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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.
I have added statement of the remarkable fact that equivariant KO-theory of the point subsumes the representations rings over $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{H}$:
$KO_G^n(\ast) \;\simeq\; \left\{ \array{ 0 &\vert& n = 7 \\ R_{\mathbb{H}}(G)/ R_{\mathbb{R}}(G) &\vert& n = 6 \\ R_{\mathbb{H}}(G)/ R_{\mathbb{C}}(G) &\vert& n = 5 \\ R_{\mathbb{H}}(G) \phantom{/ R_{\mathbb{R}}(G) } &\vert& n = 4 \\ 0 &\vert& n = 3 \\ R_{\mathbb{C}}(G)/ R_{\mathbb{H}}(G) &\vert& n = 2 \\ R_{\mathbb{R}}(G)/ R_{\mathbb{C}}(G) &\vert& n = 1 \\ R_{\mathbb{R}}(G) \phantom{/ R_{\mathbb{R}}(G)} &\vert& n =0 } \right.$Is this what was meant
$c_1(V) = c_1(\wedge^n V)$?
Yes, that’s indeed what is meant (but we may want to add superscripts to make it seem less surprising):
First we define $c_1$ for line bundles/1d reps, then we define $c_1$ on any vector bundle/rep by saying that it’s the previously defined $c_1$ of the determinant line bundle/top exterior power.
Have expanded the respective paragraph to now read like so:
For 1-dimensional representations $V$ the first Chern class of $\widehat{V}$ is just the canonical isomorphism of 1-dimensional characters with group cohomology of $G$ and then with ordinary cohomology of the classifying space $B G$
$c_1\left(\widehat{(-)}\right) \;\colon\; Hom(G, U(1)) \overset{\simeq}{\longrightarrow} H^2_{grp}(G,\mathbb{Z}) \simeq H^2(B G, \mathbb{Z}) \,,$while for any $n$-dimensional representation $V$ the first Chern class is this isomorphism applied to the $n$th exterior power $\wedge^n V$ of $V$ (which is a 1-dimensional representation, namely the “determinant line bundle” of $widehat{V}$, to which the previous definition of $c_1$ applies):
$c_1(V) = c_1(\wedge^n V) \,.$I think I’m struggling with the grammar. So first Chern class is an equivalence, so a map? Then “the first Chern class of $X$” is the image of this map applied to $X$? But above you’re saying this image for $\widehat{V}$ is an isomorphism.
Sorry for being unclear. How about this:
For 1-dimensional representations $V$ their first Chern class $c_1(\widehat{V}) \in H^2(B G, \mathbb{Z})$ is their image under the canonical isomorphism from 1-dimensional characters in $Hom_{Grp}(G,U(1))$ to the group cohomology $H^2_{grp}(G, \mathbb{Z})$ and further to the ordinary cohomology $H^2(B G, \mathbb{Z})$ of the classifying space $B G$:
$c_1\left(\widehat{(-)}\right) \;\colon\; Hom_{Grp}(G, U(1)) \overset{\simeq}{\longrightarrow} H^2_{grp}(G,\mathbb{Z}) \overset{\simeq}{\longrightarrow} H^2(B G, \mathbb{Z}) \,.$More generally, for $n$-dimensional representations $V$ their first Chern class $c_1(\widehat V)$ is the previously defined first Chern-class of the line bundle $\widehat{\wedge^n V}$ corresponding to the $n$-th exterior power $\wedge^n V$ of $V$. The latter is a 1-dimensional representation, corresponding to the determinant line bundle $det(\widehat{V}) = \widehat{\wedge^n V}$:
$c_1(\widehat{V}) \;=\; c_1(det(\widehat{V})) \;=\; c_1( \widehat{\wedge^n V} ) \,.$Much clearer!
appended to the previous discussion the explicit formula for $c_1$ of an $n$-dimensional representation $V$ as a polynomial in its character values (here):
$c_1(V) = \chi_{\left(V^{\wedge^n}\right)} \;\colon\; g \;\mapsto\; \underset{ { k_1,\cdots, k_n \in \mathbb{n} } \atop { \underoverset{\ell = 1}{n}{\sum} \ell k_\ell = n } }{\sum} \underoverset{ l = 1 }{ n }{\prod} \frac{ (-1)^{k_l + 1} }{ l^{k_l} k_l ! } \left(\chi_V(g^l)\right)^{k_l}$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 $G$-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 $G$-orbit category this is discussed in
added pointer to:
added pointer to
for another construction of the representing $G$-space for equivariant K-theory
added pointer to
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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
added pointer to:
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 pointer to
which already gives a classifying $G$-space for $G$-equivariant K-theory
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 ]
added pointer to:
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