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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.