  We associate to a compact spin manifold M a real-valued invariant \tau(M) by
taking the supremum over all conformal classes over the infimum inside each
conformal class of the first positive Dirac eigenvalue, normalized to volume 1.
This invariant is a spinorial analogue of Schoen's $\sigma$-constant, also
known as the smooth Yamabe number. We prove that if N is obtained from M by
surgery of codimension at least 2, then $\tau(N) \geq
\min\{\tau(M),\Lambda_n\}$ with $\Lambda_n>0$. Various topological conclusions
can be drawn, in particular that \tau is a spin-bordism invariant below
$\Lambda_n$. Below $\Lambda_n$, the values of $\tau$ cannot accumulate from
above when varied over all manifolds of a fixed dimension.
