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2. Computations in algebraic K-theory

    1. The motivic filtration on the algebraic $K$-theory of the sphere spectrum

          Due to work of Rognes, Blumberg–Mandell, and others, we understand the algebraic K-theory of the sphere spectrum fairly well. Recent work of Hahn–Raksit–Wilson [arXiv:2206.11208] gives a motivic filtration on TC of $\mathbb{E}_{\infty}$-rings which is closely related to the motivic filtration on algebraic K-theory in motivic homotopy theory.

      Add remark

      Problem 2.1.

      [Jeremy Hahn] Can one understand the motivic filtration (or motivic spectral sequence) on the $\mathrm{TC}$ of the sphere spectrum? What about for the motivic filtration on related invariants such as $\mathrm{THH}$, $\mathrm{TP}$?

        • The $K$-theory of Johnson–Wilson theory

              The algebraic $K$-theory of Johnson–Wilson theory $E(n)$ is closely related to the topological cyclic homology of the connective cover of the two periodic Johnson–Wilson theory $E'(n)$.

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          Problem 2.2.

          [Ishan Levy] Compute something about $\mathrm{TC}$ (and related invariants such as $\mathrm{TP}, \mathrm{THH}$) of $\tau_{\geq0}E'(n)$, where $E'(n)$ is the $2$-periodic Johnson–Wilson theory.

            • Detecting Greek letter families in algebraic $K$-theory


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              Problem 2.3.

              [Maxwell Johnson] Prove Conjecture 1.1 of [arXiv:1810.10088]. This is a version of redshift for the Greek letter family in the homotopy groups of spheres: it asks that the $K$-theory of an $\mathbb{E}_{\infty}$-ring detecting the $\alpha^{(n)}$-family detects the $\alpha^{(n+1)}$-family.

                • Atiyah–Segal completion thoerem for iterated $K$-theory


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                  Problem 2.4.

                  [Mark Behrens] Can one prove an Atiyah–Segal completion theorem for the algebraic $K$-theory of topological $K$-theory?

                    • Motivic filtered Poitou–Tate duality


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                      Problem 2.5.

                      [Jeremy Hahn] Is there a duality at the level of motivic filtered spectra $\mathrm{fil}_{\text{mot}}\mathrm{TC}(\mathcal{O}_K)$ that recovers Poitou–Tate duality at the level of associated graded?

                        • Devissage for equivariant/hermitian $K$-theory

                              We understand devissage for the $K$-theory of stable categories fairly well now due to work of Quillen, Barwick, and Burklund–Levy (see [arXiv:2112.14723]). See section 2.2 of [arXiv:2009.07225] for what is known for hermitian $K$-theory.

                          Add remark

                          Problem 2.6.

                          [Ishan Levy] What is the optimal devissage result for equivariant/hermitian $K$-theory?

                            • Motivic filtration on $\mathrm{TR}$

                                  [arXiv:2206.11208] gives a motivic filtration on $\mathrm{THH}, \mathrm{TC}, \mathrm{TC^{-}}, \mathrm{TP}$ of ring spectra generalizing previously defined motivic filtrations such as in [arXiv:1802.03261]. In remark 1.13 of loc. cit. it is mentioned that their techniques can be made to additionally define a motivic filtration on $\mathrm{TR}$ of discrete rings.

                              Add remark

                              Problem 2.7.

                              [Dylan Wilson] Can one define a well behaved motivic filtration on the $\mathrm{TR}$ of ring spectra? Can one compute it in good cases?

                                • Calculation of $\mathrm{TP}(\mathbb{Z})$

                                      Bokstedt and Madsen computed $\mathrm{TC}(\mathbb{Z})$, without calculating $\mathrm{TP}(\mathbb{Z})$. However, Hesselholt and Madsen calculated (in [MR1317575]) $(L_{K(1)} \mathbb{S})^{tS^1}$. Their results suggest the following:

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                                  Conjecture 2.8.

                                  [Sanath Devalapurkar] Fix a prime $p$. Let $j = \tau_{\geq 0} L_{K(1)} \mathbb{S}$. Then, $\mathrm{TP}(\mathbb{Z}_p) \simeq j^{tS^1}$, for the trivial $S^1$-action on $j$.
                                    1. Remark. [Sanath Devalapurkar] This would follow if, for instance, $\mathrm{TR}(\mathbb{Z}_p) \simeq j$.

                                        • $L$-theory of integers

                                              The symmetric and normal $L$-theory of the integers was computed in [arXiv:2004.06889], and the Tate cohomology of $Q_8$ was computed by Atiyah in [MR0148722]. The normal $L$-theory of $\mathbb{Z}$ looks very similar to the Tate cohomology of $Q_8$; similarly, the symmetric $L$-theory of $\mathbb{Z}$ looks very similar to the Tate cohomology of $\mathrm{Pin}(2)$.

                                          Add remark

                                          Conjecture 2.9.

                                          [Sanath Devalapurkar] Equip $\mathrm{Pin}(2)$ (resp. $Q_8$) with the canonical $\mathbb{Z}/2$-action (resp. the action given by conjugation by $i$). Then, the $E_\infty$-map $L^s(\mathbb{Z})_{(2)} \to L^n(\mathbb{Z})$ can be identified by taking strict $\mathbb{Z}/2$-fixed points of the map $\mathbb{Z}_{(2)}^{t\mathrm{Pin}(2)} \to \mathbb{Z}^{tQ_8}$. (This map is induced by the inclusion $Q_8\subseteq \mathrm{Pin}(2)$.)

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