and
rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)
Examples of Sullivan models in rational homotopy theory:
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?
group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
While the rationalization of complex topological K-theory has lost most interesting information (being just a direct sum of ordinary rational cohomology in every even degree) the rationalization of its enhancement to equivariant cohomology – equivariant K-theory – is still a comparatively rich object, controlled by the rational representation theory of the given equivariance group.
Rational equivariant K-theory appears in a variety of different but equivalent concrete incarnations, such as Bredon cohomology with coefficients in the representation ring/class functions or as Chen-Ruan cohomology (here shown over the complex numbers, see Lück-Oliver 01 for discussion over the rational numbers):
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 |
As any Chern-Dold character, the equivariant Chern character on equivariant K-theory takes values in its rationalization. Hence, together with the above variety of presentations of rational equivariant K-theory, there is a corresponding variety of realizations of the equivariant Chern character.
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)
With focus on commutative ring-structures:
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)
Last revised on October 4, 2021 at 11:57:25. See the history of this page for a list of all contributions to it.