BV-algebras in topology and homotopy algebra

This Licentiate thesis consists of two papers. The first paper is called A-infinity-algebras derived from associative algebras with a non-derivation differential. The second paper is joint work with A. Berglund and is called Free loop space homology of highly connected manifolds. The first paper introduces a non-commutative variant of generalized BV-algebras. We prove the existence of an A-infinity-structure for any graded associative algebra equipped with a differential that is not necessarily a derivation. In a sense this structure measures the failure of the differential to be a derivation. The second paper is concerned with the homology of the free loop space of a manifold that is (n-1)-connected of dimension at most 3n-2. This is computed together with the Chas-Sullivan loop product and loop bracket. In characteristic zero we also determine the BV-operator. The methods used are Hochschild cohomology and the theory of Koszul algebras.