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3. Motivic, equivariant, and synthetic spectra

    1. Odd primary analog of Wood cofibre sequences

          The Wood cofibre sequence is the cofibre sequence $\Sigma \mathrm{KO} \xrightarrow{\eta} \mathrm{KO} \to \mathrm{KU}$ (and similarly for the connective versions). There is also a version relating $\mathrm{BP}_{\mathbb{R}}\langle n \rangle^{C_2}$ to $\mathrm{BP}_{\mathbb{R}}\langle n \rangle$.

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      Problem 3.1.

      [Hood Chatham] What are odd primary analogs of the Wood cofibre sequence? For example, one might expect a $p$-stage filtration on $E_{p-1}$ with associated graded $E_{p-1}^{hC_p}$. Is there an analog for $E_{(p-1)p^{n-1}m}$?

        • a inverted Artin–Tate $\mathbb{R}$-motives

              The category of Artin–Tate $\mathbb{R}$-motives has a topological model as a $1$-parameter deformation of $C_2$-equivariant homotopy theory [arXiv:2010.10325]. Let $\Sp_{\mathbb{R}}^{AT}$ denote this category.

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          Problem 3.2.

          [Mark Behrens] What category do you get when you invert the class $a$ in $\Sp_{\mathbb{R}}^{AT}$?

            • Mahowald invariants increase chromatic height


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              Problem 3.3.

              [Mark Behrens] Given a $v_n$-periodic family in the stable homotopy groups of spheres, does its Mahowald invariant give rise to an (eventually) $v_{n+1}$-periodic family?
                  There is an equivariant formulation of the Mahowald invariant.

                • Equivariant and synthetic


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                  Problem 3.4.

                  [Mark Behrens] Can one profitably mix synthetic spectra with equivariant homotopy theory?
                    1. Remark. [Sanath Devalapurkar] Along these lines, what is the $\mathbf{Z}/2$-equivariant analogue of the even filtration (where $\tau_{\geq 2\ast}$ is replaced by $\tau_{\geq \rho\ast}$, and evenness is replaced by slice-even)?

                        • Motivic modular forms (mmf)


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                          Problem 3.5.

                          [Bert Guillou] Is there a connective version of motivic modular forms, which should be named $\mathrm{mmf}$, and is a lift of $\mathrm{tmf}$?

                              Cite this as: AimPL: Equivariant techniques in stable homotopy theory, available at http://aimpl.org/equivstable.