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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.

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Joyce writes on various kinds of bordism and cobordism for orbifolds in his book available at people.maths.ox.ac.uk/joyce/dmbook.pdf, Section 13.3. He also introduces bordism and cobordism for what he calls d-orbifolds. Using those, you should be able to treat the spaces M_g,n (X,A) even if they are not smooth. –  Timo Schürg Dec 5 '12 at 20:32
In the 1960's, Conner and Floyd used cobordism theory to study actions of finite groups on compact manifolds. The theory is explained in Differentiable Periodic Maps, LNM 738, 1979. In my understanding, the main problem in the equivariant setting (and therefore for orbifolds) is that transversality does not always hold, so homotopy alone does not capture the full story. –  Daniel Moskovich Dec 5 '12 at 23:25

2 Answers 2

Andres Angel has written some papers on this subject, what I know is a preprint entitled "Orbifold cobordism" where he computes some orbifold cobordism rings in terms of bordism groups of classifying spaces of groups.

Papers available here: http://www.math.uni-bonn.de/people/aangel79/papers.html

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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.

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