added pointer to:
am adding these pointer:
Anna Marie Bohmann, Christy Hazel, Jocelyne Ishak, Magdalena Kędziorek, Clover May, Naive-commutative structure on rational equivariant K-theory for abelian groups (arXiv:2002.01556)
Anna Marie Bohmann, Christy Hazel, Jocelyne Ishak, Magdalena Kędziorek, Clover May, Genuine-commutative ring structure on rational equivariant K-theory for finite abelian groups (arXiv:2104.01079)
Will add them also to rational equivariant stable homotopy theory
[ edit: oh, and of course this also goes to rational equivariant K-theory ]
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added pointer to
which already gives a classifying -space for -equivariant K-theory
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Jose Cantarero, Equivariant K-theory, groupoids and proper actions, Thesis 2009 (ubctheses:1.0068026)
Jose Cantarero, Equivariant K-theory, groupoids and proper actions, Journal of K-Theory , Volume 9 , Issue 3 , June 2012 , pp. 475 - 501 (arXiv:0803.3244, doi:10.1017/is011011005jkt173)
added the corollary (here) on equivariant complex orientation of equivariant complex K-theory
]]>added statement of the equivariant K-theory of projective -spaces (here)
]]>for the proof of equivariant Bott periodicty I have added pointer to page and verse in
added a brief remark on equivariant Bott periodicity, with pointer to section 5 in:
Where is Atiyah’s original proof?
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for another construction of the representing -space for equivariant K-theory
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I have added a References subsection (here) on equivariant topological K-theory being represented by a naive G-spectrum.
Currently it reads as follows:
That -equivariant topological K-theory is represented by a topological G-space is
This is enhanced to a representing naive G-spectrum in
Review includes:
In its incarnation (under Elmendorf’s theorem) as a Spectra-valued presheaf on the -orbit category this is discussed in
Fixing a typo in the coefficient groups of equivariant KO for n = 6; Greenlees’ reference has the correct group.
Arun Debray
]]>appended to the previous discussion the explicit formula for of an -dimensional representation as a polynomial in its character values (here):
]]>Much clearer!
]]>Sorry for being unclear. How about this:
For 1-dimensional representations their first Chern class is their image under the canonical isomorphism from 1-dimensional characters in to the group cohomology and further to the ordinary cohomology of the classifying space :
More generally, for -dimensional representations their first Chern class is the previously defined first Chern-class of the line bundle corresponding to the -th exterior power of . The latter is a 1-dimensional representation, corresponding to the determinant line bundle :
]]>I think I’m struggling with the grammar. So first Chern class is an equivalence, so a map? Then “the first Chern class of ” is the image of this map applied to ? But above you’re saying this image for is an isomorphism.
]]>Have expanded the respective paragraph to now read like so:
For 1-dimensional representations the first Chern class of is just the canonical isomorphism of 1-dimensional characters with group cohomology of and then with ordinary cohomology of the classifying space
while for any -dimensional representation the first Chern class is this isomorphism applied to the th exterior power of (which is a 1-dimensional representation, namely the “determinant line bundle” of , to which the previous definition of applies):
]]>Yes, that’s indeed what is meant (but we may want to add superscripts to make it seem less surprising):
First we define for line bundles/1d reps, then we define on any vector bundle/rep by saying that it’s the previously defined of the determinant line bundle/top exterior power.
]]>Is this what was meant
]]>?
added (towards the end of this subsection) the expression for of a complex representation regarded as a vector bundle over (from the appendix of Atiyah 61)
]]>I have added statement of the remarkable fact that equivariant KO-theory of the point subsumes the representations rings over , and :
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