Low-dimensional surgery and the Yamabe invariant

TL;DR

This paper establishes explicit lower bounds for the Yamabe invariant after low-dimensional surgeries on manifolds, advancing understanding in cases where previous methods were ineffective.

Contribution

It derives new explicit lower bounds for the Yamabe invariant in specific low-dimensional surgery cases using surgery and bordism theory.

Findings

Explicit lower bounds for  in certain dimensions

Identification of gap phenomena for Yamabe invariants

Extension of methods to previously inaccessible cases

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.