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The Exotic $K(2)$-Local Picard Group at the Prime $2$

Agnes Beaudry, Irina Bobkova, Paul G. Goerss, Hans-Werner Henn, Viet-Cuong Pham, Vesna Stojanoska

Abstract

We calculate the group $κ_2$ of exotic elements in the $K(2)$-local Picard group at the prime $2$ and find it is a group of order $2^9$ isomorphic to $(\mathbb{Z}/8)^2 \times (\mathbb{Z}/2)^3$. In order to do this we must define and exploit a variety of different ways of constructing elements in the Picard group, and this requires a significant exploration of the theory. The most innovative technique, which so far has worked best at the prime $2$, is the use of a $J$-homomorphism from the group of real representations of finite quotients of the Morava stabilizer group to the $K(n)$-local Picard group.

The Exotic $K(2)$-Local Picard Group at the Prime $2$

Abstract

We calculate the group of exotic elements in the -local Picard group at the prime and find it is a group of order isomorphic to . In order to do this we must define and exploit a variety of different ways of constructing elements in the Picard group, and this requires a significant exploration of the theory. The most innovative technique, which so far has worked best at the prime , is the use of a -homomorphism from the group of real representations of finite quotients of the Morava stabilizer group to the -local Picard group.
Paper Structure (29 sections, 77 theorems, 384 equations, 5 figures)

This paper contains 29 sections, 77 theorems, 384 equations, 5 figures.

Table of Contents

  1. Introduction
  2. $K(n)$-Local Results for General $n$ and $p$
  3. The $K(n)$-local category
  4. Orientations and Filtrations
  5. Orientations
  6. The naturality of orientations
  7. The descent filtration
  8. The $\mathbb{G}_n^1$-orientable elements of the Picard group
  9. The determinant, the subgroup $\mathbb{G}_n^1$ and the class $\zeta$
  10. A description of $\kappa(\mathbb{G}_n^1)$ in terms of homotopy groups
  11. A comparison of filtrations
  12. The $J$-construction
  13. Truncating spectral sequences
  14. $K(2)$-Local Computations at $p=2$
  15. The subgroup filtration of $\kappa_2$
  16. ...and 14 more sections

Key Result

Proposition 3.1

Let $X \in \kappa_n$ and let $\iota_X \in E_0X$ be a $\mathbb{G}_n$-invariant generator. Then $X$ is in $\kappa_n(K)$ if and only if $\iota_X$ is a permanent cycle in the homotopy fixed point spectral sequence

Figures (5)

  • Figure 1: The $E_3$ and $E_5$-pages of the homotopy fixed point spectral sequences $H^s(\mathbb{G}_2^1, E_t) \Rightarrow \pi_{t-s}E^{h\mathbb{G}_2^1}$ (top) and $H^s(\mathbb{G}_2, E_t) \Rightarrow \pi_{t-s}E^{h\mathbb{G}_2}$ (bottom). Here, $\blacksquare=\mathbb{Z}_{2}$, $\bullet=\mathbb{Z}/2$, $+=\mathbb{Z}/4$ and $\times=\mathbb{Z}/8$. If, instead, we let $\blacksquare =\mathbb{W}$, $\bullet=\mathbb{W}/2$, $+=\mathbb{W}/4$ and $\times=\mathbb{W}/8$, then this is the $E_3$ and $E_5$-pages for $H^s(\mathbb{S}_2^1, E_t) \Rightarrow \pi_{t-s}E^{h\mathbb{S}_2^1}$ (top) and $H^s(\mathbb{S}_2, E_t) \Rightarrow \pi_{t-s}E^{h\mathbb{S}_2}$ (bottom). These computations were done in BBGHPScoh.
  • Figure 2: The $E_2$-page of the Topological Duality spectral sequence for $S^0$. The vertical axis is the Topological Duality spectral sequence filtration $s$ and the horizontal axis is $t-s$. Further, $\blacksquare = \mathbb{W}$, $\bullet=\mathbb{F}_4$, and the circled bullet represents $E_0^{3,3} \cong \mathbb{W}/8$. Horizontal lines are $\eta$-multiplications and curved lines are $\nu$-multiplications.
  • Figure 3: The cohomology ring $H^s(C_2,\pi_t S^0)$ in a range. The vertical axis is $s$ and the horizontal axis is $t-s$.
  • Figure 4: The spectral sequence for the cofiber sequence of towers ${{\mathbf{X}}}^7_3 {\longrightarrow} {{\mathbf{X}}}_{\leq 7} {\longrightarrow} {{\mathbf{X}}}_{\leq 2}$. Classes in blue are in the spectral sequence for ${{\mathbf{X}}}_{\leq 2}$, classes in black are in that of ${{\mathbf{X}}}^7_3$. The classes combined give the $E_2$-page of the spectral sequence of ${{\mathbf{X}}}_{\leq 7}$. A $\blacksquare$ is a copy of $\mathbb{Z}_{2}$, a $\bullet$ is a $\mathbb{Z}/2$. The circled $\bullet$ represent $\mathbb{Z}/8$. We have only drawn one differential in the spectral sequence of ${{\mathbf{X}}}_{\leq 7}$.
  • Figure 5: The homotopy fixed point spectral sequence \ref{['eq:tmf-hfpss']} for $\pi_*E^{hG_{48}}$ in a small range. The $x$-axis is $t-s$ and the $y$-axis is $s$. The circles represent $j{\mathbb{F}}_2[[j]]$ and bullets represent ${\mathbb{F}}_2$. $n$ bullets connected by a vertical line represent $\mathbb{Z}/{2^n}$. Lines of slope 1 and 1/3 represent multiplication by $\eta$ and $\nu$, respectively.

Theorems & Definitions (190)

  • Remark 2.1
  • Definition 3.1
  • Remark 3.1
  • Definition 3.2
  • Remark 3.2
  • Remark 3.3: Subgroup filtrations
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 180 more