Derived induction and restriction theory
| Autor: | Justin Noel, Akhil Mathew, Niko Naumann |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
Homotopy colimit Brauer's theorem Group cohomology 19L47 Artin's theorem 19A22 01 natural sciences Spectrum (topology) Mathematics::Algebraic Topology group cohomology Mathematics::K-Theory and Homology 55N34 K–theory 0103 physical sciences tensor triangulated categories FOS: Mathematics Algebraic Topology (math.AT) Category Theory (math.CT) Mathematics - Algebraic Topology 0101 mathematics Representation Theory (math.RT) Quillen's F–isomorphism theorem induction Mathematics Homotopy 010102 general mathematics equivariant homotopy theory 18G40 Mathematics - Category Theory K-theory 20J06 topological modular forms 55P91 Cohomology 55N91 Topological modular forms spectral sequences Spectral sequence 55P42 010307 mathematical physics Geometry and Topology Mathematics - Representation Theory |
| Zdroj: | Geom. Topol. 23, no. 2 (2019), 541-636 |
| Popis: | Let $G$ be a finite group. To any family $\mathscr{F}$ of subgroups of $G$, we associate a thick $\otimes$-ideal $\mathscr{F}^{\mathrm{Nil}}$ of the category of $G$-spectra with the property that every $G$-spectrum in $\mathscr{F}^{\mathrm{Nil}}$ (which we call $\mathscr{F}$-nilpotent) can be reconstructed from its underlying $H$-spectra as $H$ varies over $\mathscr{F}$. 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 $E\in \mathscr{F}^{\mathrm{Nil}}$ 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 $\mathcal{F}_p$-isomorphism theorem when $E$ is a homotopy commutative $G$-ring spectrum. We show that the subcategory $\mathscr{F}^{\mathrm{Nil}}$ contains many $G$-spectra of interest for relatively small families $\mathscr{F}$. These include $G$-equivariant real and complex $K$-theory as well as the Borel-equivariant cohomology theories associated to complex oriented ring spectra, any $L_n$-local spectrum, 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 such that these results hold. Comment: 63 pages. Many edits and some simplifications. Final version, to appear in Geometry and Topology |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|