The spinorial τ-invariant and 0-dimensional surgery

Abstract

Let M be a compact manifold with a metric g and with a fixed spin structure χ. Let be the first non-negative eigenvalue of the Dirac operator on (M,g,χ).

We set

where the infimum runs over all metrics g of volume 1 in a conformal class [g0] on M and where the supremum runs over all conformal classes [g0] on M.

Let be obtained from (M,χ) by 0-dimensional surgery. We prove that

.

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

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The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of Crelle’s Journal. In the 190 years of its existence, Crelle’s Journal has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals.

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