Title: "Equivariant" Topology of Orbifolds

Abstract:

Orbifolds are a class of singular spaces which are locally given as
the quotient space of a smooth manifold modulo a finite group. Such
objects appear naturally in geometry and topology. For example, the
reduced space of a symplectic reduction is often naturally an
(symplectic) orbifold. Algebraic varieties with quotient singularity
are also examples of orbifolds (e.g. Calabi-Yau orbifolds). Orbifolds
also appear in the theories in physics, e.g. orbifold topological sigma
models.

An orbifold $X$ is called "good" if it is globally the quotient space
of a manifold $Y$ modulo a discrete group $G$. There are examples of
"bad" orbifolds (e.g. Thurston's teardrop orbifold), and there are
currently no effective ways to tell if a given orbifold is good,
unless it is given a priori as such an orbifold. A good orbifold $X=Y/G$
apparently carries an extra structure --- the "equivariant" topology or
geometry of the $G$-manifold $Y$.

We will discuss an ongoing project which aims at making sense of
the said "equivariant" aspect of good orbifolds for general orbifolds.
At the heart of this project is a class of natural morphisms between
orbifolds, generalizing equivariant maps between $G$-manifolds. If time
permits, we will relate this work with some recent studies of orbifolds
which are related to string theory.