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Macquarie University  Department of Mathematics

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Workshop on Categorical Methods in Algebra, Geometry and Mathematical Physics

Satellite to the StreetFest conference in honour of Ross Street's sixtieth birthday

July 18-21 2005, Australian National University, Canberra

Mon, 18 July:  10:10 - 10:50

Model categories and homotopy colimits in toric topology
Panov, Taras (Moscow State University)

\thanks{Homotopy-theoretical aspects of toric topology is a joint project with Nigel Ray and Rainer Vogt.}
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Since the pioneering work of Davis and Januszkiewicz, algebraic topologists have been drawn increasingly
towards the study of spaces which arise from well-behaved actions of the torus $T^n$. Investigations are no longer confined to the properties of Davis and Januszkiewicz's toric manifolds, but have extended to related geometrical structures, such as moment-angle complexes, subspace arrangements or torus manifolds of Hattori and Masuda, as well as the homotopy types of associated spaces and their rationalisations and localisations. We refer to this enlarged field of activity as {\it toric topology}.

From the viewpoint of applications in combinatorics and commutative algebra it is important to understand the topology of loop spaces of different spaces arising in toric topology. Many of these spaces (e.g., toric manifolds, moment-angle complexes and their Borel
constructions) admit a simple presentation as a colimit of a certain diagram of spaces over the face category of a simplicial complex. This opens a way to construct good algebraic and topological models for the associated loop spaces by studying the behavior of the loop and classifying space functors, and their algebraic analogues,
with respect to homotopy colimits in different algebraic and topological model categories (spaces, topological monoids, non-commutative DGAs etc.). This study leads to better understanding homotopy colimits themselves in these categories, which may be of independent interest in homotopy theory. Our main applications concern Stanley's ``Combinatorial commutative algebra",
in particular, we apply our models to effective calculation of Ext-cohomology of Stanley--Reisner face rings for several important classes of simplicial complexes, as well as to different combinatorial problems
related to the numbers of faces in triangulations (known to
combinatorial geometers as {\it f-vectors} of simplicial complexes).

Typeset PDF of this abstract.