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Duals of higher real $K$-theories at $p=2$

Juan C. Moreno Del Angel

Abstract

We study $\mathrm{K}(h)$-local Spanier-Whitehead duality for $C_{2^n}$-equivariant Lubin-Tate spectra, $E_h$, at the prime $2$ and heights $h$ divisible by $2^{n-1}$. We determine a $C_{2^n}$-equivariant equivalence $DE_h\simeqΣ^{-V_h} E_h$, for an explicit $C_{2^n}$-representation, $V_h$. We then study the $\mathrm{RO}(C_{2^n})$-periodicities of $E_h$ at some low heights. With these ingredients, we determine the self-duality of some higher real $K$-theories up to a specified suspension shift, at some low-heights. In particular, we show that $DE_4^{hC_8}\simeq Σ^{112}E_4^{hC_8}$.

Duals of higher real $K$-theories at $p=2$

Abstract

We study -local Spanier-Whitehead duality for -equivariant Lubin-Tate spectra, , at the prime and heights divisible by . We determine a -equivariant equivalence , for an explicit -representation, . We then study the -periodicities of at some low heights. With these ingredients, we determine the self-duality of some higher real -theories up to a specified suspension shift, at some low-heights. In particular, we show that .

Paper Structure

This paper contains 16 sections, 55 theorems, 224 equations, 1 figure.

Table of Contents

  1. Introduction
  2. An equivariant dual of Lubin-Tate spectra
  3. Some conditions on Lubin-Tate spectra
  4. Dualizing a pro-Galois extension
  5. Dualizing Lubin-Tate spectra
  6. The dualizing $C_{2^n}$-representation
  7. The Real bordism theories
  8. Real bordism Lubin-Tate spectra
  9. The $C_{2^n}$-orientation
  10. Equivariant periodicities and integer shifts
  11. $\text{RO}(G)$-periodicities and where to find them
  12. Orientation classes for Lubin-Tate spectra
  13. A repository of periodicities for Lubin-Tate spectra
  14. The integer shifts
  15. Conclusions
  16. ...and 1 more sections

Key Result

Theorem A

Let $\mathbb{G}_h$ be the Morava stabilizer group at the prime $2$ and height $h=2^{n-1}m$, for $m$ odd. Let $V_h$ be the $p$-adic lie algebra of $\mathbb{G}_h$, viewed as a $C_{2^n}$-representation by the adjoint action of $C_{2^n}\leq\mathcal{O}^\times_h\leq \mathbb{G}_h$. Then $V_h$ is isomorphic

Figures (1)

  • Figure 1: Examples where $E_h$ at $p=2$ (left) and odd $p$ (right) satisfies $DE_h\simeq_G\Sigma^{s_h}E_h$.

Theorems & Definitions (131)

  • Theorem A: \ref{['thedualizingrep']}
  • Theorem B: \ref{['C4shifts']}
  • Corollary B: \ref{['C4shiftshfpcor']}
  • Remark 1.1
  • Theorem C: \ref{['height4p', 'C8shift']}
  • Corollary C: \ref{['C8shiftshfpcor']}
  • Theorem D: \ref{['hmtpyClosedEthy']}
  • Definition 2.1.1
  • Proposition 2.1.2
  • Remark 2.1.3
  • ...and 121 more