#### Algebraic and Geometric Topology 5 (2005),
paper no. 3, pages 31-51.

## On Davis-Januszkiewicz homotopy types I; formality and rationalisation

### Dietrich Notbohm, Nigel Ray

**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.
**Keywords**.
Colimit, formality, Davis-Januszkiewicz space, homotopy colimit,
model category, rationalisation, Stanley-Reisner algebra

**AMS subject classification**.
Primary: 55P62, 55U05.
Secondary: 05E99.

**DOI:** 10.2140/agt.2005.5.31

**E-print:** `arXiv:math.AT/0311167`

Submitted: 21 May 2004.
(Revised: 23 December 2004.)
Accepted: 5 January 2005.
Published: 7 January 2005.

Notes on file formats
Dietrich Notbohm, Nigel Ray

Department of Mathematics and Computer Science, University of
Leicester

University Road, Leicester LE1 7RH,
UK

and

Department of Mathematics, University of
Manchester

Oxford Road, Manchester M13 9PL, UK

Email: dn8@mcs.le.ac.uk, nige@ma.man.ac.uk

AGT home page

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