Abstract
Let G be a finite group. To any family ℱ of subgroups of G, we associate a thick ⊗–ideal ℱNil of the category of G–spectra with the property that every G–spectrum in ℱNil (which we call ℱ–nilpotent) can be reconstructed from its underlying H–spectra as H varies over ℱ. A similar result holds for calculating G–equivariant homotopy classes of maps into such spectra via an appropriate homotopy limit spectral sequence. In general, the condition E∈ℱNil implies strong collapse results for this spectral sequence as well as its dual homotopy colimit spectral sequence. As applications, we obtain Artin- and Brauer-type induction theorems for G–equivariant E–homology and cohomology, and generalizations of Quillen’s ℱp–isomorphism theorem when E is a homotopy commutative G–ring spectrum.
We show that the subcategory ℱNil contains many G–spectra of interest for relatively small families ℱ. These include G–equivariant real and complex K–theory as well as the Borel-equivariant cohomology theories associated to complex-oriented ring spectra, the Ln–local sphere, the classical bordism theories, connective real K–theory and any of the standard variants of topological modular forms. In each of these cases we identify the minimal family for which these results hold.
Citation
Akhil Mathew. Niko Naumann. Justin Noel. "Derived induction and restriction theory." Geom. Topol. 23 (2) 541 - 636, 2019. https://doi.org/10.2140/gt.2019.23.541
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