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?
A kind of generalization of group characters to chromatic homotopy theory.
Let be a topological quotient stack. Its free loop stack restricts to those loops that are constant as continuous maps, and contain only possibly the transition data with values in (i.e. the “twisted loop space” retaining “twisted sector” data), this is the inertia stack
The geometric realization of is denoted in (Hopkins-Kuhn-Ravenel 00, Stapleton 13, p. 2).
Regarding as the circle group, there is a canonical infinity-action on any free loop space (by rigid rotation of loops), and it restricts to . Hence there is the homotopy quotient stack
The geometric realization of is denoted in (Stapleton 13, p. 2).
Now with and given some prime, write for the th Morava E-theory and for the th Morava K-theory. Then there is a homomorphism of Borel equivariant cohomology theories
where denotes Bousfield localization. This is the twisted transchromatic character map (Stapleton 13, p. 5), shown here for the special case , in the notation there.
Here is a ring such that… (Stapleton 13, p. 3)
Introduction and review:
Nathaniel Stapleton, An Introduction to HKR Character Theory (pdf)
Arpon Raksit, Characters in global equivariant homotopy theory, 2015 (pdf, pdf)
Original articles:
Michael Hopkins, Nicholas Kuhn, Douglas Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000) 553-594 [doi:10.1090/S0894-0347-00-00332-5, pdf]
Nathaniel Stapleton, Transchromatic generalized character maps, Algebr. Geom. Topol. 13 (2013) 171-203 (arXiv:1110.3346, euclid:agt/1513715495)
Nathaniel Stapleton, Transchromatic twisted character maps, J. Homotopy Relat. Struct. 10, 29–61 (2015). (arXiv:1304.5194, doi:10.1007/s40062-013-0040-9)
Nathaniel Stapleton, Transchromatic generalized character maps (and more!) (pdf)
Tomer Schlank, Nathaniel Stapleton, A transchromatic proof of Strickland’s theorem (arXiv:1404.0717)
Takeshi Torii, HKR characters, p-divisible groups and the generalized Chern character, (pdf)
Last revised on June 21, 2024 at 08:31:27. See the history of this page for a list of all contributions to it.