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We use factorization homology and higher Hochschild (co)chains to study various problems in algebraic topology and homotopical algebra, notably brane topology, centralizers of $E_n$-algebras maps and iterated bar constructions. In particular, we obtain an $E_{n+1}$-algebra model on the shifted integral chains of the mapping space of the n-sphere into an orientable closed manifold $M$. We construct and use $E_infty$-Poincare duality to identify higher Hochschild cochains, modeled over the $n$-sphere, with the chains on the above mapping space, and then relate Hochschild cochains to the deformation complex of the $E_infty$-algebra $C^*(M)$, thought of as an $E_n$-algebra. We invoke (and prove) the higher Deligne conjecture to furnish $E_n$-Hochschild cohomology, and all that is naturally equivalent to it, with an $E_{n+1}$-algebra structure. We prove that this construction recovers the sphere product. In fact, our approach to the Deligne conjecture is based on an explicit description of the $E_n$-centralizers of a map of $E_infty$-algebras $f:Ato B$ by relating it to the algebraic structure on Hochschild cochains modeled over spheres, which is of independent interest and explicit. More generally, we give a factorization algebra model/description of the centralizer of any $E_n$-algebra map and a solution of Deligne conjecture. We also apply similar ideas to the iterated bar construction. We obtain factorization algebra models for (iterated) bar construction of augmented $E_m$-algebras together with their $E_n$-coalgebras and $E_{m-n}$-algebra structures, and discuss some of its features. For $E_infty$-algebras we obtain a higher Hochschild chain model, which is an $E_n$-coalgebra. In particular, considering an n-connected topological space $Y$, we obtain a higher Hochschild cochain model of the natural $E_n$-algebra structure of the chains of the iterated loop space of $Y$.
A first goal of this paper is to precisely relate the homotopy theories of bialgebras and $E_2$-algebras. For this, we construct a conservative and fully faithful $infty$-functor from pointed conilpotent homotopy bialgebras to augmented $E_2$-algebra
Let an n-algebra mean an algebra over the chain complex of the little n-cubes operad. We give a proof of Kontsevichs conjecture, which states that for a suitable notion of Hochschild cohomology in the category of n-algebras, the Hochschild cohomology
Building on work of Livernet and Richter, we prove that E_n-homology and E_n-cohomology of a commutative algebra with coefficients in a symmetric bimodule can be interpreted as functor homology and cohomology. Furthermore we show that the associated Yoneda algebra is trivial.
In this paper we construct a graded Lie algebra on the space of cochains on a $mathbbZ_2$-graded vector space that are skew-symmetric in the odd variables. The Lie bracket is obtained from the classical Gerstenhaber bracket by (partial) skew-symmetri
Quantum Drinfeld Hecke algebras are generalizations of Drinfeld Hecke algebras in which polynomial rings are replaced by quantum polynomial rings. We identify these algebras as deformations of skew group algebras, giving an explicit connection to Hoc