Properties and Examples of $A$-Landweber Exact Spectra

TL;DR

This paper extends the classical Landweber exactness concept to $A$-Landweber spectra for compact abelian Lie groups, demonstrating that such spectra also lack nontrivial phantom maps and providing counterexamples to a conjecture of May.

Contribution

It introduces the notion of $A$-Landweber exact spectra and proves an analogous phantom map result, along with showing that May's conjecture on $KU_G$ is false.

Findings

$A$-Landweber spectra admit no nontrivial phantom maps.

Counterexample to May's conjecture on $KU_G$.

Results apply to $MU_A$, $KU_A$, their $p$-localizations, and $BP_A$.

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.