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nLab > Latest Changes: equivariant complex oriented cohomology theory

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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?

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2020

    added this pointer:

    diff, v2, current

    • 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

    Added

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

    diff, v3, current

    • 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.

    diff, v4, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeNov 12th 2020

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

    diff, v8, current

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

    I have changed the statement on the allegedly conjectural universal property of equivariant MUMU:

    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 MUMU read as follows:

    For an abelian compact Lie group GG, equivariant complex cobordism theory MU GMU_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 GMU_G is universal in that there is a bijection between equivariant complex orientations (in degree 2) on some cohomology theory E GE_G and homotopy ring homomorphisms of GG-spectra MU G→E GMU_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 MUMU” and not all of them are related as one might hope. Will try to sort this out…

    diff, v9, current

    • 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.

1 to 9 of 9

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