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Let $G$ be a finite group. By the transversality results of Wasserman $G$-equivariant bordism (say real) should be a naive homology theory, and as such it should be represented by a naive G-spectrum.

Is there a description of this spectrum as some sort of Thom spectrum?

I did find a construction in Costenoble's "An introduction to equivariant cobordism", but I do not understand in which sense his spectrum represents geometric $G$-cobordism. I have the feeling that it could be just the restriction of $MO_G$ to the trivial universe, but I'm getting really confused about this.

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  • $\begingroup$ Maybe section V.5 of Stefan Schwede's book project on global homotopy theory is useful for you. math.uni-bonn.de/people/schwede/global.pdf $\endgroup$ – archipelago May 25 '15 at 22:01
  • $\begingroup$ @Archipelago: Thanks! That's more than I was hoping for. $\endgroup$ – Emanuele Dotto May 25 '15 at 22:23
  • $\begingroup$ Did this answer your question? $\endgroup$ – archipelago May 27 '15 at 19:32
  • $\begingroup$ Completely. That's a really thorough discussion of equivariant bordism, the most complete reference I've seen. I don't know what the math overflow protocol is in this case, I'd say the reference you gave answers my question. (and no, it's not the restriction of MO to the trivial universe) $\endgroup$ – Emanuele Dotto Jun 1 '15 at 14:34
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Since my comment answered Emanuele Dotto's answer, I post it as an answer:

Stefan Schwede discusses equivariant bordism in his book project about global homotopy theory in detail.

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