Algebraic and Geometric Topology 5 (2005), paper no. 3, pages 31-51.

On Davis-Januszkiewicz homotopy types I; formality and rationalisation

Dietrich Notbohm, Nigel Ray

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(K)), 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(K) 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(K) is unique for a special family of complexes K. In a sequel, we will consider the uniqueness of c(K) at each prime separately, and apply Sullivan's arithmetic square to produce global results for this family.

Keywords. Colimit, formality, Davis-Januszkiewicz space, homotopy colimit, model category, rationalisation, Stanley-Reisner algebra

AMS subject classification. Primary: 55P62, 55U05. Secondary: 05E99.

DOI: 10.2140/agt.2005.5.31

E-print: arXiv:math.AT/0311167

Submitted: 21 May 2004. (Revised: 23 December 2004.) Accepted: 5 January 2005. Published: 7 January 2005.

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Dietrich Notbohm, Nigel Ray
Department of Mathematics and Computer Science, University of Leicester
University Road, Leicester LE1 7RH, UK
Department of Mathematics, University of Manchester
Oxford Road, Manchester M13 9PL, UK


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