Scalar Curvature Question [?28]: For instance, letX be homomorphic to CP 2 and letvol(Uδ(S)) ≥δ2 for allT(X)-non-spin surfacesS ⊂X and 0<δ ≤1

Definitions and mathematical statement

The source says that $X$ is "homomorphic to $\mathbb{CP}^2$", which is not a specified topological equivalence of manifolds. The likely homeomorphism reading must be checked; it is not strengthened to diffeomorphism. In the natural metric setting, let $(X,g)$ be a smooth Riemannian four-manifold with the intended topological relation to $\mathbb{CP}^2$, and for a surface $S\subset X$ put \[ U_\delta(S)=\{x\in X:d_g(x,S)<\delta\}. \] The requested numerical implication is \[ \left[\ \forall\ T(X)\text{-non-spin surfaces }S\subset X\ \forall\delta\in(0,1]:\quad \operatorname{Vol}_g(U_\delta(S))\ge\delta^2\ \right] \quad\Longrightarrow\quad \operatorname{spin\text{-}area}(X)\ge10^{-6}. \] Both the universal surface condition and every real radius $0<\delta\le1$ are retained; the constant is exactly $1/1{,}000{,}000$.

The excerpt does not define $T(X)$-non-spin. One plausible tangent-bundle reading is that the pullback $i^*TX$, $i:S\hookrightarrow X$, has no spin structure; for an oriented ambient tangent bundle this is detected by $i^*w_2(TX)\ne0$. That differs from saying the surface's own tangent bundle is non-spin. The source does not specify the surface regularity, closedness, embedding versus mapping, or whether this is the intended obstruction criterion.

Most importantly, $\operatorname{spin\text{-}area}$ and its normalization are not defined in the supplied record. It is not equated with a particular K-area invariant merely because the summary mentions that program. The metric convention and the meaning of the fixed scale $1$ are also essential for the numerical assertion. The source summary calls the estimate open and says no published verification of this specific $\mathbb{CP}^2$/$10^{-6}$ claim was found; that is a supplied review report, not independent current verification.

Short English statement

If every surface detecting the intended non-spin tangent-bundle obstruction has tube volume at least $\delta^2$ for all $0<\delta\le1$, must the spin-area be at least $10^{-6}$? The topological wording and both specialized invariant definitions need confirmation from the original source.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $6700028$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• Misha Gromov, 101 Questions, Problems and Conjectures around Scalar Curvature (2017), printed tag [?28], occurrence 29, PDF page 38: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf. Supplied normalized PDF-text extraction accessed 2026-07-29; described as incomplete.
• No separate definition of spin-area, tangent-bundle-non-spin surfaces, or publication verifying this quantitative estimate is supplied.