# The spinorial τ-invariant and 0-dimensional surgery by Bernd Ammann, Emmanuel Humbert

The spinorial τ-invariant and 0-dimensional surgery (.dvi, .ps,.ps.gz or .pdf)
J. reine angew. Math. 624, 27-50 (2008)
DOI 10.1515/CRELLE.2008.079

### Abstract

Let M be a compact manifold with a metric g and with a fixed spin structure χ. Let λ1+(g) be the first non-negative eigenvalue of the Dirac operator on (M,g,χ). We set
τ(M,χ):= \sup \inf λ1+(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##) be obtained from (M,χ) by 0-dimensional surgery. We prove that

τ(M##)\geq τ(M,χ)

As a corollary we can calculate τ(M,χ) for any Riemann surface M.

### Mathematics Subject Classification

53C27 (Primary) 58J05, 57R65 (Secondary)
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The Paper was written on 27.7.2006
Last update 27.7.2006