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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.