Surgery stable curvature conditions

Abstract

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, ${{\mathrm{O}}}(n)$-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 $S^{c-1} \times \mathbb {R}^{n-c+1}$, $c \ge 3$. 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 $M^n$, $n \ge 5$, 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 $R > - \epsilon \left\| R\right\|$.