Small eigenvalues of the Dirac operator, Surgeries and Bordism Theory

In the project we want to study several topics about small eigenvalues of the Dirac operator and its connections to bordism theory. We will study the following topics: (1) Bordism theory for manifolds without harmonic spinors: Harmonic spinors are spinors to the eigenvalue 0. This eigenvalue plays a very particular role, mainly due to its topological significance from Atiyah-Singer index theory. We want to extend recent progress on such questions in order to obtain a bordism theory for manifolds without harmonic spinors. (2) Equivariant harmonic L2-spinors.1 We study harmonic L2-spinors and Fredholmness of the Dirac operator on possibly non-compact coverings of compact manifolds. (3) Prescribing small eigenvalues of the Dirac operator using surgery techniques. (4) ó-type invariants of conformally covariant operators and its invariance under bordisms. (5) Manifolds with boundaries. (6) Relations to general relativity. For all these subjects bordisms and surgeries play a central role for proving the existence of metrics with certain properties.