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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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(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

Keywords
colimit, formality, Davis–Januszkiewicz space, homotopy colimit, model category, rationalisation, Stanley–Reisner algebra
Mathematical Subject Classification 2000
Primary: 55P62, 55U05
Secondary: 05E99
References
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Publication
Received: 21 May 2004
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