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Universal Toda brackets of ring spectra
, 2006
"... Abstract. We construct and examine the universal Toda bracket of a highly structured ring spectrum R. This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of Rmodule spectra. It determines ..."
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Abstract. We construct and examine the universal Toda bracket of a highly structured ring spectrum R. This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of Rmodule spectra. It determines for example all triple Toda brackets of R and the first obstruction to realizing a module over the homotopy groups of R by an Rmodule spectrum. For periodic ring spectra, we study the corresponding theory of higher universal Toda brackets. The real and complex Ktheory spectra serve as our main examples. 1.
Higher homotopy operations and cohomology
"... Abstract. We explain how higher homotopy operations, defined topologically, may be identified under mild assumptions with (the last of) the DwyerKanSmith cohomological obstructions to rectifying homotopycommutative diagrams. ..."
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Abstract. We explain how higher homotopy operations, defined topologically, may be identified under mild assumptions with (the last of) the DwyerKanSmith cohomological obstructions to rectifying homotopycommutative diagrams.
Moduli spaces of homotopy theory
 Contemp. Math
"... Abstract. The moduli spaces refered to are topological spaces whose path components parametrize homotopy types. Such objects have been studied in two separate contexts: rational homotopy types, in the work of several authors in the late 1970’s; and general homotopy types, in the work of DwyerKan an ..."
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Abstract. The moduli spaces refered to are topological spaces whose path components parametrize homotopy types. Such objects have been studied in two separate contexts: rational homotopy types, in the work of several authors in the late 1970’s; and general homotopy types, in the work of DwyerKan and their collaborators. We here explain the two approaches, and show how they may be related to each other. 1.
REALIZING MODULES OVER THE HOMOLOGY OF A DGA
, 708
"... Abstract. Let A be a DGA over a field and X a module over H∗(A). Fix an A∞structure on H∗(A) making it quasiisomorphic to A. We construct an equivalence of categories between An+1module structures on X and length n Postnikov systems in the derived category of Amodules based on the bar resolution ..."
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Abstract. Let A be a DGA over a field and X a module over H∗(A). Fix an A∞structure on H∗(A) making it quasiisomorphic to A. We construct an equivalence of categories between An+1module structures on X and length n Postnikov systems in the derived category of Amodules based on the bar resolution of X. This implies that quasiisomorphism classes of Anstructures on X are in bijective correspondence with weak equivalence classes of rigidifications of the first n terms of the bar resolution of X to a complex of Amodules. The above equivalences of categories are compatible for different values of n. This implies that two obstruction theories for realizing X as the homology of an Amodule coincide. 1.