Surgery and the spinorial tau-invariant

Abstract

We associate to a compact spin manifold M a real-valued invariant τ(M) by taking the supremum over all conformal classes over 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 $\sigma$-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 $\tau (N)\ge \min \{\tau (M),{\Lambda }_n\}$ with ${\Lambda }_n>0$. Various topological conclusions can be drawn, in particular that τis a spin-bordism invariant below ${\Lambda }_n$. Below ${\Lambda }_n$, the values of $\tau$ cannot accumulate from above when varied over all manifolds of a fixed dimension.

to appear in CPDE

References in corpus (3)

• Surgery and Harmonic Spinors
• The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions
• The spinorial τ-invariant and 0-dimensional surgery

Cited by in corpus (2)

• Smooth Yamabe invariant and surgery
• Relations between threshold constants for Yamabe type bordism invariants