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2005 On Davis–Januszkiewicz homotopy types I; formality and rationalisation
Dietrich Notbohm, Nigel Ray
Algebr. Geom. Topol. 5(1): 31-51 (2005). DOI: 10.2140/agt.2005.5.31

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

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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

Information

Received: 21 May 2004; Revised: 23 December 2004; Accepted: 5 January 2005; Published: 2005
First available in Project Euclid: 20 December 2017

zbMATH: 1065.55006
MathSciNet: MR2135544
Digital Object Identifier: 10.2140/agt.2005.5.31

Subjects:
Primary: 55P62, 55U05
Secondary: 05E99

Rights: Copyright © 2005 Mathematical Sciences Publishers

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