Low$_2$ computably enumerable sets have hyperhypersimple supersets

Abstract: A longstanding question is to characterize the lattice of supersets (modulo finite sets), $L<sup>*(A)$, of a low$_2$ computably enumerable (c.e.) set. The conjecture is that $L^*(A)\cong {E}<sup>*$ the lattice of all c.e. sets. In spite of claims in the literature, this longstanding question/conjecture remains open. We contribute to this problem by solving one of the main test cases. We show that if c.e. $A$ is low$_2$ then $A$ has an atomless, hyperhypersimple superset.