group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
Equivariant K-theory is the equivariant cohomology version of the generalized cohomology theory K-theory.
To the extent that K-theory is given by equivalence classes of virtual vector bundles (topological K-theory, operator K-theory), equivariant K-theory is given by equivalence classes of virtual equivariant bundles or generalizations to noncommutative topology thereof, as in equivariant operator K-theory, equivariant KK-theory.
The Bott periodicity of plain K-theory generalizes to equivariant K-theory:
Complex equivariant K-theory is invariant under smashing with representation spheres of complex representations (Atiyah 68, Theorem 4.3), while real equivariant K-theory is invariant under smashing with representation spheres of real 8d reps with spin structure (Atiyah 68, Theorem 6.1).
Review in Karoubi 05, Section 5.
Equivariant complex K-theory is an equivariant complex oriented cohomology theory (Greenlees 01, Sec. 10).
(equivariant K-theory of projective G-space)
For $G$ an abelian compact Lie group, let
be a finite-dimensional direct sum of complex 1-dimensional linear representations.
The $G$-equivariant K-theory ring $K_G(-)$ of the corresponding projective G-space $P(-)$ is the following quotient ring of the polynomial ring in one variable $L$ over the complex representation ring $R(G)$ of $G$:
where
$L \,=\, \big[ \mathcal{L}_{ \underset{i}{\oplus} \mathbf{1}_{V_i} }\big]$ is the K-theory class of the tautological equivariant line bundle on the given projective G-space;
$1_{{}_{V_i}} L \;=\; \big[ \mathbf{1}_{V_i} \boxtimes \mathcal{L}_{ \underset{i}{\oplus} \mathbf{1}_{V_i} } \big]$ is the class of its external tensor product of equivariant vector bundles with the given linear representation.
(Greenlees 01, p. 248 (24 of 39))
(equivariant complex orientation of equivariant K-theory)
For $G$ an abelian compact Lie group and $\mathbf{1}_V \,\in\, G Representations_{\mathbb{C}}^{fin}$ a complex 1-dimensional linear representation, the corresponding representation sphere is the projective G-space $S^{\mathbf{1}_V} \,\simeq\, P\big( \mathbf{1}_V \oplus \mathbf{1} \big)$ (this Prop.) and so, by Prop. ,
is generated by the Bott element $(1 - L)$ over $P\big( \mathbf{1}_V \oplus \mathbf{1} \big)$. By the nature of the tautological equivariant line bundle, this Bott element is the restriction of that on infinite complex projective G-space $P\big(\mathcal{U}_G\big)$. The latter is thereby exhibited as an equivariant complex orientation in equivariant complex K-theory.
(Greenlees 01, p. 248 (24 of 39))
The Green-Julg theorem identifies, under some conditions, equivariant K-theory with operator K-theory of corresponding crossed product algebras.
The representation ring of $G$ over the complex numbers is the $G$-equivariant K-theory of the point, or equivalently by the Green-Julg theorem, if $G$ is a compact Lie group, the operator K-theory of the group algebra (the groupoid convolution algebra of the delooping groupoid of $G$):
The first isomorphism here follows immediately from the elementary definition of equivariant topological K-theory, since a $G$-equivariant vector bundle over the point is manifestly just a linear representation of $G$ on a complex vector space.
(e.g. Greenlees 05, section 3, Wilson 16, example 1.6 p. 3)
Under the identification (2) and the Atiyah-Segal completion map
one may ask for the Chern character of the K-theory class $\widehat{V} \in KU(B G)$ expressed in terms of the actual character of the representation $V$. For more see at Chern class of a linear representation.
There is a closed formula at least for the first Chern class (Atiyah 61, appendix):
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 $B G$:
More generally, for $n$-dimensional linear 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 $det(\widehat{V}) = \widehat{\wedge^n V}$:
(Atiyah 61, appendix, item (7))
More explicitly, via the formula for the determinant as a polynomial in traces of powers (see there) this means that the first Chern class of the $n$-dimensional representation $V$ is expressed in terms of its character $\chi_V$ as
For example, for a representation of dimension $n = 2$ this reduces to
(see also e.g. tom Dieck 09, p. 45)
$\,$
An isomorphism analogous to (2) identifies the $G$-representation ring over the real numbers with the equivariant orthogonal $K$-theory of the point in degree 0:
But beware that equivariant KO, even of the point, is much richer in higher degree (Wilson 16, remark 3.34).
In fact, equivariant KO-theory of the point subsumes the representation rings over the real numbers, the complex numbers and the quaternions:
Accordingly the construction of an index (push-forward to the point) in equivariant K-theory is a way of producing $G$-representations from equivariant vector bundles. This method is also called Dirac induction.
Specifically, applied to equivariant complex line bundles on coadjoint orbits of $G$, this is a K-theoretic formulation of the orbit method.
For $X$ a topological space equipped with a $G$-action for $G$ a topological group, write $X//G$ for the homotopy type of the corresponding homotopy quotient. A standard model for this is the Borel construction
The ordinary topological K-theory of $X//G$ is also called the Borel-equivariant K-theory of $X$, denoted
There is a canonical map
from the genuine equivariant K-theory to the Borel equivariant K-theory. In terms of the Borel construction this is given by the composite
where the first map is pullback along the projection $X \times E G \to X$ and the first equivalence holds because the $G$-action on $X \times E G$ is free.
This map from genuine to Borel equivariant K-theory is not in general an isomorphism.
Specifically for $X$ the point, then $K_G(\ast) \simeq R(G)$ is the representation ring and $K_G^{Bor}(\ast) \simeq K(B G)$ is the topological K-theory of the classifying space $B G$ of $G$-principal bundles. In this case the above canonical map is of the form
This is never an isomorphism, unless $G$ is the trivial group. But the Atiyah-Segal completion theorem says that the map identifies $K(B G)$ as the completion of $R(G)$ at the ideal of virtual representations of rank 0.
(equivariant) cohomology | representing spectrum | equivariant cohomology of the point $\ast$ | cohomology of classifying space $B G$ |
---|---|---|---|
(equivariant) ordinary cohomology | HZ | Borel equivariance $H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})$ | |
(equivariant) complex K-theory | KU | representation ring $KU_G(\ast) \simeq R_{\mathbb{C}}(G)$ | Atiyah-Segal completion theorem $R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)$ |
(equivariant) complex cobordism cohomology | MU | $MU_G(\ast)$ | completion theorem for complex cobordism cohomology $MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)$ |
(equivariant) algebraic K-theory | $K \mathbb{F}_p$ | representation ring $(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)$ | Rector completion theorem $R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Rector+completion+theorem">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)$ |
(equivariant) stable cohomotopy | $K \mathbb{F}_1 \overset{\text{<a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a>}}{\simeq}$ S | Burnside ring $\mathbb{S}_G(\ast) \simeq A(G)$ | Segal-Carlsson completion theorem $A(G) \overset{\text{<a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$ |
Incarnations of rational equivariant K-theory:
cohomology theory | definition/equivalence due to | |
---|---|---|
$\simeq K_G^0\big(X; \mathbb{C} \big)$ | rational equivariant K-theory | |
$\simeq H^{ev}\Big( \big(\underset{g \in G}{\coprod} X^g\big)/G; \mathbb{C} \Big)$ | delocalized equivariant cohomology | Baum-Connes 89, Thm. 1.19 |
$\simeq H^{ev}_{CR}\Big( \prec \big( X \!\sslash\! G\big);\, \mathbb{C} \Big)$ | Chen-Ruan cohomology of global quotient orbifold | Chen-Ruan 00, Sec. 3.1 |
$\simeq H^{ev}_G\Big( X; \, \big(G/H \mapsto \mathbb{C} \otimes Rep(H)\big) \Big)$ | Bredon cohomology with coefficients in representation ring | Ho88 6.5+Ho90 5.5+Mo02 p. 18, Mislin-Valette 03, Thm. 6.1, Szabo-Valentino 07, Sec. 4.2 |
$\simeq K_G^0\big(X; \mathbb{C} \big)$ | rational equivariant K-theory | Lück-Oliver 01, Thm. 5.5, Mislin-Valette 03, Thm. 6.1 |
There is an equivariant Chern character map from equivariant K-theory to rational equivariant ordinary cohomology above
(e.g. Stefanich, Sati-Schreiber 20, Sec. 3.4)
The idea of equivariant topological K-theory and the Atiyah-Segal completion theorem goes back to
Michael Atiyah, Characters and cohomology of finite groups, Publications Mathématiques de l’IHÉS, Volume 9 (1961) , p. 23-64 (numdam:PMIHES_1961__9__23_0)
Michael Atiyah, Friedrich Hirzebruch, Vector bundles and homogeneous spaces, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, 3, 7–38 (pdf)
Graeme Segal, Equivariant K-theory, Inst. Hautes Etudes Sci. Publ. Math. No. 34 (1968) p. 129-151 (numdam:PMIHES_1968__34__129_0)
Michael Atiyah, Graeme Segal, Equivariant K-theory and completion, J. Differential Geometry 3 (1969), 1–18.
(euclid:jdg/1214428815, MR 0259946)
and for algebraic K-theory to
with construction via permutative categories in
See also at algebraic K-theory – References – On quotient stacks.
Introductions and surveys:
John Greenlees, Equivariant version of real and complex connective K-theory, Homology Homotopy Appl. Volume 7, Number 3 (2005), 63-82. (Euclid:1139839291)
N. C. Phillips, Equivariant K-theory for proper actions, Pitman Research Notes in Mathematics Series 178, Longman, Harlow, UK, 1989.
Bruce Blackadar, section 11 of K-Theory for Operator Algebras
Alexander Merkujev, Equivariant K-theory (pdf)
Zachary Maddock, An informal discourse on equivariant K-theory (pdf)
Dylan Wilson, Equivariant K-theory, 2016 (pdf, pdf)
Textbook accounts:
See also
Jose Cantarero, Equivariant K-theory, groupoids and proper actions, Thesis 2009 (ubctheses:1.0068026, pdf)
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)
(short version of Cantarero 09)
On Bott periodicity in equivariant K-theory:
Michael Atiyah, Bott periodicity and the index of elliptic operators, The Quarterly Journal of Mathematics, Volume 19, Issue 1, 1968, Pages 113–140 (doi:10.1093/qmath/19.1.113)
Max Karoubi, Bott Periodicity in Topological, Algebraic and Hermitian K-Theory, In: Friedlander E., Grayson D. (eds) Handbook of K-Theory, Springer 2005 (doi:10.1007/978-3-540-27855-9_4)
Basic computations:
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)
Max Karoubi, Equivariant K-theory of real vector spaces and real vector bundles, Topology and its Applications, 122, (2002) 531-456 (arXiv:math/0509497)
The equivariant Chern character is discussed in
German Stefanich, Chern Character in Twisted and Equivariant K-Theory (pdf)
Hisham Sati, Urs Schreiber, Sec. 3.4 of: The character map in equivariant twistorial Cohomotopy (arXiv:2011.06533)
Discussion relating to K-theory of homotopy quotients/Borel constructions is in
Discussion of the twisted ad-equivariant K-theory of compact Lie groups:
Discussion of K-theory of orbifolds is for instance in section 3 of
Discussion of differential K-theory of orbifolds:
Discussion of combined twisted and equivariant and real K-theory
El-kaïoum M. Moutuou, Twistings of KR for Real groupoids (arXiv:1110.6836)
El-kaïoum M. Moutuou, Graded Brauer groups of a groupoid with involution, J. Funct. Anal. 266 (2014), no.5 (arXiv:1202.2057)
Daniel Freed, Lectures on twisted K-theory and orientifolds, lectures at ESI Vienna, 2012 (pdf)
Daniel Freed, Gregory Moore, Section 7 of: Twisted equivariant matter, Ann. Henri Poincaré (2013) 14: 1927 (arXiv:1208.5055)
Kiyonori Gomi, Freed-Moore K-theory (arXiv:1705.09134, spire:1601772)
Discussion in the context of equivariant complex oriented cohomology theory:
That $G$-equivariant topological K-theory is represented by a topological G-space is due to:
Takao Matumoto, Equivariant K-theory and Fredholm operators, J. Fac. Sci. Tokyo 18 (1971/72), 109-112 (pdf, pdf)
Michael Atiyah, Graeme Segal, Sec. 6 and Corollary A3.2 in: Twisted K-theory, Ukrainian Math. Bull. 1, 3 (2004) (arXiv:math/0407054, journal page, published pdf)
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)
This is enhanced to a representing naive G-spectrum in:
In its incarnation (under Elmendorf's theorem) as a Spectra-valued presheaf on the $G$-orbit category this is discussed in
Review:
Valentin Zakharevich, Section 2.2 of: K-Theoretic Computation of the Verlinde Ring, thesis 2018 (hdl:2152/67663, pdf, pdf)
Michael L. Ortiz, Theorem 2.2 in: Differential Equivariant K-Theory (arXiv:0905.0476)
The proposal that D-brane charge on orbifolds is given by equivariant K-theory goes back to
but it was pointed out that only a subgroup or quotient group of equivariant K-theory can be physically relevant, in
For further references see at fractional D-brane.
On Chern classes of linear representations:
L. Evens, On the Chern classes of representations of finite groups, Trans. Am. Math. Soc. 115, 180-193 (1965) (doi:10.2307/1994264)
F. Kamber, Ph. Tondeur, Flat Bundles and Characteristic Classes of Group-Representations, American Journal of Mathematics Vol. 89, No. 4 (Oct., 1967), pp. 857-886 (doi:10.2307/2373408)
Peter Symonds, A splitting principle for group representations, Comment. Math. Helv. (1991) 66: 169 (doi:10.1007/BF02566643)
Last revised on March 15, 2021 at 06:58:16. See the history of this page for a list of all contributions to it.