Surgery and the Spinorial Τ-invariant

We associate to a compact spin manifoldM a real-valued invariant τ(M) by taking the supremum over all conformal classes of the infimum inside each conformal class of the first positive Dirac eigenvalue, when the metrics are normalized to unit volume. This invariant is a spinorial analogue of Schoen’s σ-constant, also known as the smooth Yamabe invariant. We prove that if N is obtained from M by surgery of codimension at least 2 then τ(N) ≥ min{τ(M),Λn}, where Λn is a positive constant depending only on n = dimM . Various topological conclusions can be drawn, in particular that τ is a spin-bordism invariant below Λn. Also, below Λn the values of τ cannot accumulate from above when varied over all manifolds of dimension n.