Dirac spectra and the topology of spaces of metrics

Let M be a closed connected spin manifold of dimension n≥3. 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 C⁰-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 KOₙ₊₁; for n≥5, 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 Sp/Sp(a) for n≡48, and for n≡58, where a=|indHD⁺| and 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.