Title:
Colimits, Stanley-Reisner algebras, and loop spaces
Authors:
Taras Panov, Nigel Ray, and Rainer Vogt
Addresses:
Department of Mathematics and Mechanics, Moscow State
University, 119899 Moscow, Russia;
Department of Mathematics, University of Manchester,
Manchester M13 9PL, England;
Fachbereich Mathematik/Informatik, Universitaet Osnabrueck,
D-49069 Osnabrueck, Germany.
E-mail addresses:
tpanov@mech.math.msu.su
nige@ma.man.ac.uk
rainer@mathematik.uni-osnabrueck.de
Arxiv:
math.AT/0202081
Abstract:
We study diagrams associated with a finite simplicial complex K,
in various algebraic and topological categories. We relate their
colimits to familiar structures in algebra, combinatorics,
geometry and topology. These include: right-angled Artin and
Coxeter groups (and their complex analogues, which we call
circulation groups); Stanley-Reisner algebras and coalgebras;
Davis and Januszkiewicz's spaces DJ(K) associated with
toric manifolds and their generalisations; and coordinate subspace
arrangements. When K is a flag complex, we extend well-known
results on Artin and Coxeter groups by confirming that the
relevant circulation group is homotopy equivalent to the space of
loops $\Omega DJ(K)$. We define homotopy colimits for
diagrams of topological monoids and topological groups, and show
they commute with the formation of classifying spaces in a
suitably generalised sense. We deduce that the homotopy colimit of
the appropriate diagram of topological groups is a model for
$\Omega DJ(K)$ for an arbitrary complex K, and that
the natural projection onto the original colimit is a homotopy
equivalence when K is flag. In this case, the two models are
compatible.