Preprint
Inserted: 23 oct 2026
Last Updated: 23 oct 2026
Pages: 126
Year: 2025
Doi: 10.5281/zenodo.23546912
Notes:
Version 3, updated 21 October 2026. First published 24 November 2025. MSC 2020: 58J50, 53C27, 58J20, 58D17, 57R65.
Links: Zenodo — version 3
Abstract:
Let $M$ be a closed connected spin manifold of dimension $n\ge3$. We prescribe arbitrary admissible finite Dirac spectra in bounded open symmetric intervals, including multiplicities and nullity. Kernel dimensions are unbounded on each $M$. Nonzero eigenvalues generically have the least multiplicity allowed by the spinor structures, also at fixed density.
Compatible multiplication perturbations of Dirac operators are approximated in norm resolvent by metric Dirac operators, including first parameter derivatives for compact smooth families. The metrics are arbitrarily $C^0$-close, preserve density, and agree with the original metric outside a prescribed neighborhood of the support. In dimensions four and five, such changes in any nonempty open set realize the first admissible nullity above the index minimum.
If invertible metrics exist, ordinary real index difference identifies their components at fixed density with $KO_{n+1}$; for $n\ge5$, each contains a positive-scalar-curvature metric whenever one exists.
With density and the metric outside any nonempty open set fixed, the Dirac-minimal space has weak homotopy type $\mathrm{Sp}/\mathrm{Sp}(a)$ for $n\equiv4\pmod8$, and $\Omega\mathrm{Sp}$ for $n\equiv5\pmod8$, where $a=\lvert \operatorname{ind}_{\mathbb H}D^+\rvert$ and $\mathrm{Sp}$ is the stable compact symplectic group. On compact connected spin $n$-manifolds, invertible boundary metrics admit invertible cylindrical extensions exactly when the relative real index vanishes.
Keywords: positive scalar curvature, Spin Dirac operator, norm-resolvent approximation, generic simplicity, harmonic spinor, spectral prescription, real index difference, spaces of metrics
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