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4. Norms and the slice filtration

    1. The associated graded of the localized slice spectral sequence tower

          The localized slice spectral sequence gives a way to break up the slice spectral sequence into pieces that see phenomena only on certain geometric fixed points. For the group C_{2^n}, there is a localized slice spectral sequence for each subgroup.

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      Problem 4.1.

      [Mark Behrens] For the group C_{2^n} one understand more easily the fibres between the localized slice spectral sequence for two adjacent subgroups H and H'?

        • The image of J in equivariant truncated Brown–Peterson spectra


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          Problem 4.2.

          [Hood Chatham] What is the image of the J homomorphism in the fixed points of \mathrm{BP}^{((G))}\langle m\rangle?

            • The Hurewicz image of equivariant truncated Brown–Peterson spectra


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              Problem 4.3.

              [Mark Behrens] What is the Hurewicz image of the truncated Brown–Peterson spectrum \mathrm{BP}^{((G))}\langle m \rangle?

                • Understanding n-slices

                      [arXiv:1711.03472] gives a description of n-slices, the objects that appear in the associated graded of the slice filtration.

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                  Problem 4.4.

                  [Bert Guillou] Is there a way to get a good computational grasp of n-slices? For example understanding their Postnikov towers, homotopy groups etc.

                    • Heights of slice differentials

                          The differentials in the slice spectral sequence for \mathrm{BP}^{((G))}\langle n \rangle look like they are stratified by height.

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                      Problem 4.5.

                      [XiaoLin Danny Shi] Is it possible to make sense of the height of a differential in the slice spectral sequence of \mathrm{BP}^{((G))}\langle n \rangle so that it has a well defined height?

                        • The G\cdot \overline{v}_n-Bockstein spectral sequence for \mathrm{BP}^{((G))}\langle n\rangle

                              \mathrm{BP}^{((G))}\langle n-1\rangle is the quotient of \mathrm{BP}^{((G))}\langle n\rangle by G\cdot \overline{v}_n.

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                          Problem 4.6.

                          [Mark Behrens] Can one understand the G\cdot \overline{v}_n-Bockstein spectral sequence for this quotient map?

                            • Connecting slice differentials via power operations

                                  We would like a systematic way of understanding slice differentials in the slice spectral sequence for \mathrm{BP}\langle n \rangle^{((G))}.

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                              Problem 4.7.

                              [XiaoLin Danny Shi] Are there power operations connecting differentials in the slice spectral sequence for \mathrm{BP}\langle n \rangle^{((G))}
                                  Andy Senger has thought about this problem, you probably should talk to him if you are interested.

                                • Relations in \mathrm{BP}^{((G))}


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                                  Problem 4.8.

                                  [Mike Hill] Is 8\sigma = 0 in the fixed points of \mathrm{BP}^{((G))}?

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