TITLE: Cobordism of orbifolds?
QUESTION [27 upvotes]: Is it possible to setup classical cobordism theory in the context of orbifolds?  For example, let's consider the free abelian group generated by oriented smooth orbifolds and quotient by those which are null cobordant (that is, they are the boundary of a smooth oriented orbifold with boundary).  Is the resulting ring expressible as the homotopy groups of some Thom spectrum?  Can we prove this just by smoothly embedding an orbifold in $\mathbb R^n/S_n$ and following the classical proof?
My motivation is that in Gromov--Witten theory, the moduli space of stable maps $\bar M_{g,n}(X,A)$ is a smooth oriented orbifold (assuming it is cut out transversally, and assuming we have smooth charts for gluings) and it is defined up to cobordism.  Thus instead of taking its fundamental class and pushing forward to $H_\ast(\bar M_{g,n}\times X^n)$ to get Gromov--Witten invariants, we could consider the class it represents in the generalized cohomology theory which we might call "oriented orbifold cobordism" of $\bar M_{g,n}\times X^n$, and get a slightly more refined invariant.

REPLY [6 votes]: Here are a couple papers on oriented orbifold cobordism. The first gives rational invariants and generators and also shows that rationally odd dimensional orbifolds bound.
 The second paper develops machinery for handling the torsion (all dimensions) and applies that to show that every oriented three-orbifold bounds.
K.S. Druschel.  Oriented Orbifold Cobordism, Pacific J. Math.,  164(2) (1994), 299-319.
K.S. Druschel. The Cobordism of Oriented Three Dimensional Orbifolds, Pacific J. Math., bf 193(1) (2000), 45-55.