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Topology/Geometry Seminar
Tuesday, May 5, 2009
4:00 pm,
135 Hawthorn
Dorette Pronk
Dalhousie University
Equivariant Homotopy Theory for Orbifolds.
Abstract: Orbifolds are paracompact spaces which can locally be described as the quotient of an open subset of Euclidean space by the action of a finite group. An orbifold is called representable if it can be described as the quotient of a manifold by the action of Lie group (in this case, the group does not need to be finite, but the isotropy groups are required to be finite.) Since representable orbifolds have the same underlying geometric structure as smooth G-spaces we would like to extend equivariant homotopy theory to obtain new orbifold homotopy invariants. In general, maps
between orbifolds are more general than maps between G-spaces. So we need to conditions on equivariant invariants to be orbifold invariants.
In order to understand what maps between orbifolds should be, one needs to represent orbifolds by Lie groupoids. These Lie groupoid representations for a given orbifold are only unique up to Morita equivalence. So in order for us to be able to import equivariant homotopy invariants into orbifold homotopy theory, we need to know how maps between representable orbifolds are related to equivariant maps and we need a manageable description of what it means to be invariant under Morita equivalence. In this talk I will explain how this works. As an application, I will show how one can define orbifold Bredon cohomology and (geometric) orbifold K-theory.
This is joint work with Laura Scull from Fort Lewis College in Durango, Colorado.
We meet weekly on Friday afternoons at Penn State Altoona (and sometimes at Penn State University Park). If you are interested in giving a talk in our seminar please contact one of the coordinators for the Spring 2009 semester: Wojtek Dorabiala and Aissa Wade.
