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35
A combinatorial method for computing Steenrod squares
, 1999
"... We present here a combinatorial method for computing Steenrod squares of a simplicial set X . This method is essentially based on the determination of explicit formulae for the component morphisms of a higher diagonal approximation (i.e., a family of morphisms measuring the lack of commutativity of ..."
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We present here a combinatorial method for computing Steenrod squares of a simplicial set X . This method is essentially based on the determination of explicit formulae for the component morphisms of a higher diagonal approximation (i.e., a family of morphisms measuring the lack of commutativity of the cup product on the cochain level) in terms of face operators of X.A generalization of this method to Steenrod reduced powers is sketched. c 1999 Elsevier Science B.V. All rights reserved.
Corings over operads characterize morphisms, math.AT/0505559
"... objects in M, with its composition monoidal structure. Let R be a Pcoring, ..."
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Cited by 7 (1 self)
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objects in M, with its composition monoidal structure. Let R be a Pcoring,
VIRTUAL OPERAD ALGEBRAS AND REALIZATION OF HOMOTOPY TYPES
, 1999
"... 1.1. Let k be a base commutative ring, C(k) be the category of complexes of kmodules. The category of operads Op(k) in C(k) admits a closed model category (CMC) structure with quasiisomorphisms as weak equivalences and surjective maps as fibrations (see [H], Sect. 6 and also Section 2 below). ..."
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Cited by 5 (2 self)
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1.1. Let k be a base commutative ring, C(k) be the category of complexes of kmodules. The category of operads Op(k) in C(k) admits a closed model category (CMC) structure with quasiisomorphisms as weak equivalences and surjective maps as fibrations (see [H], Sect. 6 and also Section 2 below).
Quillen stratification for the Steenrod algebra
, 1999
"... this paper we describe the cohomology of the Steenrod algebra, modulo nilpotent elements, according to the recipe in Quillen's result: as the inverse limit of the cohomology rings of the "elementary abelian" subHopf algebras of A. We hope that this leads to further study in several directions. Firs ..."
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Cited by 5 (1 self)
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this paper we describe the cohomology of the Steenrod algebra, modulo nilpotent elements, according to the recipe in Quillen's result: as the inverse limit of the cohomology rings of the "elementary abelian" subHopf algebras of A. We hope that this leads to further study in several directions. First of all, according to the point of view of axiomatic stable homotopy theory in [7], the representation theory of any cocommutative Hopf algebra has formal similarities to stable homotopy theory; viewed this way, our main result is an # Research partially supported by National Science Foundation grant DMS9407459. 422 JOHN H. PALMIERI analogue of Nishida's theorem, and one can hope that it will lead to results like the thick subcategory theorem of Hopkins and Smith. For instance, see Conjecture 1.4 for a suggested classification of thick subcategories of finite
The motivic Adams spectral sequence
"... We present some data on the cohomology of the motivic Steenrod algebra over an algebraically closed field of characteristic 0. Our results are based on computer calculations and a motivic version of the May spectral sequence. We discuss features of the associated Adams spectral sequence, and use t ..."
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Cited by 5 (1 self)
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We present some data on the cohomology of the motivic Steenrod algebra over an algebraically closed field of characteristic 0. Our results are based on computer calculations and a motivic version of the May spectral sequence. We discuss features of the associated Adams spectral sequence, and use these tools to give new proofs of some results in classical algebraic topology. We also consider a motivic AdamsNovikov spectral sequence. The investigations reveal the existence of some stable motivic homotopy classes that have no classical analogue.
On the existence of the self map v 9 2 on the SmithToda complex V (1) at the prime 3, Contemp
 Math
"... Abstract. Let V (1) be the SmithToda complex at the prime 3. We prove that there exists a map v 9 2: Σ144 V (1) → V (1) that is a K(2) equivalence. This map is used to construct various v2periodic infinite families in the 3primary stable homotopy groups of spheres. Contents ..."
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Cited by 4 (3 self)
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Abstract. Let V (1) be the SmithToda complex at the prime 3. We prove that there exists a map v 9 2: Σ144 V (1) → V (1) that is a K(2) equivalence. This map is used to construct various v2periodic infinite families in the 3primary stable homotopy groups of spheres. Contents
The Motivic DGA
, 2001
"... Abstract. The main goal of this paper is to associate to each smooth quasiprojective scheme X over any field an E ∞ differential graded algebra whose cohomology groups are the higher Chow groups of X. Modulo torsion, this provides a strictly commutative differential graded algebra. Such a construct ..."
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Abstract. The main goal of this paper is to associate to each smooth quasiprojective scheme X over any field an E ∞ differential graded algebra whose cohomology groups are the higher Chow groups of X. Modulo torsion, this provides a strictly commutative differential graded algebra. Such a construction is currently known only for the case X itself is a field. Needless to say there are several applications of this result, some of which are considered here: for example, we are able to construct a category of relative mixed Tatemotives associated to any such scheme under the hypothesis that the DGA we obtain is connected. We also show that this holds for all smooth linear projective varieties over a field k, if the BeilinsonSoulé vanishing condition holds for the field. We also establish certain cohomology operations on modp motivic cohomology by this procedure. 1.