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 an abelian compact Lie group, let
be a finite-dimensional direct sum of complex 1-dimensional linear representations.
The -equivariant K-theory ring of the corresponding projective G-space is the following quotient ring of the polynomial ring in one variable over the complex representation ring of :
where
is the K-theory class of the tautological equivariant line bundle on the given projective G-space;
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 an abelian compact Lie group and a complex 1-dimensional linear representation, the corresponding representation sphere is the projective G-space (this Prop.) and so, by Prop. ,
is generated by the Bott element over . By the nature of the tautological equivariant line bundle, this Bott element is the restriction of that on infinite complex projective G-space . 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 over the complex numbers is the -equivariant K-theory of the point, or equivalently by the Green-Julg theorem, if is a compact Lie group, the operator K-theory of the group algebra (the groupoid convolution algebra of the delooping groupoid of ):
The first isomorphism here follows immediately from the elementary definition of equivariant topological K-theory, since a -equivariant vector bundle over the point is manifestly just a linear representation of 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 expressed in terms of the actual character of the representation . 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 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 linear 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 :
(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 -dimensional representation is expressed in terms of its character as
For example, for a representation of dimension this reduces to
(see also e.g. tom Dieck 09, p. 45)
An isomorphism analogous to (2) identifies the -representation ring over the real numbers with the equivariant orthogonal -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 -representations from equivariant vector bundles. This method is also called Dirac induction.
Specifically, applied to equivariant complex line bundles on coadjoint orbits of , this is a K-theoretic formulation of the orbit method.
For a topological space equipped with a -action for a topological group, write for the homotopy type of the corresponding homotopy quotient. A standard model for this is the Borel construction
The ordinary topological K-theory of is also called the Borel-equivariant K-theory of , 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 and the first equivalence holds because the -action on is free.
This map from genuine to Borel equivariant K-theory is not in general an isomorphism.
Specifically for the point, then is the representation ring and is the topological K-theory of the classifying space of -principal bundles. In this case the above canonical map is of the form
This is never an isomorphism, unless is the trivial group. But the Atiyah-Segal completion theorem says that the map identifies as the completion of at the ideal of virtual representations of rank 0.
Incarnations of rational equivariant K-theory:
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. 34 (1968) 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
Robert Bruner, John Greenlees, Connective Real K-Theory of Finite Groups, Mathematical Surveys and Monographs 169 AMS 2010 (ISBN:978-0-8218-5189-0)
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:
For formulation and proof of the McKay correspondence:
That -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 -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)
Discussion of rational equivariant K-theory (see also the references at equivariant Chern character):
Alain Connes, Paul Baum, Chern character for discrete groups, A Fête of Topology, Papers Dedicated to Itiro Tamura 1988, Pages 163-232 (doi:10.1016/B978-0-12-480440-1.50015-0)
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)
Guido Mislin, Alain Valette, Theorem 6.1 in: Proper Group Actions and the Baum-Connes Conjecture, Advanced Courses in Mathematics CRM Barcelona, Springer 2003 (doi:10.1007/978-3-0348-8089-3)
and with emphasis of commutative ring-structure:
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)
The proposal that D-brane charge on orbifolds is given by equivariant K-theory (see at D-brane charge quantization in 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)
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