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27
SECONDARY ALGEBRAS ASSOCIATED TO RING SPECTRA
, 2006
"... Abstract. Homotopy groups of a connective ring spectrum R form angraded algebra π∗R which is commutative if R is commutative. We describe a secondary algebra π∗,∗R which enriches the structure of the algebra π∗R in a new unexpected way. The algebra π∗,∗R encodes secondary homotopy operations in π∗R ..."
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Abstract. Homotopy groups of a connective ring spectrum R form angraded algebra π∗R which is commutative if R is commutative. We describe a secondary algebra π∗,∗R which enriches the structure of the algebra π∗R in a new unexpected way. The algebra π∗,∗R encodes secondary homotopy operations in π∗R, such as Toda brackets, and the first Postnikov invariant of R as a ring spectrum. Moreover, π∗,∗R represents a cohomology class in the third Mac Lane cohomology of the algebra π∗R. If R is commutative then π∗,∗R has an E∞structure and encodes the cupone squares in π∗R. Contents
THE ALGEBRA OF SECONDARY HOMOTOPY OPERATIONS IN RING SPECTRA
, 2007
"... Abstract. The primary algebraic model of a ring spectrum R is the ring π∗R of homotopy groups. We introduce the secondary model π∗,∗R which has the structure of a secondary analogue of a ring. The homology of π∗,∗R is π∗R and triple Massey products in π∗,∗R coincide with Toda brackets in π∗R. We als ..."
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Abstract. The primary algebraic model of a ring spectrum R is the ring π∗R of homotopy groups. We introduce the secondary model π∗,∗R which has the structure of a secondary analogue of a ring. The homology of π∗,∗R is π∗R and triple Massey products in π∗,∗R coincide with Toda brackets in π∗R. We also describe the secondary model of a commutative ring spectrum Q from which we derive the cupone square operation in π∗Q. As an application we obtain for each ring spectrum R new derivations of the ring π∗R. Contents
A curious example of two model categories and some associated differential graded algebras, in preparation
"... Abstract. The paper gives a new proof that the model categories of stable modules for the rings Z/p 2 and Z/p[ɛ]/(ɛ 2) are not Quillen equivalent. The proof uses homotopy endomorphism ring spectra. Our considerations lead to an example of two differential graded algebras which are derived equivalent ..."
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Abstract. The paper gives a new proof that the model categories of stable modules for the rings Z/p 2 and Z/p[ɛ]/(ɛ 2) are not Quillen equivalent. The proof uses homotopy endomorphism ring spectra. Our considerations lead to an example of two differential graded algebras which are derived equivalent but whose associated model categories of modules are not Quillen equivalent. As a bonus, we also obtain derived equivalent dgas with nonisomorphic Ktheories. Contents
Teleman C., Openclosed field theories, string topology, and Hochschild homology
 in Alpine Perspectives on Algebraic Topology, Editors
"... and Hochschild homology ..."
THE PORDER OF TOPOLOGICAL TRIANGULATED CATEGORIES
"... p annihilates objects of the form Y/p. In this paper we show that the porder of a topological ..."
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p annihilates objects of the form Y/p. In this paper we show that the porder of a topological
The norder of algebraic triangulated categories
"... Abstract. We quantify certain features of algebraic triangulated categories using the ‘norder’, an invariant that measures how strongly n annihilates objects of the form Y/n. We show that the norder of an algebraic triangulated category is infinite, and that the porder of the plocal stable homoto ..."
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Abstract. We quantify certain features of algebraic triangulated categories using the ‘norder’, an invariant that measures how strongly n annihilates objects of the form Y/n. We show that the norder of an algebraic triangulated category is infinite, and that the porder of the plocal stable homotopy category is exactly p − 1, for any prime p. In particular, the plocal stable homotopy category is not algebraic.
A QUILLEN MODEL CATEGORY STRUCTURE ON SOME CATEGORIES OF COMONOIDS
, 807
"... Abstract. We prove that for certain monoidal (Quillen) model categories, the category of comonoids therein also admits a model structure. ..."
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Abstract. We prove that for certain monoidal (Quillen) model categories, the category of comonoids therein also admits a model structure.
THE FUNDAMENTAL GROUP OF A pCOMPACT GROUP
"... The notion of pcompact group [10] is a homotopy theoretic version of the geometric or analytic notion of compact Lie group, although the homotopy theory differs from the geometry is that there are parallel theories of pcompact groups, one for each prime number p. A key ..."
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The notion of pcompact group [10] is a homotopy theoretic version of the geometric or analytic notion of compact Lie group, although the homotopy theory differs from the geometry is that there are parallel theories of pcompact groups, one for each prime number p. A key
ATG Equivalences of monoidal model categories
, 2003
"... Abstract We construct Quillen equivalences between the model categories of monoids (rings), modules and algebras over two Quillen equivalent model categories under certain conditions. This is a continuation of our earlier work where we established model categories of monoids, modules and algebras [S ..."
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Abstract We construct Quillen equivalences between the model categories of monoids (rings), modules and algebras over two Quillen equivalent model categories under certain conditions. This is a continuation of our earlier work where we established model categories of monoids, modules and algebras [SS00]. As an application we extend the DoldKan equivalence to show that the model categories of simplicial rings, modules and algebras are Quillen equivalent to the associated model categories of connected differential graded rings, modules and algebras. We also show that our classification results from [SS03] concerning stable model categories translate to any one of the known symmetric monoidal model categories of spectra. AMS Classification 55U35; 18D10, 55P43, 55P62
TORSION INVARIANTS FOR TRIANGULATED CATEGORIES
"... The most commonly known triangulated categories arise from chain complexes in an abelian category by passing to chain homotopy classes or inverting quasiisomorphisms. Such examples are called ‘algebraic ’ because they have underlying abelian (or at least additive) categories. Stable homotopy theory ..."
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The most commonly known triangulated categories arise from chain complexes in an abelian category by passing to chain homotopy classes or inverting quasiisomorphisms. Such examples are called ‘algebraic ’ because they have underlying abelian (or at least additive) categories. Stable homotopy theory produces examples of triangulated categories by quite different means, and in this context the underlying categories are usually very ‘nonadditive ’ before passing to homotopy classes of morphisms. We call such triangulated categories topological, compare Definition 3.1; this class includes the algebraic triangulated categories. The purpose of this paper is to explain some systematic differences between these two kinds of triangulated categories. There are certain properties – defined entirely in terms of the triangulated structure – which hold in all algebraic examples, but which can fail in general. These differences are all torsion phenomena, and rationally every topological triangulated category is algebraic (at least under mild size restrictions). Our main tool is a new numerical invariant, the norder of an object in a triangulated category, for n a natural number (see Definition 1.1). The norder is a nonnegative integer (or infinity), and an object Y has positive norder if and only if n · Y = 0; the norder can be thought of