  We give a simple criterion for a pointwise curvature condition to be stable
under surgery. Namely, a curvature condition $C$, which is understood to be an
open, convex, O(n)-invariant cone in the space of algebraic curvature
operators, is stable under surgeries of codimension at least $c$ provided it
contains the curvature operator corresponding to $S^{c-1} \times
\reals^{n-c+1}$, $c \geq 3$.
  This is used to generalize the well-known classification result of positive
scalar curvature in the simply-connected case in the following way: Any
simply-connected manifold $M^n$, $n \geq 5$, which is either spin with
vanishing $\alpha$-invariant or else is non-spin admits for any $\epsilon > 0$
a metric such that the curvature operator satisfies $R > - \epsilon \norm{R}$.
