Dirac spectra and the topology of spaces of metrics

Nifa, Anass

doi:10.5281/zenodo.24122704

Preprint · v4

Dirac spectra and the topology of spaces of metrics

Anass NifaPublished: 24 November 2025Updated: 11 November 2026

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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 $a=|\operatorname{ind}_{\mathbb H}D^+|$ 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.

MSC 2020

58J50: Spectral problems; spectral geometry; scattering theory on manifolds
53C27: Spin and Spin^c geometry
58J20: Index theory and related fixed-point theorems on manifolds
58D17: Manifolds of metrics (especially Riemannian)
57R65: Surgery and handlebodies

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.24122704
All versions: 10.5281/zenodo.23546911
Subjects: Spin Dirac operator; norm-resolvent approximation; generic simplicity; harmonic spinor; spectral prescription; real index difference; spaces of metrics; positive scalar curvature
Language: English

Cite this paper

Anass Nifa. Dirac spectra and the topology of spaces of metrics. Preprint, Zenodo, v4 (2025). doi:10.5281/zenodo.24122704

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