Low-dimensional surgery and the Yamabe invariant

Abstract

Assume that M is a compact n-dimensional manifold and that N is obtained by surgery along a k-dimensional sphere, k ≤ n − 3. The smooth Yamabe invariants σ(M) and σ(N) satisfy σ(N) ≥ min(σ(M), Λ) for a constant Λ > 0 depending only on n and k. We derive explicit positive lower bounds for Λ in dimensions where previous methods failed, namely for (n,k) ∈ {(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.