ELLIPTIC COHOMOLOGY
, 712
"... Abstract. This is a historical talk about the recent confluence of two lines of research in equivariant elliptic cohomology, one concerned with connected Lie groups, the other with the finite case. These themes come together in (what seems to me remarkable) work of N. Ganter, relating replicability ..."
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Abstract. This is a historical talk about the recent confluence of two lines of research in equivariant elliptic cohomology, one concerned with connected Lie groups, the other with the finite case. These themes come together in (what seems to me remarkable) work of N. Ganter, relating replicability of McKayThompson series to the theory of exponential cohomology operations.
The plus construction, Bousfield localization, and derived completion
, 2009
"... We define a plusconstruction on connective augmented algebras over operads in symmetric spectra using Quillen homology. For associative and commutative algebras, we show that this plusconstruction is related to both Bousfield localization and Carlsson’s derived completion. 1 ..."
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We define a plusconstruction on connective augmented algebras over operads in symmetric spectra using Quillen homology. For associative and commutative algebras, we show that this plusconstruction is related to both Bousfield localization and Carlsson’s derived completion. 1
ON REALIZING DIAGRAMS OF ΠALGEBRAS
, 2006
"... Abstract. Given a diagram of Πalgebras (graded groups equipped with an action of the primary homotopy operations), we ask whether it can be realized as the homotopy groups of a diagram of spaces. The answer given here is in the form of an obstruction theory, of somewhat wider application, formulate ..."
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Abstract. Given a diagram of Πalgebras (graded groups equipped with an action of the primary homotopy operations), we ask whether it can be realized as the homotopy groups of a diagram of spaces. The answer given here is in the form of an obstruction theory, of somewhat wider application, formulated in terms of generalized Πalgebras. This extends a program begun in [DKS1, BDG] to study the realization of a single Πalgebra. In particular, we explicitly analyze the simple case of a single map, and provide a detailed example, illustrating the connections to higher homotopy operations. A recurring problem in algebraic topology is the rectification of homotopycommutative diagrams: given a diagram F: D → ho T ∗ (i.e., a functor from a small category to the homotopy category of topological spaces), we ask whether F lifts to ˆ F: D → T∗, and if so, in how many ways.
Documenta Math. 271 Duality for Topological Modular Forms
, 2011
"... Abstract. It has been observed that certain localizations of the spectrum of topologicalmodular forms are selfdual (MahowaldRezk, GrossHopkins). We provide an integral explanation of these results that is internal to the geometry of the (compactified) moduli stack of elliptic curves M, yet is onl ..."
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Abstract. It has been observed that certain localizations of the spectrum of topologicalmodular forms are selfdual (MahowaldRezk, GrossHopkins). We provide an integral explanation of these results that is internal to the geometry of the (compactified) moduli stack of elliptic curves M, yet is only true in the derived setting. When 2 is inverted, a choice of level 2 structure for an elliptic curve provides a geometrically wellbehaved cover of M, which allows one to consider Tmf as the homotopy fixed points of Tmf(2), topological modular forms with level 2 structure, under a natural action by GL2(Z/2). As a result of GrothendieckSerre duality, we obtain that Tmf(2) is selfdual. The vanishing of the associated Tate spectrum then makes Tmf itself Anderson selfdual.
4. Categories of kinvariants and EilenbergMacLane objects 9
"... Abstract. We give a functorial construction of kinvariants for ring spectra, and use these to classify extensions in the Postnikov tower of a ring spectrum. Contents ..."
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Abstract. We give a functorial construction of kinvariants for ring spectra, and use these to classify extensions in the Postnikov tower of a ring spectrum. Contents