Scalar Curvature Question [?39]: Let $B=B\Gamma$ be the classifying space of a discrete countable group, and let $f:X\to B$ be a continuous map

Definitions and mathematical statement

Let $\Gamma$ be a discrete countable group, let $B=B\Gamma$ be a classifying space, and let $f:X\to B$ be continuous with $X$ a Riemannian manifold. The source first asks for a compact subset $B_0\subset B$ containing $f(X)$, and then asks for arbitrarily small curvature in nonzero integer multiples of an indicated pullback K-theory class.

If $L_0$ is the intended fixed bundle and all categories are specified, its displayed target has the quantifier pattern \[ \exists B_0\subset B\text{ compact},\ f(X)\subseteq B_0:\quad \forall\varepsilon>0\ \exists N\in\mathbb Z\setminus\{0\}\ \exists(L,\nabla)\text{ over }X: \quad\|\operatorname{curv}(\nabla)\|\le\varepsilon, \quad [L]=Nf^*[L_0]. \] The integer and bundle may depend on $\varepsilon$; no single fixed multiple for all errors is required by the excerpt. For positive $N$, the usual finite-dimensional Whitney sum has class $[f^*L_0^{\oplus N}]=Nf^*[L_0]$. Negative $N$ is meaningful as an additive K-theory class, not as an ordinary negative Whitney sum, unless an appropriate virtual/Fredholm model is supplied.

The source does not define Fredholm bundle, Fredholm coarea, K-theoretic equivalence, or the curvature norm. A common finite-rank curvature norm would use the supremum of the fiber-operator norm on unit tangent two-planes, but the intended Fredholm setting may differ. The bundle $L_0$, its base $B$ or $B_0$, and whether its class is given, arbitrary, or existential are also missing. In particular a nonzero integer does not by itself require its multiple of a K-class to remain nonzero; that additional nontriviality must not be silently imposed.

No compactness hypothesis on $X$ or relative compactness of $f(X)$ is stated, so existence of the initial compact $B_0$ is not an automatic consequence of the supplied assumptions. The classifying-space model and any required finite subcomplex condition need confirmation. These gaps prevent the displayed conditional pattern from being a fully specified theorem.

The summary describes this as an open Fredholm K-area/coarea question related to almost-flat bundles, Novikov theory, and scalar-curvature obstructions, with no dedicated resolution located. No precise auxiliary theorem or definition is supplied, and no independently verified status claim is made.

Short English statement

Can suitable nonzero multiples of the intended pullback K-class be represented by Fredholm bundles of arbitrarily small curvature, after locating the image in a compact part of $B\Gamma$? The compactness assumption, bundle quantifier, Fredholm definitions, and treatment of negative multiples are missing.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $6700036$, 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 [?39], occurrence 38, PDF page 45: 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.
• The summary mentions almost-flat bundles and Novikov-related work without supplying exact definitions, original papers, or a resolution reference.