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, 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 $R > - ε\norm{R}$.

Cited by in corpus (2)

• Positive isotropic curvature and self-duality in dimension 4
• Path-connectedness of the moduli spaces of metrics with positive isotropic curvature on four-manifolds