On Davis-Januszkiewicz homotopy types I; formality and rationalisation

Notbohm, Dietrich and Ray, Nigel (2005) On Davis-Januszkiewicz homotopy types I; formality and rationalisation. Algebraic and Geometric Topology, 5. pp. 31-51. ISSN 1472-2747

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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.

Item Type:	Article
Uncontrolled Keywords:	Colimit, formality, Davis-Januszkiewicz space, homotopy colimit, model category, rationalisation, Stanley-Reisner algebra
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 05 Combinatorics MSC 2010, the AMS's Mathematics Subject Classification > 52 Convex and discrete geometry MSC 2010, the AMS's Mathematics Subject Classification > 55 Algebraic topology
Depositing User:	Ms Lucy van Russelt
Date Deposited:	15 Aug 2006
Last Modified:	20 Oct 2017 14:12
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/503

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