The spinorial τ-invariant and 0-dimensional surgery

Let $M$ be a compact manifold with a metric $g$ and with a fixed spin structure $χ$. Let $λ\_1^+(g)$ be the first non-negative eigenvalue of the Dirac operator on $(M,g,χ)$. We set $τ(M,χ):= \sup \inf λ\_1^+(g)$ where the infimum runs over all metrics $g$ of volume 1 in a conformal class $[g\_0]$ on $M$ and where the supremum runs over all conformal classes $[g\_0]$ on $M$. Let $(M^#,χ^#)$ be obtained from $(M,χ)$ by 0-dimensional surgery. We prove that $$τ(M^#,χ^#)\geq τ(M,χ).$$ As a corollary we can calculate $τ(M,χ)$ for any Riemann surface $M$.

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