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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
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