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, $k\leq n-3$. The smooth Yamabe invariants $\sigma(M)$ and $\sigma(N)$ satisfy $\sigma(N)\geq \min (\sigma(M),\Lambda)$ for a constant $\Lambda$ > 0 depending only on $n$ and $k$. We derive explicit positive 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.