Cobordisms of orbifold

We define and study various geometric cobordism rings of orbifolds. Our main results include a complete calculation of the unoriented cobordism ring of effective orbifolds with isotropy groups of odd order and the rational cobordism ring of effective oriented orbifolds. These rings are expressed in terms of smooth bordism groups of classifying spaces of various groups. Along the way we construct a spectral sequence that converges to the cobordism groups of orbifolds with any given isotropy representations. This also allows us to calculate various other cobordism groups of orbifolds in low dimensions.