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Our objective is to show a possibly interesting structure of homotopic nature appearing in persistent (co)homology. Assuming that the filtration of a simplicial set embedded in Rn induces a multiplicative filtration on the dg algebra of simplicial cochains, we use a result by Kadeishvili to get a unique A∞-algebra structure on the complete persistent cohomology of the filtered simplicial set. We then construct of a (pseudo)metric on the set of all barcodes of all cohomological degrees enriched with the A∞-algebra structure stated before, refining the usual bottleneck metric, and which is also independent of the particular A∞-algebra structure chosen. We also compute this distance for some basic examples. As an aside, we give a simple proof of a result relating the barcode structure between persistent homology and cohomology, that was observed by de Silva, Morozov, and Vejdemo-Johansson under some restricted assumptions which we do not suppose. [ABSTRACT FROM AUTHOR]

The classical Borsuk Ulam theorem can be stated as: there exists no equivariant map S n → S n − 1 , relative to the antipodal actions on the spheres. Let G = Z 2 act freely on a finitistic space X with mod 2 cohomology ring isomorphic to that of the product of a projective space (real, complex or quaternionic) and the 3-sphere. In this paper, we show that the Volovikov's index of R P m × S 3 is any one of the integers 2, 4, m + 3 or m + 4 . In case of C P m × S 3 , this index is 3, 4 or 2 m + 4 and that of H P m × S 3 is 4, 5, 8 or 9. We apply this to determine the possibilities of nonexistence of equivariant maps X → S n or S n → X . [ABSTRACT FROM AUTHOR]

In this paper we transport Steenrod's cup-iproducts,i ≥ 0, from the singular cochains on the free loop space Maps(S1,BG) to Hochschild's original cochain complex Homk(k[G]⊗*,k[G]) defining Hochschild cohomology. HereGis a discrete group,kan arbitrary (commutative) coefficient ring, andBGthe classifying space ofG. This induces a natural action of the (mod 2) Steenrod algebra on the Hochschild cohomology of a group ring. For cochains supported onBG, we prove that Gerstenhaber's cup product agrees with the simplicial cup product and Gerstenhaber's pre-Lie product agrees with Steenrod's cup-one product. As a consequence, for cocyclesfandgsupported onBG, the Gerstenhaber bracket [f,g] = 0 inHH*(k[G]; k[G]). This is interpreted in terms of the Batalin–Vilkovisky structure onHH*(k[G]; k[G]). [ABSTRACT FROM PUBLISHER]

Let M be an exact symplectic manifold with contact type boundary such that c( M) = 0. Motivated by noncommutative symplectic geometry and string topology, we show that the cyclic cohomology of the Fukaya category of M has an involutive Lie bialgebra structure. [ABSTRACT FROM AUTHOR]

We consider quiver algebrasAqover a fieldkdefined by two cycles and a quantum-like relation depending on a nonzero elementqink. We determine the Hochschild cohomology ring ofAqand give necessary and sufficient conditions forAqto have the finitely generated Hochschild cohomology ring. [ABSTRACT FROM PUBLISHER]

For a class of co-chain complexes in the category of pre-Hilbert $$A$$ -modules, we prove that their cohomology groups equipped with the canonical quotient topology are pre-Hilbert $$A$$ -modules, and derive the Hodge theory and, in particular, the Hodge decomposition for them. As an application, we show that $$A$$ -elliptic complexes of pseudodifferential operators acting on sections of finitely generated projective $$A$$ -Hilbert bundles over compact manifolds belong to this class if the images of the continuous extensions of their associated Laplace operators are closed. Moreover, we prove that the cohomology groups of these complexes share the structure of the fibers, in the sense that they are also finitely generated projective Hilbert $$A$$ -modules. [ABSTRACT FROM AUTHOR]

The aim of this paper is to characterize the first graded Hochschild cohomology of a hereditary algebra whose Gabriel quiver is admitted to have oriented cycles. The interesting conclusion we have obtained shows that the standard basis of the first graded Hochschild cohomology depends on the genus of a quiver as a topological object. In this paper, we overcome the limitation of the classical Hochschild cohomology for hereditary algebra where the Gabriel quiver is assumed to be acyclic. As preparation, we first investigate the graded differential operators on a path algebra and the associated graded Lie algebra. [ABSTRACT FROM AUTHOR]

We apply the transfer algorithm to transfer an A-algebra structure that cannot be computed using the classical basic perturbation lemma. We construct a space X whose topology induces a nontrivial 2-in/2-out operation $ \omega_2^2 $ on loop cohomology $ {H^{*}}\left( {\Omega X;{{\mathbb{Z}}_2}} \right) $. [ABSTRACT FROM AUTHOR]

In this paper we present the Sweedler cohomology for a cocommutative weak Hopf algebra H. We show that the second cohomology group classifies completely weak crossed products, having a common preunit, of H with a commutative left H-module algebra A. [ABSTRACT FROM AUTHOR]

Abstract: We calculate the Batalin–Vilkovisky, and the induced Gerstenhaber structures of the Hochschild cohomology of the singular R-cochain complex of n-dimensional K-projective spaces, where and and any field. In the special case that , and , we show that this structure cannot be identified with the BV-structure of the integral loop homology of the 2-dimensional sphere computed by L. Menichi, but their induced Gerstenhaber structures can still be identified. Combined with the work of Y. Félix and J. Thomas, the main result of the present paper calculates the BV-structure of the rational loop homology of projective spaces. [Copyright &y& Elsevier]

Abstract: Let be a diagram in which p is a fibration and the pair of the maps is relatively formalizable. Then, we show that the rational cohomology algebra of the pullback of the diagram is isomorphic to the torsion product of algebras and over . Let M be a space which admits an action of a Lie group G. The isomorphism of algebras enables us to represent the cohomology of the Borel construction of the space of free (resp. based) loops on M in terms of the torsion product if M is equivariantly formal (resp. G-formal). Moreover, we compute explicitly the -equivariant cohomology of the space of the based loops on the complex projective space , where the -action is induced by a linear action of on . [Copyright &y& Elsevier]

The present paper continues a cycle of papers of the author (same of them joint), in which the Yoneda algebras are calculated for several families of algebras of dihedral and semidihedral type (in K. Erdmam's classification). In this paper, the Yaneda algebra is described in terms of quivers with relations for algebras of semidihedral type of the family SD(2 $$ \mathcal{B} $$). Bibliography: 28 titles. [ABSTRACT FROM AUTHOR]