Twisted Dirac Kernels Under Homotopy Equivalence

Definitions and mathematical statement

Fix closed Riemannian manifolds $M'$ and $M$ and a homotopy equivalence $f:M'\to M$. Let $(E,\nabla)$ range over the vector bundles with connection on $M$ intended by the source. Its curvature is the endomorphism-valued two-form $F_\nabla=\nabla^2$; a curvature norm requires the given Riemannian metric together with a bundle metric and a specified norm convention. Let $D_E$ and $D_{f^*E}$ denote the corresponding twisted operators on $M$ and $M'$.

The proposed assertion is \[ \exists\delta>0\ \forall(E,\nabla):\qquad \|F_\nabla\|\le\delta \quad\Longrightarrow\quad \ker D_E\cong\ker D_{f^*E}. \] The threshold may depend on the fixed manifolds, metrics, map, operator framework, and norm conventions, but the source quantifies over every allowed bundle with that same threshold. No rank-dependent bound is inserted. The inequality is non-strict as supplied.

The operator $D$ is not defined in the extraction. The title says twisted Dirac kernels, while a supplied reference concerns twisted signature complexes. A spin Dirac operator would require spin and bundle data not stated here; a signature-type operator has different structural requirements. Neither is silently chosen. The pullback connection would ordinarily be $f^*\nabla$, requiring a sufficiently smooth representative of $f$; that regularity is likewise not specified.

The meaning of $\cong$ must also be recovered: an abstract vector-space isomorphism, a graded isomorphism, a module isomorphism, or a natural identification are distinct requests. If these are elliptic operators on finite-rank bundles over closed manifolds and ungraded scalar vector spaces are intended, the kernels are finite-dimensional and the request is equality of kernel dimensions. This conditional clarification does not replace the question by index equality, which is a different invariant. The supplied report assessment records an expectation against the assertion, without supplying a counterexample.

Short English statement

The question asks for one positive curvature threshold, uniform over the allowed bundles, that makes twisted kernels invariant under the fixed homotopy equivalence. The twisted operator, bundle category, curvature norm, and type of kernel isomorphism are missing.

Sources

• Local UnsolvedMath registry, permanent ID 30000440, source code catalogue entry, "Twisted Dirac Kernels Under Homotopy Equivalence." Proposed year 2006; proposer unspecified.
• Analysis and Topology in Interaction (2006), Oberwolfach Reports, pp. 729--804, DOI 10.4171/owr/2006/13; https://doi.org/10.4171/owr/2006/13.
• Supplied reference "twisted signature complexes and homotopy invariance": https://digital.library.adelaide.edu.au/items/5ed1c700-1185-4e0f-9120-32cb796eee69.
• Statement status corrected_verified; the supplied note says map, parameter, connection, kernel, and pullback notation was enclosed in balanced delimiters, preserving hypotheses.
• Literature triage checked 2026-08-21: open. It says the report records the question with an expectation against it and the supplied search located no theorem or counterexample settling the stated small-curvature kernel invariance. This is a dated assessment requiring independent verification. Created 2026-08-25; updated 2026-09-16.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying report and literature provenance remain as supplied. Unreviewed statement-only draft.