# Surgery and the spinorial τ-invariant

by

Bernd Ammann, Mattias Dahl, Emmanuel Humbert

**Surgery and the spinorial τ-invariant**
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*Comm. Part. Diff. Eq.* **34**, * 1147-1179* (2009)

We associate to a compact spin manifold M 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, normalized to volume 1. This invariant is a spinorial analogue of Schoen's σ-constant, also known as the smooth Yamabe number.

We prove that if N is obtained from M by surgery of codimension at least 2, then τ(N) ≥ min {τ(M),Λ_{n}} with Λ_{n}>0.

Various topological conclusions can be drawn, in particular that τ is a spin-bordism invariant below Λ_{n}. Below Λ_{n}, the values of τ cannot accumulate from above when varied over all manifolds of a fixed dimension.
53C27 (Primary) 55N22, 57R65 (Secondary)

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The Paper was written on 2.11.2007

Last update 7.11.2007