On Davis–Januszkiewicz homotopy types II : Completion and globalisation

For any finite simplicial complex $K$ , Davis and Januszkiewicz 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, which they showed to be homotopy equivalent to the original examples. It is therefore natural to investigate the extent to which the homotopy type of a space $X$ is determined by such a cohomology ring. Having analysed this problem rationally in Part I, we here consider it prime by prime, and utilise Lannes’ $T$ –functor and Bousfield–Kan type obstruction theory to study the $p$ –completion of $X$ . We find the situation to be more subtle than for rationalisation, and confirm the uniqueness of the completion whenever $K$ is a join of skeleta of simplices. We apply our results to the global problem by appealing to Sullivan’s arithmetic square, and deduce integral uniqueness whenever the Stanley–Reisner algebra is a complete intersection.

Keywords

arithmetic square, completion, Davis–Januszkiewicz space, homotopy colimit, homotopy type, Stanley–Reisner algebra, $T$–functor, $p$–completion

Mathematical Subject Classification 2000

Primary: 55P15, 55P60

Secondary: 05E99

References

Publication

Received: 16 December 2008

Revised: 8 April 2009

Accepted: 11 April 2009

Published: 29 August 2010

Authors

Dietrich Notbohm
Department of Mathematics Vrije Universiteit Amsterdam De Boelelaan 1081a 1081 HV Amsterdam The Netherlands

Nigel Ray
School of Mathematics University of Manchester Oxford Road Manchester M13 9PL United Kingdom

Volume 10, issue 3 (2010)

Dietrich Notbohm and Nigel Ray

Abstract

Keywords

Mathematical Subject Classification 2000

References

Publication

Authors