We give a simple criterion for whether a pointwise curvature condition is 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 toThis is used to generalize the well-known classification result of positive scalar curvature in the simply-connected case in the following way: Any simplyconnected manifold M n , n ≥ 5, which is either spin with vanishing α-invariant or else is non-spin admits for any ǫ > 0 a metric such that the curvature operator satisfies R > −ǫ R .Here, R S c−1 ×R n−c+1 = π 2 R c−1 : 2 R n → 2 R n corresponds to the curvature operator of S c−1 × R n−c+1 equipped with its canoncial product metric.

No abstract