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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 7th 2020

Am splitting this off from complex oriented cohomology theory. For the moment just collecting references

How about equivariant elliptic cohomolgy? When is it equivariantly complex orientable?

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeNov 8th 2020

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeNov 8th 2020

Can we take it that equivariant complex cobordism theory is the “universal” equivariant complex oriented cohomology theory?

Let’s see

• William Abram, Equivariant Complex Cobordism, 2013, (PhD thesis)

speaks of Greenlees’ Conjecture III.2 and takes some steps, but it’s certainly not resolved there.

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeNov 8th 2020

It has been conjectured that equivariant complex cobordism theory is the universal equivariant complex oriented cohomology theory.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeNov 8th 2020
• (edited Nov 8th 2020)

Thanks. I have slightly rephrased to make it clearer that it’s known to be an example, just not known to be universal as such.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeNov 12th 2020

added the argument (here) for the canonical equivariant complex orientation of equivariant complex K-theory

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeNov 20th 2020
• (edited Nov 20th 2020)

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…

• CommentRowNumber8.
• CommentAuthorDavid_Corfield
• CommentTimeNov 20th 2020

Is anything known for nonabelian groups?

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeNov 20th 2020

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.