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 $$R > - \epsilon \left\| R\right\| $$ .