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The character of the total power operation. (English) Zbl 1360.55004
The authors compute the total power operation for the Morava \(E\)-theory of any finite group up to torsion.
Let \(X\) be a topological space, \(\Sigma_m\) the symmetric group of \(m\) elements, and \(E\Sigma_m \times_{\Sigma_m}X^m\) the Borel construction for the canonical action of the symmetric group \(\Sigma_m\) on \(X^m\) by permutations. Let \(p\) be a fixed prime, \(n\) a natural number, \(\kappa\) a perfect field of characteristic \(p\), \(\Gamma\) a height \(n\) formal group law. Define the \(E_\infty\)-ring spectrum \(E_n\) in Morava \(K\)-theory. The coefficient ring \(E^0_n\) carries a universal deformation \(\mathbb G\) of \(\Gamma\). The total power operations \[ \mathbb P_m: E^0_n(X) \to E^0_n(E\Sigma_m \times_{\Sigma_m}X^m), \] for all \(m>0\), are natural multiplicative and nonadditive. The restriction of the power operations along the diagonal \(X \hookrightarrow X^m\) produces the maps \[ P_m: E^0_n(X) \to E^0_n(B\Sigma_m \times X) \cong E^0_n(B\Sigma_m) \otimes_{E^0_n}E^0_n(X). \] Let \(\mathbb L = (\mathbb Z_p)^n\) and \(\mathbb T= \mathbb L^* \cong (\mathbb Q_p/\mathbb Z_p)^n\), \(C_0\) a \(p^{-1}E^0_n\)-algebra that corepresents isomorphisms of \(p\)-divisible groups between \(\mathbb T\) and \(\mathbb G\) in the sense \(\mathrm{hom}(C_0,R) \cong \mathrm{Iso}(R \otimes \mathbb T, R \otimes \mathbb G)\) and there is a natural action of \(\mathrm{Aut}(\mathbb T)\) on the quotient of \(\mathrm{hom}(\mathbb L, G)\) by the conjugation action of \(G\) and also on the set \(\mathrm{Cl}_n(G,C_0)\) of \(C_0\)-valued functions on \(\mathrm{hom}(\mathbb L, G)/\sim\). Let us consider the generalised Hopkins, Kuhn and Ravenel character \(\chi: E^0_n(BG) \to \mathrm{Cl}_n(G,C_0)\) and the induced map isomorphism \[ C_0 \otimes \chi: C_0 \otimes _{E^0_n} E^0_n(BG)\stackrel{\cong}{\longrightarrow} \mathrm{Cl}_n(G,C_0) \] and \(\mathrm{Aut}(\mathbb T)\)-equivariant isomorphism \(\chi^{\mathrm{Aut}(\mathbb T)}: p^{-1}E^0_n(BG) \stackrel{\cong}{\longrightarrow} Cl_n(G,C_0)^{\mathrm{Aut}(\mathbb T)}\).
Let \(\mathrm{Isog}(\mathbb T) \twoheadrightarrow \mathrm{Sub}(\mathbb T)\) be the \(\mathrm{Aut}(\mathbb T)\)-principal bundle from the monoid \(\mathrm{Isog}(\mathbb T)\) of endoisogenies of \(\mathbb T\), i.e. endomorphisms with finite kernel, on the set of finite subgroups of \(\mathbb T\). For each section \(\Phi\) of this principal bundle one constructs a multiplicative natural transformation \[ \mathrm{Cl}_n(-,C_0) \stackrel{\mathbb P^\Phi_m}{\longrightarrow} \mathrm{Cl}_n(-\wr \Sigma_m,C_0). \] The main results of the paper are the following commutative diagrams (natural in \(G\)): \[ \begin{tikzcd} E^0_n(BG) \ar[r, "\mathbb P_m"] \ar[d, "\chi" left] & E^0_n(BG\wr \Sigma_m) \ar[d, "\chi"]\\ \mathrm{Cl}_n(G,C_0) \ar[r, "\mathbb P^\Phi_m"] & \mathrm{Cl}_n(G\wr \Sigma_m,C_0)\end{tikzcd} \] (Theorems A & 9.1) \[ \begin{tikzcd} E^0_n(BG) \ar[r, "\mathbb P_m"] \ar[d, "\chi" left] & E^0_n(BG\wr \Sigma_m) \ar[d, "\chi"]\\ p^{-1}E^0_n(BG) \ar[r, "\mathbb P_m^\mathbb Q"] & p^{-1}E^0_n(BG,\wr \Sigma_m,C_0)\end{tikzcd} \] (Theorem B & 10.1) and that the vertical character map \(\chi: E^0_n(B-) \to p^{-1}E^0_n(B-)\) is a map of global power functors (Theorem C resp. 10.4).
MSC:
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
55S25 \(K\)-theory operations and generalized cohomology operations in algebraic topology
55P42 Stable homotopy theory, spectra
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