# nLab transchromatic character

Contents

cohomology

### Theorems

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

A kind of generalization of group characters to chromatic homotopy theory.

Let $X \sslash G$ be a topological quotient stack. Its free loop stack $\mathcal{L}(X\sslash G) = hom_{Stacks}(S^1, X//G)$ restricts to those loops that are constant as continuous maps, and contain only possibly the transition data with values in $X \sslash G$ (i.e. the “twisted loop space” retaining “twisted sector” data), this is the inertia stack

$\mathcal{L}_{const}( X \sslash G ) \simeq hom_{Stacks}(\mathbf{B}\mathbb{Z}, X \sslash G) \,.$

The geometric realization of $\mathcal{L}_{const}(X \sslash G)$ 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

$\mathcal{L}_{const}( X \sslash G ) \sslash S^1$

The geometric realization of $\mathcal{L}_{const}(X \sslash G) \sslash S^1$ 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

$E_n^\bullet(X//G) \longrightarrow B_{n-1}^\ast \underset{L_{K(n-1)} E_n^\bullet(B \mathbb{Q}_p/\mathbb{Z}_p)}{\otimes} L_{K(n-1)} E^\bullet_n( \mathcal{L}_{const}( X//G ) )// S^1) \,,$

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)

## References

Introduction and review:

Original articles:

Last revised on July 3, 2021 at 21:42:42. See the history of this page for a list of all contributions to it.