nLab
equivariant Chern character

Idea

The equivariant Chern character is the generalization of the Chern character from topological K-theory to equivariant topological K-theory, equivalently the specialization of the equivariant Chern-Dold character for generalized equivariant cohomology to equivariant topological K-theory.

The equivariant Chern character has a variety of different but equivalent concrete incarnations, depending on the choice of presentation of the rational equivariant K-theory that it takes values in:

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

Related concepts

• twisted Chern character, twisted equivariant Chern character

• equivariant Chern-Dold character

• equivariant rationalization

• delocalized equivariant cohomology

• Chen-Ruan cohomology

References

• Jolanta Słomińska, On the Equivariant Chern homomorphism, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astronom. Phys, Vol. 24 (1976), 909-913.

• 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)

• Michel Duflo, Michèle Vergne, Cohomologie équivariante et descente, Astérisque, no. 215 (1993) (numdam:AST_1993__215__5_0)

• Wolfgang Lück, Bob Oliver, 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, pdf)

• Alejandro Adem, Yongbin Ruan, Section 5 of: Twisted Orbifold K-Theory, Commun. Math. Phys. 237 (2003) 533-556 (arXiv:math/0107168)

• Ieke Moerdijk, p. 18 of: Orbifolds as Groupoids: an Introduction, in: Alejandro Adem, Jack Morava, Yongbin Ruan (eds.) Orbifolds in Mathematics and Physics, Contemporary Math 310 , AMS (2002), 205–222 (arXiv:math.DG/0203100)

• 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)

• Ulrich Bunke, Markus Spitzweck, Thomas Schick, Inertia and delocalized twisted cohomology, Homotopy, Homology and Applications, vol 10(1), pp 129-180 (2008) (arXiv:math/0609576, doi:10.4310/HHA.2008.v10.n1.a6)

• Pierre Albin, Richard Melrose, Delocalized equivariant cohomology and resolution (arXiv:1012.5766)

Review in:

• Ulrich Bunke, Section 3.1 in: Orbifold index and equivariant K-homology, Math. Ann. 339, 175–194 (2007) (arXiv:math/0701768, doi:10.1007/s00208-007-0111-5)

• German Stefanich, Chern Character in Twisted and Equivariant K-Theory (pdf)