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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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