nLab transchromatic character

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)

Related concepts

• Chern character

• higher chromatic Chern character

• Morava E-theoretic Chern character

• free loop stack, Huan's inertia orbifold

• free loop space of classifying space

References

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)