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5. Rational Homotopy of Invariants


    1. Add remark

      Problem 5.1.

      What is $\pi_*(E^{h\Gamma}) \otimes \mathbf{Q}$ for closed subgroups $\Gamma \subset \mathbf{G}_n$?
          If $\Gamma$ is open this could be understood using finite covers of $\mathcal{H}^{n-1}$. If the image of $\Gamma$ under $\text{det}$ is finite, then this becomes harder because the argument for why the group cohomology of $H^*(\Gamma, \pi_{2k} E_n) \otimes \mathbf{Q}$ vanishes for $k \neq 0$ no longer applies. This may require computation of cohomology with coeffients in nontrivial line bundles.

      A general expectation can be stated for $\mathbf{G}_n^1 = \text{ker}(\mathbf{G}_n \overset{\det}{\longrightarrow} \mathbf{Z}_p^\times \rightarrow \mathbf{Z}_p)$. The composite $\mathbf{G}_n \rightarrow\mathbf{Z}_p$ produces an extension class in degree $1$ called $\zeta_1$. Then $$H^*(\mathbf{G}_n^1, \pi_* E) \otimes \mathbf{Q} \cong (H^*(\mathbf{G}_n, \pi_*E) \otimes \mathbf{Q} )/\zeta_1 \quad \text{(conjecturally).}$$ Although this is known from explicit computations at height $2$, it does not immediately follow from the methods of [arxiv: 2402.00960].


        • Add remark

          Problem 5.2.

          What is the connection between a choice of maximal torus of $\mathbf{G}_n$ of dimension $n$ and there being $n$ exterior generators?

          Can this be related to the identification (say $p \neq 2$) $$\text{THH}(\mathbf{Z}_p) \cong t_{\ge 0} (\text{im}(J))^{tC_p}?$$ What about $\text{THH}(\text{BP}\langle n \rangle)$. Is it $t_{\ge 0}(E^{hT^\text{ur}}_{n+1})^{tC_p}$? Here $T^\text{ur}$ is the maximal unramified torus in $\mathbf{G}_n$.

              Cite this as: AimPL: Chromatic homotopy theory and p-adic geometry, available at http://aimpl.org/chromhomotopy.