Surgery and the spinorial tau-invariant

TL;DR

This paper introduces a new spinorial invariant for compact spin manifolds, analogous to the Yamabe number, and studies its behavior under surgeries, establishing lower bounds and invariance properties.

Contribution

It defines the au-invariant based on the first positive Dirac eigenvalue and proves its stability under surgeries of codimension at least 2, revealing topological invariance below a positive threshold.

Findings

au is a spin-bordism invariant below \\Lambda_n.

au(N) \\geq \\min\\\{ au(M),\\Lambda_n\\\} for surgeries of codimension \\geq 2.

Values of \\tau can only accumulate from above below \\Lambda_n.

Abstract

We associate to a compact spin manifold M a real-valued invariant \tau(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 $\tau(N) \geq \min\{\tau(M),\Lambda_n\}$ with $\Lambda_n>0$. Various topological conclusions can be drawn, in particular that \tau is 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.