Derived A-infinity Algebras: Combinatorial models and obstruction theory

Let R be a commutative ring, and let A be a derived A_\infty-algebra over R with structure maps m_{ij} for all i\geq 0, j \geq 1. In this thesis we construct a collection of based topological spaces V_{ij} which give rise to the notion of a DA_\infty-space. The structure of these spaces gives new insight into the structure of a derived A_\infty-algebra. We study the cell structure of these spaces via a combinatorial model using partitioned trees. We will prove that the singular chain complex on a DA_\infty-space gives rise to a derived A_\infty-algebra. We go on to consider obstruction theories to the existence of the structure maps of a derived A_\infty-algebra. The bigrading on A leads to choices of the order in which we develop the derived A_\infty-structure. We give three different definitions of a “partial” derived A_\infty-structure and in light of these definitions provide two different obstruction theories to extend a dA´_{ij}-structure to a dA_{ij} structure, plus an obstruction theory to extend a dA_{r-1}-structure to a dA_{r+1}-structure. In each case, the obstruction lies in a particular class of the Hochschild cohomology of the homology of A.