On Davis–Januszkiewicz homotopy types I; formality and rationalisation

Abstract

For an arbitrary simplicial complex $K$, Davis and Januszkiewicz have defined a family of homotopy equivalent CW–complexes whose integral cohomology rings are isomorphic to the Stanley–Reisner algebra of $K$. Subsequently, Buchstaber and Panov gave an alternative construction (here called $c\left(K\right)$), which they showed to be homotopy equivalent to Davis and Januszkiewicz’s examples. It is therefore natural to investigate the extent to which the homotopy type of a space is determined by having such a cohomology ring. We begin this study here, in the context of model category theory. In particular, we extend work of Franz by showing that the singular cochain algebra of $c\left(K\right)$ is formal as a differential graded noncommutative algebra. We specialise to the rationals by proving the corresponding result for Sullivan’s commutative cochain algebra, and deduce that the rationalisation of $c\left(K\right)$ is unique for a special family of complexes $K$. In a sequel, we will consider the uniqueness of $c\left(K\right)$ at each prime separately, and apply Sullivan’s arithmetic square to produce