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Transchromatic twisted character maps. (English) Zbl 1318.55011
In earlier work [N. Stapleton, Algebr. Geom. Topol. 13, No. 1, 171–203 (2013; Zbl 1300.55011)], the author constructed a transchromatic generalized character map for the Morava $$E$$-theories $$E_n$$. It takes values in a height $$t$$ cohomology theory where $$0 \leq t \leq n$$. For $$t=0$$, his construction recovers the generalized character map introduced in influential work by Hopkins, Kuhn, and Ravenel [M. J. Hopkins et al., J. Am. Math. Soc. 13, No. 3, 553–594 (2000; Zbl 1007.55004)].
In the paper under review, the author constructs a twisted transchromatic generalized character map $E_n^*(EG \times_G X) \to B_t^*\otimes_{L_{K(t)} E_n^*(B \mathbb Q_p/\mathbb Z_p^{n-t})}L_{K(t)}E_n^*({\text{Twist}}_{n-t}(X))$ that refines (a completed version of) the transchromatic generalized character map from his earlier work. Here $$L_{K(t)}$$ is the localization with respect to height $$t$$ Morava $$K$$-theory and $$B_t$$ is a certain universal $$L_{K(t)} E_n^0$$-algebra over which the formal group associated to $$E_n$$ is a non-trivial extension of a height $$t$$ formal group by a height $$n-t$$ constant étale $$p$$-divisible group. The space $${\text{Twist}}_{n-t}(X)$$ is a variant of the target space for the earlier generalized character maps that takes torus actions into account. The main theorem states that for a finite group $$G$$, the twisted transchromatic generalized character map induces an isomorphism when tensored up with $$B_t$$.

##### MSC:
 55P42 Stable homotopy theory, spectra 55N91 Equivariant homology and cohomology in algebraic topology 55P48 Loop space machines and operads in algebraic topology
##### Keywords:
generalized character theory; Morava $$E$$-theory
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##### References:
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