On Davis–Januszkiewicz homotopy types I ; formality and rationalisation

Notbohm, Dietrich; Ray, Nigel

doi:10.2140/agt.2005.5.31

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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

DOI: 10.2140/agt.2005.5.31

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

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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

Volume 5, issue 1 (2005)