Scalar Curvature Question [?22]: It seems not impossible, at least for compactX, that, in fact, K-area(X×R) =K-area(X×S1)

Definitions and mathematical statement

The source tentatively proposes, at least for compact $X$, \[ \mathrm{K\!\text{-}area}(X\times\mathbb R) =\mathrm{K\!\text{-}area}(X\times S^1). \] This is a metric-dependent statement. If $(X,g)$ is the intended compact Riemannian manifold and $S^1_\ell$ is a circle of circumference $\ell>0$, an explicit product-metric version would compare \[ \mathrm{K\!\text{-}area}(X\times\mathbb R,g+dt^2) \quad\text{and}\quad \mathrm{K\!\text{-}area}(X\times S^1_\ell,g+ds^2). \] The source does not specify $\ell$ or whether the assertion concerns all circle sizes, sufficiently small sizes, or a limiting value as $\ell\to0$.

In a conventional curvature-based K-area construction, one takes a supremum of reciprocal curvature norms of admissible Hermitian bundles with unitary connections and nontrivial characteristic data. The admissibility, nontriviality, parity/stabilization, and conditions at infinity are essential to the definition. They are not supplied here, and the summary calls the invariant stable K-area without giving a stabilization convention. Consequently no particular bundle formula is substituted for the source's invariant. The intended equality should be formulated with one specified K-area variant, the same conventions on both factors, and explicit product metrics.

Short English statement

For compact $X$, compare K-area after adjoining a line with K-area after adjoining a circle. Fix the invariant variant, conditions at infinity, product metrics, and circle-size quantifier before asserting equality.

Sources

• Misha Gromov, 101 Questions, Problems and Conjectures around Scalar Curvature (2017), printed tag [?22], occurrence 23, PDF page 32, https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf; accessed 2026-07-29, PDF-text-normalized extraction. The provenance describes the public author PDF as incomplete.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $6700023$, source code catalogue entry, Scalar Curvature Question [?22]. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.