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Am splitting this off from complex oriented cohomology theory. For the moment just collecting references
John Greenlees, Equivariant formal group laws and complex oriented cohomology theories, Homology Homotopy Appl. Volume 3, Number 2 (2001), 225-263 (euclid:hha/1139840255)
Michael Cole, John Greenlees, Igor Kriz, The universality of equivariant complex bordism, Math Z 239, 455–475 (2002) (doi:10.1007/s002090100315)
How about equivariant elliptic cohomolgy? When is it equivariantly complex orientable?
added this pointer:
Can we take it that equivariant complex cobordism theory is the “universal” equivariant complex oriented cohomology theory?
Let’s see
speaks of Greenlees’ Conjecture III.2 and takes some steps, but it’s certainly not resolved there.
Added
It has been conjectured that equivariant complex cobordism theory is the universal equivariant complex oriented cohomology theory.
I have changed the statement on the allegedly conjectural universal property of equivariant $MU$:
The main theorem of Cole-Greenlees-Kriz 02 does prove the expected universal property.
I guess what may remain conjectural is the version in terms of the equivariant Lazard ring away from the Noetherian case, and/or the generalization away from orientations in degree 2 (?).
Dunno yet, will try to add clarifications once I see through this. But for the moment, for the entry to not be misleading and to be more informative, I have made the bit on equivariant $MU$ read as follows:
For an abelian compact Lie group $G$, equivariant complex cobordism theory $MU_G$ is an equivariant complex oriented cohomology theory (Greenlees 01, Sec. 13).
Much as in the non-equivariant case (see at universal complex orientation on MU), $MU_G$ is universal in that there is a bijection between equivariant complex orientations (in degree 2) on some cohomology theory $E_G$ and homotopy ring homomorphisms of $G$-spectra $MU_G \to E_G$ (Cole-Greenlees-Kriz 02, Theorem 1.2).
For the analogous statement on the equivariant Lazard ring see Greenlees 01, Sec. 13, Cole-Greenlees-Kriz 02, Theorem 1.3.
Another point that needs clarification here is that there are different versions of “equivariant cobordism theory” and of “equivariant $MU$” and not all of them are related as one might hope. Will try to sort this out…
Is anything known for nonabelian groups?
Good question. Most authors focus on abelian compact Lie groups. I gather extension from abelian to general compact Lie groups has more recently been a topic of investigation for elliptic cohomology theories, following the indications in Lurie’s “survey”; and I gather David Gepner et al. have been working on this. The current endpoint of this development might be Gepner-Meier 20, but I haven’t really read it yet.
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