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Abstract. Motivated by ideas from stable homotopy theory we study the space of strongly homotopy associative multiplications on a two-cell chain complex. In the simplest case this moduli space is isomorphic to the set of orbits of a group of invertible power series acting on a certain space. The Hochschild cohomology rings of resulting A∞-algebras have an interpretation as totally ramified extensions of discrete valuation rings. All A∞-algebras are supposed to be unital and we give a detailed analysis of unital structures which is of independent interest. Keywords: A∞-algebra, derivation, Hochschild cohomology, formal power series. 1.

Abstract. This paper builds a general framework in which to study cohomology theories of strongly homotopy algebras, namely A∞, C ∞ and L∞-algebras. This framework is based on noncommutative geometry as expounded by Connes and Kontsevich. The developed machinery is then used to establish a general form of Hodge decomposition of Hochschild and cyclic cohomology of C∞-algebras. This generalizes and puts in a conceptual framework previous work by Loday and Gerstenhaber-Schack.

In this paper, working over Z (p) and using algebra perturbation results from [18], p-minimal homological models of twisted tensor products (TTPs) of Cartan 's elementary complexes are obtained. Moreover, making use of the notion of indecomposability of a TTP, we deduce that a homological model of a indecomposable p-minimal TTP of length # (# 2) of exterior and divided power algebras is a tensor product of k-indecomposable (k #) p-minimal TTPs of exterior and divided power algebras.

Abstract. Let p be an odd prime. When n ≥ 3, we show that each tensor factor of form E ⊗ Γ in H ∗ (Z, n; Zp) is an A∞-infinity bialgebra with nontrivial structure. We give explicit formulas for the structure maps and the quadratic relations among them. Thus E ⊗Γ is a naturally occurring example of an A∞-bialgebra whose internal structure is well-understood. 1.

Abstract. In this paper we will study deformations of A∞-algebras. We will also answer questions relating to Moore algebras which are one of the simplest nontrivial examples of an A∞-algebra. We will compute the Hochschild cohomology of odd Moore algebras and classify them up to a unital weak equivalence. We will construct miniversal deformations of particular Moore algebras and relate them to the universal odd and even Moore algebras. Finally we will conclude with an investigation of formal one-parameter deformations of an A∞-algebra. 1.

Abstract. In this paper we establish the existence of certain structures on the ordinary and equivariant homology of the free loop space on a manifold or, more generally, a formal Poincaré duality space. These structures; namely the loop product, the loop bracket and the string bracket, were introduced and studied by Chas and Sullivan under the general heading ‘string topology’. Our method is based on obstruction theory for C∞-algebras and rational homotopy theory. The resulting string topology operations are manifestly homotopy invariant.

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We study cohomology theories of strongly homotopy algebras, namely A∞, C ∞ and L∞-algebras and establish the Hodge decomposition of Hochschild and cyclic cohomology of C∞-algebras thus generalising previous work by Loday and Gerstenhaber-Schack. These results are then used to show that a C∞-algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic C∞-algebra (an∞-generalisation of a commutative Frobenius algebra introduced by Kontsevich). As another application, we show that the ‘string topology ’ operations (the loop product, the loop bracket and the string bracket) are homotopy invariant and can be defined on the homology or equivariant homology of an arbitrary Poincaré