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The biderivative and A∞bialgebras
 Homology Homotopy Appl
"... Abstract. An A∞bialgebra is a DGM H equipped with structurally compatible operations { ω j,i: H ⊗i → H ⊗j} such that ( H, ω 1,i) is an A∞algebra and ( H, ω j,1) is an A∞coalgebra. Structural compatibility is controlled by the biderivative operator Bd, defined in terms of two kinds of cup products ..."
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Cited by 4 (1 self)
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Abstract. An A∞bialgebra is a DGM H equipped with structurally compatible operations { ω j,i: H ⊗i → H ⊗j} such that ( H, ω 1,i) is an A∞algebra and ( H, ω j,1) is an A∞coalgebra. Structural compatibility is controlled by the biderivative operator Bd, defined in terms of two kinds of cup products on certain cochain algebras of pemutahedra over the universal PROP U = End(TH). To Jim Stasheff on the occasion of his 68th birthday. 1.
Twisting Elements in Homotopy Galgebra
"... We study the notion of twisting elements da = a ⌣1 a with respect to ⌣1 product when it is a part of homotopy Gerstenhaber algebra structure. This allows to bring to one context the two classical concepts, the theory of deformation of algebras of M. Gerstenhaber, and A(∞)algebras of J. Stasheff. 1 ..."
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Cited by 2 (2 self)
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We study the notion of twisting elements da = a ⌣1 a with respect to ⌣1 product when it is a part of homotopy Gerstenhaber algebra structure. This allows to bring to one context the two classical concepts, the theory of deformation of algebras of M. Gerstenhaber, and A(∞)algebras of J. Stasheff. 1
Cochain operations defining Steenrod ⌣iproducts in the bar construction
 Georgian Math. J
"... the bar construction ..."
THE LOOP COHOMOLOGY OF A SPACE WITH THE POLYNOMIAL COHOMOLOGY ALGEBRA
, 810
"... Abstract. We prove that for a simply connected space X with the polynomial cohomology algebra H ∗ (X; k) = S(U), the loop space cohomology H ∗ (ΩX; k) = Λ(s −1 U) is the exterior algebra if k is a commutative unital ring without 2torsion, or if and only if k = Z2 and the Steenrod operation Sq1 is ..."
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Abstract. We prove that for a simply connected space X with the polynomial cohomology algebra H ∗ (X; k) = S(U), the loop space cohomology H ∗ (ΩX; k) = Λ(s −1 U) is the exterior algebra if k is a commutative unital ring without 2torsion, or if and only if k = Z2 and the Steenrod operation Sq1 is multiplicatively decomposable on H ∗ (X; Z2). The last statement in fact contains a converse of a theorem of A. Borel. 1.
THE BITWISTED CARTESIAN MODEL FOR THE FREE LOOP FIBRATION
, 707
"... To Jim Stasheff on the occasion of his 70th birthday To Murray Gerstenhaber on the occasion of his 80th birthday Abstract. Using the notion of truncating twisting function from a simplicial set to a cubical set a special, bitwisted, Cartesian product of these sets is defined. For the universal trunc ..."
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To Jim Stasheff on the occasion of his 70th birthday To Murray Gerstenhaber on the occasion of his 80th birthday Abstract. Using the notion of truncating twisting function from a simplicial set to a cubical set a special, bitwisted, Cartesian product of these sets is defined. For the universal truncating twisting function, the (co)chain complex of the corresponding bitwisted Cartesian product agrees with the standard Cartier (Hochschild) chain complex of the simplicial (co)chains. The modelling polytopes Fn are constructed. An explicit diagonal on Fn is defined and a multiplicative model for the free loop fibration is obtained. 1.
FILTERED HIRSCH ALGEBRAS
, 707
"... Abstract. Motivated by the cohomology theory of loop spaces we consider a concept of a special class of higher order homotopy commutative differential graded algebras. For such algebras the socalled filtered Hirsch model is constructed. As an application we converse a theorem of Borel by proving th ..."
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Abstract. Motivated by the cohomology theory of loop spaces we consider a concept of a special class of higher order homotopy commutative differential graded algebras. For such algebras the socalled filtered Hirsch model is constructed. As an application we converse a theorem of Borel by proving that for a simply connected space X with the mod 2 polynomial cohomology algebra its loop space cohomology is exterior if and only if the Steenrod operation Sq1 is multiplicatively decomposable on H ∗ (X; Z2). 1.
ON THE BETTI NUMBERS OF A LOOP SPACE
, 905
"... Abstract. We show that if A is a special homotopy Galgebra over a commutative unital ring k with both H(A) and TorA i (k, k), i ≥ 0, finitely generated kmodules, then the integers, τi(A), the cardinality of the minimal generator set of the kmodule TorA i (k, k), grow unbounded if and only if ˜ H( ..."
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Abstract. We show that if A is a special homotopy Galgebra over a commutative unital ring k with both H(A) and TorA i (k, k), i ≥ 0, finitely generated kmodules, then the integers, τi(A), the cardinality of the minimal generator set of the kmodule TorA i (k, k), grow unbounded if and only if ˜ H(A) requires at least two algebra generators. The case A = C ∗ (X; k), the simplicial cochain complex of a simply connected finite CWcomplex X, implies a similar statement for the ”Betti numbers ” of the loop space ΩX; this, in particular, unifies the existing proofs when k is a field of zero or positive characteristic. 1.