Stable K-Area and Nonvanishing Rosenberg Indices

Definitions and mathematical statement

Let $M$ be a closed smooth spin manifold of dimension $n$, and put $\Gamma=\pi_1(M)$ in the connected convention. The first question is the proposed implication \[ M\text{ has infinite stable }K\text{-area} \quad\Longrightarrow\quad\alpha_{\mathrm{max}}(M)\ne0. \] The second asks for an analogous nonvanishing statement for $\alpha_{\mathrm{red}}(M)$, at least in specified special cases such as area-enlargeability. These are retained as separate maximal and reduced questions.

In the usual Rosenberg-index setup, $\alpha_{\mathrm{max}}(M)$ is the index of the spin Dirac operator twisted by the Mishchenko bundle with fiber the maximal group $C^*$-algebra of $\Gamma$; the reduced index uses the reduced group $C^*$-algebra. The natural quotient \[ q:C^*_{\mathrm{max}}\Gamma\longrightarrow C^*_{\mathrm{red}}\Gamma \] induces $q_*\alpha_{\mathrm{max}}(M)=\alpha_{\mathrm{red}}(M)$ in the compatible index theory. This does not imply that a nonzero maximal class has nonzero reduced image. The source does not specify real $KO_n$ versus complex $K_n$ coefficients, which can affect nonvanishing; neither is silently substituted for the other.

Infinite stable $K$-area is not defined in the record. Usual $K$-area notions detect nontrivial characteristic data with bundles of arbitrarily small curvature, while "stable" requires a prescribed stabilization and detection convention. The allowed bundles, curvature norm, characteristic pairing, and stabilization are missing, so a particular torus product or finite-rank versus Hilbert-module definition is not imposed.

For orientation, a common area-enlargeability condition asks, for every $\varepsilon>0$, for a suitable covering $\widetilde M_\varepsilon\to M$ and a nonzero-degree map to $S^n$ whose differential contracts two-dimensional areas, $\|\Lambda^2df\|\leq\varepsilon$, with an appropriate constant-outside-compact-set condition for a noncompact cover. The precise allowed covers, target/degree conventions and dimension hypotheses must come from the source. This is not replaced by contraction of lengths, which is the stronger ordinary enlargeability-type condition. Nor is the relation of area-enlargeability to the missing stable-$K$-area definition assumed without its hypotheses.

The literature assessment checked 2026-08-21 reports maximal-index nonvanishing for enlargeable spin manifolds and related positive-scalar-curvature obstructions, while leaving the full stable-$K$-area implications and reduced-index statement unresolved. The draft keeps the reported enlargeability result distinct from the specifically mentioned area-enlargeability question. Its partially-solved label and notation-only verification are dated metadata, not independently verified current results.

Short English statement

Does infinite stable $K$-area force a nonzero maximal Rosenberg index, and can reduced-index nonvanishing be proved under that or relevant special hypotheses such as area-enlargeability? The two completions are distinct: the quotient map can kill a nonzero class. Stable $K$-area, coefficient theory and covering conventions are missing.

Sources

• Local UnsolvedMath registry, permanent ID 30000593, source code catalogue entry, "Stable K-Area and Nonvanishing Rosenberg Indices," including both maximal/reduced questions and the area-enlargeability special-case request.
• Topologie (2007), Oberwolfach Reports, pp. 2579--2630, DOI 10.4171/owr/2006/43: https://doi.org/10.4171/owr/2006/43. The citation uses 2007, while the DOI and proposed-year metadata use 2006.
• Supplied reference "Hanke--Schick, enlargeability and index theory": https://www.uni-math.gwdg.de/schick/publ/Enlargeability_and_index_theory_final_for_JDG.pdf. The URL is cited as supplied; no PDF has been opened.
• Literature triage checked 2026-08-21; record updated 2026-09-16. No external source has been visited or independently checked here.