I will show decompositions of several bordism groups of orbifolds in terms of usual smooth bordism theory, these decompositions involve information around the singular sets and provide a way to define invariants for orbifolds.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/3946