On Davis-Januszkiewicz Homotopy Types I;
Formality and Rationalisation
by Dietrich Notbohm} and Nigel Ray
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 {\it 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.