Small eigenvalues of the Dirac operator, Surgeries and Bordism Theory

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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.