Surgery stable curvature conditions
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 $α$-invariant or else is non-spin admits for any $ε> 0$ a metric such that the curvature operator satisfies $R > - ε\norm{R}$.