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\begin{center}{\huge Martin's Invariance Conjecture 
and Low Sets}\end{center}
\medskip
\begin{center}{\large Leo Harrington (University of 
California, Berkeley)\\
and\\
Robert I. Soare (University of Chicago)}\end{center}
\bigskip

  A class of degrees of computably enumerable (c.e.) sets is {\em
    invariant} if it is the set of degrees of some class of c.e. sets
  invariant under automorphisms of $\mathcal{E}$, the c.e. sets under
  inclusion.  For example, Martin proved that \high{1}, the high$_1$
  degrees, are invariant, being the degrees of maximal sets.
  Lachlan and Shoenfield proved that \lowbar{2}, the complement of the
  low$_2$ degrees are invariant, being the degrees of atomless
  sets.  These results and his work at the time on projective
  determinacy led Martin to make the following conjecture.
\medskip

\noindent
{\bf Conjecture 1.1 (Martin's Invariance Conjecture)} {\em 
Among all the 
jump classes $\high{n}$ and $\low{n}$ for $n>0$, and their complements
\highbar{n} and \lowbar{n}, the invariant classes are exactly
\high{2n-1} and \lowbar{2n}.}
\medskip

After some previous results in favor of Martin's conjecture,
researchers looked particularly at \lowbar{1} because of the important
role played by low$_1$ sets in the subject and some attempted
unsuccessfully to find a property defining \lowbar{1} analagous to
\lowbar{2}.  Harrington and Soare now prove noninvariance of
\lowbar{1}, and supply other evidence for the Martin conjecture.
\medskip

\noindent
{\bf Theorem 1.2 
(Harrington and Soare)} 
{\em There is a low$_2$ nonlow$_1$ c.e.\ set $D$ such that every c.e.\ set
$A\leq_{\rm T} D$ is automorphic to a low$_1$ set $B$.
Hence, $\lowbar{1}$ is noninvariant.}

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