Homotopy Types

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, 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 X 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 X is formal as a differential graded noncommutative algebra. We then specialise to the rationals, by proving the corresponding property for Sullivan’s commutative cochain algebra; this confirms that the rationalisation of X is unique. In a sequel, we will consider the uniqueness of X at each prime separately, and apply Sullivan’s arithmetic square to produce global results in special families of cases.