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Centralizers in good groups are good. (English) Zbl 1365.55001
The authors improve the utility of character maps in stable homotopy theory by building upon previous work of Hopkins, Kuhn, and Ravenel, and upon subsequent work of the second author. What results from this are character maps that are both transchromatic (to be explained below) and more amenable to computations. Their result about good groups referred to in the title is an application of this improvement.
Chromatic homotopy theory stratifies the \(p\)-primary stable homotopy groups of a space into levels. The \(n\)th level is governed by certain cohomology theories \(E^*\) – among them, the \(n\)th Morava \(K\)-theory \(K(n)\) and the \(n\)th Morava \(E\)-theory \(E_n\) – that exhibit a periodicity analogous to the Bott periodicity of complex \(K\)-theory \(K_{\mathbb C}\). (In fact, \(K(1)\) is a wedge summand of mod \(p\) complex \(K\)-theory.) Computations become more difficult as \(n\) increases. A mathematical tool is transchromatic if it reduces a problem at the \(n\)th level to a problem at a lower level.
Given a finite group \(G\) and a cohomology theory \(E^*\) at a prime \(p\) and level \(n\) that is periodic in the sense described above, M. J. Hopkins et al. [J. Am. Math. Soc. 13, No. 3, 553–594 (2000; Zbl 1007.55004)] studied cohomology rings \(E^*(BG)\) just as one might study the complex representation ring \(R(G)\); that is, via group characters. In the latter case \(R(G)\) is (after a suitable completion) isomorphic to \(K_{\mathbb C}^*(BG)\), and so the character map is a ring homomorphism \[ \chi:K_{\mathbb C}^*(BG)\cong R(G)\to C \] where \(C\) is the ring of class functions on \(G\) taking values in a certain field. It induces an isomorphism on \(\mathbb Q\otimes R(G)\). Replacing \(K\)-theory by \(E^*\), Hopkins-Kuhn-Ravenel [loc. cit.] produce an analogous isomorphism \[ \chi_{n,p}:p^{-1}E^*(BG)\to C^{n,p} \] where \(C^{n,p}\) is a suitable character ring analog. Suffice it to say that such isomorphisms can reduce problems at level \(n\) to combinatorial problems (i.e., chromatic level 0).
In a subsequent paper [Algebr. Geom. Topol. 13, No. 1, 171–203 (2013; Zbl 1300.55011)] the second author used \(p\)-divisible groups to generalize Hopkins-Kuhn-Ravenel in the case \(E^*=E_n\), resulting in character maps on \(E_n^*(BG)\) whose targets are certain cohomology theories that can range in chromatic level anywhere between 0 and \(n\). However, the coefficient rings of these target cohomology theories lack desirable algebraic properties, making computations with them challenging.
Here enters the paper under review, in which the authors tweak the aforementioned transchromatic character maps so that for a finite \(p\)-good group (i.e., \(E_n^*(BG)\) is free and concentrated in even degrees), the character map targets are cohomology theories with algebraically “nice” coefficient rings. This is done with a combination of localizations and completions, both algebraic (in the sense of rings) and topological (in the sense of Bousfield). Many classes of finite groups are \(p\)-good, and in fact the authors apply their results to prove that certain centralizers within \(p\)-good groups are also \(p\)-good. We also note that in the authors’ work and in the other works discussed above, \(BG\) may be replaced by the more general homotopy quotient \(EG\times_G X\) for a finite \(G\)-CW-complex \(X\).
The transchromatic tools provided by the authors will likely be useful for computations. In the stable homotopy groups of spheres, for example, computations at chromatic level 3 are virtually non-existent, whereas at chromatic level 2 there has been significant computational progress over the last 15-20 years. It will be exciting to see how the authors’ modified character maps are leveraged for future computations at chromatic level \(n\geq3\).

MSC:
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
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