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 $X//G$ be a topological quotient stack. Its free loop space $\mathcal{L}(X//G) = hom_{Stacks}(S^1, X//G)$ restricts to those loops that are constant as continuous maps, and contain only possibly the transition data with values in $X//G$ (i.e. the “twisted loop space” retaining “twisted sector” data), this is
The geometric realization of $\mathcal{L}_{const}(X//G)$ is denoted $Fix(X)$ in (Hopkins-Kuhn-Ravenel 00, Stapleton 13, p. 2.
Regarding $S^1$ as the circle group, there is a canonical $S^1$ infinity-action on any free loop space $\mathcal{L}(-)$ (by rigid rotation of loops), and it restricts to $\mathcal{L}_{const}(-)$. Hence there is the homotopy quotient stack
The geometric realization of $\mathcal{L}_{const}(X//G) // S^1$ is denoted $Twist(X)$ in (Stapleton 13, p. 2).
Now with $n \in \mathbb{N}$ and given some prime, write $E_n$ for the $n$th Morava E-theory and $K(t)$ for the $t$th Morava K-theory. Then there is a homomorphism of Borel equivariant cohomology theories
where $L_{(-)}$ denotes Bousfield localization. This is the twisted transchromatic character map (Stapleton 13, p. 5), shown here for the special case $t = n-1$, in the notation there.
Here $B_{n-1}$ is a ring such that… (Stapleton 13, p. 3)
For introduction see
Nathaniel Stapleton, An Introduction to HKR Character Theory (pdf)
Arpon Raksit, Characters in global equivariant homotopy theory, 2015 (pdf)
Original articles includes
Michael Hopkins, Nicholas Kuhn, Douglas Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000), 553-594 (publisher, pdf)
Nathaniel Stapleton, Transchromatic generalized character maps, Algebr. Geom. Topol. 13 (2013) 171-203 (arXiv:1110.3346)
Nathaniel Stapleton, Transchromatic twisted character maps (arXiv:1304.5194)
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 October 10, 2018 at 04:41:21. See the history of this page for a list of all contributions to it.