Positive Scalar Curvature Information from Rosenberg Indices

Definitions and mathematical statement

Let $M$ be a closed smooth spin manifold of dimension $m$, with its spin structure fixed, and put $\Gamma=\pi_1(M)$. In the real $K$-theory convention required by the displayed $KO$-groups, use the real maximal and reduced group $C^*$-algebras. The Rosenberg indices are the indices of the spin Dirac operator twisted by the corresponding flat Mishchenko bundles: \[ \alpha_{\max}(M)\in KO_m(C^*_{\max,{\mathbb{R}}}\Gamma),\qquad \alpha_{\mathrm{red}}(M)\in KO_m(C^*_{\mathrm{red},{\mathbb{R}}}\Gamma). \] The degree is taken modulo eight. The quotient homomorphism $q:C^*_{\max,{\mathbb{R}}}\Gamma\to C^*_{\mathrm{red},{\mathbb{R}}}\Gamma$ gives the comparison \[ q_*\alpha_{\max}(M)=\alpha_{\mathrm{red}}(M). \] Thus nonvanishing of the reduced index implies nonvanishing of the maximal one; the converse requires information about this map and these classes.

A positive-scalar-curvature metric means a smooth Riemannian metric with scalar curvature strictly positive everywhere. The standard index obstruction gives the necessary condition \[ M\text{ admits such a metric}\ \Longrightarrow\ \alpha_{\max}(M)=0\quad\text{and}\quad\alpha_{\mathrm{red}}(M)=0. \] The source asks how much of the obstruction theory is captured by these indices and how the two versions compare. It does not give a precise general sufficiency statement. One must not replace that broad question by an asserted equivalence between vanishing and existence for all closed spin manifolds.

A definite comparison theorem would need to specify the dimension, fundamental-group assumptions, stable versus direct existence of a metric on $M$, and which additional obstructions or sufficiency conditions are to be tested. No surgery-range dimension assumption, torsion-free restriction, or stabilization is imposed from context. The extract also suppresses the real-algebra and connectedness conventions; for disconnected manifolds the formulation is ordinarily interpreted componentwise.

The supplied assessment checked 2026-08-21 reports important special-case index-theoretic results but no complete answer to the broad information question. It does not reproduce the hypotheses or conclusions of the cited enlargeability work, which are therefore not promoted to a general characterization.

Short English statement

Determine which positive-scalar-curvature obstructions the maximal and reduced Rosenberg indices detect, and what their vanishing can imply under precise hypotheses. The source asks a broad comparison, not a universally asserted vanishing criterion for existence.

Sources

• Local UnsolvedMath registry, permanent ID 30000592, catalogue entry, Positive Scalar Curvature Information from Rosenberg Indices.
• Topologie (2007), Oberwolfach Reports, pp. 2579--2630, DOI https://doi.org/10.4171/owr/2006/43.
• Hanke--Schick, supplied label Enlargeability and index theory: https://www.uni-math.gwdg.de/schick/publ/Enlargeability_and_index_theory_final_for_JDG.pdf.
• Supplied literature triage checked 2026-08-21; no complete general sufficiency criterion or exact special-case hypotheses are reproduced.