Surgery stable curvature conditions

2013/03/26 by Hoelzel, Sebastian
#53Cxx #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1303.6531

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 × \reals^n-c+1, c ≥ 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 Mⁿ, n ≥ 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\normR.

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