Low-dimensional surgery and the Yamabe invariant
Geometric Topology
2015-01-28 v3 Differential Geometry
Abstract
Assume that M is a compact n-dimensional manifold and that N is obtained by surgery along a k-dimensional sphere, k\le n-3. The smooth Yamabe invariants \sigma(M) and \sigma(N) satisfy \sigma(N)\ge min (\sigma(M),\Lambda) for \Lambda>0. We derive explicit lower bounds for \Lambda in dimensions where previous methods failed, namely for (n,k)\in {(4,1),(5,1),(5,2),(6,3),(9,1),(10,1)}. With methods from surgery theory and bordism theory several gap phenomena for smooth Yamabe invariants can be deduced.
Cite
@article{arxiv.1204.1197,
title = {Low-dimensional surgery and the Yamabe invariant},
author = {Bernd Ammann and Mattias Dahl and Emmanuel Humbert},
journal= {arXiv preprint arXiv:1204.1197},
year = {2015}
}
Comments
Version 2 contains new results: the case (n,k)=(6,3) is now solved, Version 3: typos corrected, final version to appear in J. Math. Soc. Japan