Low-dimensional surgery and the Yamabe invariant

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 σ(M) and σ(N) satisfy σ(N)\ge min (σ(M),Λ) for Λ>0. We derive explicit lower bounds for Λin 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.

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