Novikov–Shubin Positivity and Foliated Index Characters

Definitions and mathematical statement

For Riemannian foliations, the source asks two questions:

1. Are the Novikov--Shubin invariants always positive?
2. More generally in the source's formulation, whenever the index bundle of a leafwise elliptic operator exists, is its Chern--Connes character equal to the expected characteristic classes?

A Riemannian foliation has a transverse Riemannian metric preserved by holonomy. Novikov--Shubin invariants measure the small-spectrum decay rate near zero for the relevant leafwise operators, after accounting for their kernels. The record does not identify those operators or degrees, the trace or transverse framework, compactness and bounded-geometry hypotheses, or the normalization of the invariant. The positivity question is retained as strictly greater than zero in that intended convention, not a prescribed uniform lower bound. Treatment of a spectral gap or infinite invariant must be specified by the source.

For the second question, an index bundle must be a particular geometric or smooth family object in the report's framework, not simply an arbitrary analytic $K$-theory index class with the same name. A leafwise elliptic operator is elliptic along the foliation directions; its symbol, bundle data, and transverse regularity are missing. The Chern--Connes character requires a specified domain and target, and the "expected characteristic classes" need an actual formula and integration or pushforward operation. None is supplied.

The request is that existence of the intended index bundle should suffice for the claimed character formula without the additional hypotheses of known special cases. It is not a theorem for every elliptic symbol with an invented Dirac or spin assumption. In particular, no particular $\widehat A$, Todd, or twisting-class expression is selected for the unspecified operator class. The phrase more generally is retained as the source's relationship between the questions, not an asserted implication proved here.

Short English statement

The source asks for positivity of foliated Novikov--Shubin invariants and for a Chern--Connes character formula whenever the intended index bundle exists. The spectral setup, notion of index bundle, cohomology target, and characteristic-class formula are not specified.

Sources

• Local UnsolvedMath registry, permanent ID 30000842, code catalogue entry, "Novikov--Shubin Positivity and Foliated Index Characters." Proposed year 2007; proposer unspecified.
• Noncommutative Geometry (2008), Oberwolfach Reports, pp. 2543--2608, DOI 10.4171/owr/2007/43; https://doi.org/10.4171/owr/2007/43. The displayed report year 2008 differs from the DOI and proposed year 2007.
• Supplied "Benameur--Heitsch": https://arxiv.org/0711.0352.
• Statement status corrected_verified; the supplied note says official-report content was checked and wording/notation normalized without changing the source claim.
• Literature triage checked 2026-08-21: open. It identifies the two general conjectures and says the cited work proves special cases under stronger hypotheses, not the stated general assertions. Exact spectral and index-bundle conditions and current status require 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.