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Properties and Examples of $A$-Landweber Exact Spectra

Noah Wisdom

Abstract

It is classically known that Landweber exact homology theories (complex oriented theories which are completely determined by complex cobordism) admit no nontrivial phantom maps. Herein we propose a definition of $A$-Landweber exact spectra, for $A$ a compact abelian Lie group, and show that an analogous result on phantom maps holds. Also, we show that a conjecture of May on $KU_G$ is false. We do not prove an equivariant Landweber exact functor theorem, and therefore our result on phantom maps only applies to $MU_A$, $KU_A$, their $p$-localizations, and $BP_A$, which are shown to be $A$-Landweber exact by ad-hoc methods.

Properties and Examples of $A$-Landweber Exact Spectra

Abstract

It is classically known that Landweber exact homology theories (complex oriented theories which are completely determined by complex cobordism) admit no nontrivial phantom maps. Herein we propose a definition of -Landweber exact spectra, for a compact abelian Lie group, and show that an analogous result on phantom maps holds. Also, we show that a conjecture of May on is false. We do not prove an equivariant Landweber exact functor theorem, and therefore our result on phantom maps only applies to , , their -localizations, and , which are shown to be -Landweber exact by ad-hoc methods.
Paper Structure (15 sections, 31 theorems, 35 equations)

This paper contains 15 sections, 31 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Equivariant Landweber exactness
  3. Review of global methods
  4. Weakly global cohomology theories with Thom classes
  5. $KU_A$ is $A$-Landweber exact
  6. Observations on $A$-Landweber exact spectra
  7. Properties of $KU_A$
  8. $BP_A$ and a lift of Quillen's idempotent
  9. $p$-typicality for ordinary formal group laws
  10. $p$-typicality for equivariant formal group laws
  11. Properties of $BP_A$
  12. Phantom maps
  13. Reduction to complex generated $A$-spectra
  14. Phantom maps for complex generated $A$-spectra
  15. Acknowledgments

Key Result

Theorem 2.3

$\underline{\pi}_*(\mathbf{MU})$ is an initial object in the category of global group laws. Furthermore $MU_A^*$ is isomorphic to the $A$-equivariant Lazard ring $L_A$.

Theorems & Definitions (68)

  • Definition 2.1: Hausmann Hau22
  • Definition 2.2: Hausmann Hau22
  • Theorem 2.3: Hausmann Hau22
  • Corollary 2.4
  • Definition 2.5: Okonek Definition 1.2 Oko82
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Proposition 2.8
  • proof
  • ...and 58 more