Surgery and Harmonic Spinors
  Let M be a compact manifold with a fixed spin structure \chi. The Atiyah-Singer index theorem implies that for any metric g on M the dimension of the kernel of the Dirac operator is bounded from below by a topological quantity depending only on M and \chi. We show that for generic metrics on M this bound is attained. 
Bernd Ammann, Mattias Dahl, Emmanuel Humbert