Abstract
For an arbitrary simplicial complex , Davis and Januszkiewicz have defined a family of homotopy equivalent CW–complexes whose integral cohomology rings are isomorphic to the Stanley–Reisner algebra of . Subsequently, Buchstaber and Panov gave an alternative construction (here called ), 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 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 is unique for a special family of complexes . In a sequel, we will consider the uniqueness of at each prime separately, and apply Sullivan’s arithmetic square to produce
Citation
Dietrich Notbohm. Nigel Ray. "On Davis–Januszkiewicz homotopy types I; formality and rationalisation." Algebr. Geom. Topol. 5 (1) 31 - 51, 2005. https://doi.org/10.2140/agt.2005.5.31
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