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nLab > Latest Changes: equivariant K-theory

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 5th 2013
    • (edited Aug 13th 2014)

    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.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 2nd 2018

    slightly expanded the paragraph “Relation to representation theor” (here), adding mentioning also of KO G 0(*)≃R ℝ(G)KO^0_G(\ast) \simeq R_{\mathbb{R}}(G)

    diff, v20, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 2nd 2018
    • (edited Oct 2nd 2018)

    I have added statement of the remarkable fact that equivariant KO-theory of the point subsumes the representations rings over ℝ\mathbb{R}, ℂ\mathbb{C} and ℍ\mathbb{H}:

    KO G n(*)≃{0 | n=7 R ℍ(G)/R ℝ(G) | n=6 R ℍ(G)/R ℂ(G) | n=5 R ℍ(G)/R ℝ(G) | n=4 0 | n=3 R ℂ(G)/R ℍ(G) | n=2 R ℝ(G)/R ℂ(G) | n=1 R ℝ(G)/R ℝ(G) | n=0 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.

    diff, v22, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 9th 2018
    • (edited Oct 9th 2018)

    added (towards the end of this subsection) the expression for c 1c_1 of a complex representation regarded as a vector bundle over BGB G (from the appendix of Atiyah 61)

    diff, v24, current

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 9th 2018

    Is this what was meant

    c 1(V)=c 1(∧ nV)c_1(V) = c_1(\wedge^n V)?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 10th 2018

    Yes, that’s indeed what is meant (but we may want to add superscripts to make it seem less surprising):

    First we define c 1c_1 for line bundles/1d reps, then we define c 1c_1 on any vector bundle/rep by saying that it’s the previously defined c 1c_1 of the determinant line bundle/top exterior power.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeOct 10th 2018
    • (edited Oct 10th 2018)

    Have expanded the respective paragraph to now read like so:

    For 1-dimensional representations VV the first Chern class of V^\widehat{V} is just the canonical isomorphism of 1-dimensional characters with group cohomology of GG and then with ordinary cohomology of the classifying space BGB G

    c 1((−)^):Hom(G,U(1))⟶≃H grp 2(G,ℤ)≃H 2(BG,ℤ), 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 nn-dimensional representation VV the first Chern class is this isomorphism applied to the nnth exterior power ∧ nV\wedge^n V of VV (which is a 1-dimensional representation, namely the “determinant line bundle” of widehatVwidehat{V}, to which the previous definition of c 1c_1 applies):

    c 1(V)=c 1(∧ nV). c_1(V) = c_1(\wedge^n V) \,.

    diff, v25, current

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 10th 2018

    I think I’m struggling with the grammar. So first Chern class is an equivalence, so a map? Then “the first Chern class of XX” is the image of this map applied to XX? But above you’re saying this image for V^\widehat{V} is an isomorphism.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeOct 10th 2018
    • (edited Oct 10th 2018)

    Sorry for being unclear. How about this:

    For 1-dimensional representations VV their first Chern class c 1(V^)∈H 2(BG,ℤ)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))Hom_{Grp}(G,U(1)) to the group cohomology H grp 2(G,ℤ)H^2_{grp}(G, \mathbb{Z}) and further to the ordinary cohomology H 2(BG,ℤ)H^2(B G, \mathbb{Z}) of the classifying space BGB G:

    c 1((−)^):Hom Grp(G,U(1))⟶≃H grp 2(G,ℤ)⟶≃H 2(BG,ℤ). 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 nn-dimensional representations VV their first Chern class c 1(V^)c_1(\widehat V) is the previously defined first Chern-class of the line bundle ∧ nV^\widehat{\wedge^n V} corresponding to the nn-th exterior power ∧ nV\wedge^n V of VV. The latter is a 1-dimensional representation, corresponding to the determinant line bundle det(V^)=∧ nV^det(\widehat{V}) = \widehat{\wedge^n V}:

    c 1(V^)=c 1(det(V^))=c 1(∧ nV^). c_1(\widehat{V}) \;=\; c_1(det(\widehat{V})) \;=\; c_1( \widehat{\wedge^n V} ) \,.

    diff, v27, current

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 10th 2018

    Much clearer!

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeOct 20th 2018

    appended to the previous discussion the explicit formula for c 1c_1 of an nn-dimensional representation VV as a polynomial in its character values (here):

    c 1(V)=χ (V ∧ n):g↦∑k 1,⋯,k n∈𝕟∑ℓ=1nℓk ℓ=n∏l=1n(−1) k l+1l k lk l!(χ V(g l)) k l 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}

    diff, v29, current

  12. Fixing a typo in the coefficient groups of equivariant KO for n = 6; Greenlees’ reference has the correct group.

    Arun Debray

    diff, v33, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJun 15th 2020
    • (edited Jun 15th 2020)

    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 GG-equivariant topological K-theory is represented by a topological G-space is

    This is enhanced to a representing naive G-spectrum in

    Review includes:

    • Valentin Zakharevich, Section 2.2 of: K-Theoretic Computation of the Verlinde Ring (pdf)

    In its incarnation (under Elmendorf’s theorem) as a Spectra-valued presheaf on the GG-orbit category this is discussed in

    • James Davis, Wolfgang Lück, Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K- and L-Theory, K-Theory 15:201–252, 1998 (pdf)

    diff, v37, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeOct 1st 2020

    added pointer to:

    diff, v41, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeOct 4th 2020
    • (edited Oct 4th 2020)

    added pointer to

    • Wolfgang Lück, Bob Oliver, Section 1 of: Chern characters for the equivariant K-theory of proper G-CW-complexes, In: Aguadé J., Broto C., Casacuberta C. (eds.) Cohomological Methods in Homotopy Theory Progress in Mathematics, vol 196. Birkhäuser 2001 (doi:10.1007/978-3-0348-8312-2_15)

    for another construction of the representing GG-space for equivariant K-theory

    diff, v42, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2020

    added pointer to

    • Yimin Yang, On the Coefficient Groups of Equivariant K-Theory, Transactions of the American Mathematical Society Vol. 347, No. 1 (Jan., 1995), pp. 77-98 (jstor:2154789)

    diff, v49, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2020

    added pointer to

    • Max Karoubi, Equivariant K-theory of real vector spaces and real vector bundles, Topology and its Applications, 122, (2002) 531-456 (arXiv:math/0509497)

    diff, v49, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2020

    added a brief remark on equivariant Bott periodicity, with pointer to section 5 in:

    Where is Atiyah’s original proof?

    diff, v49, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeNov 9th 2020

    for the proof of equivariant Bott periodicty I have added pointer to page and verse in

    diff, v51, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeNov 12th 2020

    added statement of the equivariant K-theory of projective GG-spaces (here)

    diff, v52, current

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeNov 12th 2020

    added the corollary (here) on equivariant complex orientation of equivariant complex K-theory

    diff, v53, current

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