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added to equivariant K-theory comments on the relation to the operator K-theory of crossed product algebras and to the ordinary K-theory of homotopy quotient spaces (Borel constructions). Also added a bunch of references.
(Also finally added references to Green and Julg at Green-Julg theorem).
This all deserves to be prettified further, but I have to quit now.
Is this what was meant
?
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.
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):
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.
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 :
Much clearer!
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
added pointer to:
added pointer to
for another construction of the representing -space for equivariant K-theory
added pointer to
added pointer to
added a brief remark on equivariant Bott periodicity, with pointer to section 5 in:
Where is Atiyah’s original proof?
for the proof of equivariant Bott periodicty I have added pointer to page and verse in
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