nLab
transchromatic character

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Representation theory

Contents

Idea

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

Let X//GX//G be a topological quotient stack. Its free loop space (X//G)=hom Stacks(S 1,X//G)\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//GX//G (i.e. the “twisted loop space” retaining “twisted sector” data), this is

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

The geometric realization of const(X//G)\mathcal{L}_{const}(X//G) is denoted Fix(X)Fix(X) in (Hopkins-Kuhn-Ravenel 00, Stapleton 13, p. 2.

Regarding S 1S^1 as the circle group, there is a canonical S 1S^1 infinity-action on any free loop space ()\mathcal{L}(-) (by rigid rotation of loops), and it restricts to const()\mathcal{L}_{const}(-). Hence there is the homotopy quotient stack

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

The geometric realization of const(X//G)//S 1\mathcal{L}_{const}(X//G) // S^1 is denoted Twist(X)Twist(X) in (Stapleton 13, p. 2).

Now with nn \in \mathbb{N} and given some prime, write E nE_n for the nnth Morava E-theory and K(t)K(t) for the ttth Morava K-theory. Then there is a homomorphism of Borel equivariant cohomology theories

E n (X//G)B n1 *L K(n1)E n (B p/ p)L K(n1)E n ( const(X//G))//S 1), 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 ()L_{(-)} denotes Bousfield localization. This is the twisted transchromatic character map (Stapleton 13, p. 5), shown here for the special case t=n1t = n-1, in the notation there.

Here B n1B_{n-1} is a ring such that… (Stapleton 13, p. 3)

References

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

Last revised on October 10, 2018 at 04:41:21. See the history of this page for a list of all contributions to it.