#### Volume 5, issue 1 (2005)

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On Davis–Januszkiewicz homotopy types I; formality and rationalisation

### Dietrich Notbohm and Nigel Ray

Algebraic & Geometric Topology 5 (2005) 31–51
 arXiv: math.AT/0311167
##### 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

##### Keywords
colimit, formality, Davis–Januszkiewicz space, homotopy colimit, model category, rationalisation, Stanley–Reisner algebra
##### Mathematical Subject Classification 2000
Primary: 55P62, 55U05
Secondary: 05E99
##### Publication
Revised: 23 December 2004
Accepted: 5 January 2005
Published: 7 January 2005
##### Authors
 Dietrich Notbohm Department of Mathematics and Computer Science University of Leicester University Road Leicester LE1 7RH United Kingdom Nigel Ray Department of Mathematics University of Manchester Oxford Road Manchester M13 9PL United Kingdom