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4. Extensions to Profinite, Compact Lie Groups, etc.

    1.     If $G$ is the Galois group of a finite extension $L \rightarrow K$ there is a functor $Sp_G \rightarrow Sp^{Mot}_K$ given by $\Sigma^{\infty} G/H_+ \rightarrow \Sigma^{\infty} Spec(L^H)_+$. Moreover, if $K$ is a real closed field, this functor is fully faithful.

      Problem 4.1.

      [K. Ormsby] If $G$ is a profinite absolute Galois group of $K$, is there a theory of $G-spectra$ such that $Sp_{G} \rightarrow Sp^{Mot}_K$ is “rich”?
        • Problem 4.2.

          [M. Behrens] For a profinite group $G$ (having subgroups of arbitrary index), are there Wirthmüller/Adams isomomorphisms in $Sp^G$?
            •     In genuine $G$-spectra we have multiplicative norm and transfer maps when $G$ has finitely indexed subgroups. Furthermore, (via Ando-Morava) there are objects that look like what may arise from a multiplicative norm for compact Lie groups.

              Problem 4.3.

              [C. Rezk] Are there analogues of multiplicative norms or multiplicative transfers for compact Lie groups?
                •     Quillen’s theorem for $MU$ states that the map from the Lazard ring $L$ to $\pi_* MU$ is an isomorphism and so, in particular, the homotopy of $MU$ is concentrated in even degrees. Furthermore, when $G$ is a finite abelian group, it is known that the $G$-equivariant homotopy of $MU$ is also even.

                  Problem 4.4.

                  [J. Greenlees] For any finite or compact Lie group $G$, is it also the case that $\pi^G_*(MU)$ is even?
                    • Long Term

                      Problem 4.5.

                      [A. Blumberg] How should we think about equivariant stable homotopy theory in the context of infinite compact Lie groups?
                        • Problem 4.6.

                          [L. Hesselholt] What can we say about equivariant stable homotopy theory for infinite discrete groups?

                              Cite this as: AimPL: Equivariant derived algebraic geometry, available at http://aimpl.org/equideralggeom.