Scalar Curvature Question [?73]: [c] LetS be a Riemannian manifold homeomorphic to the connected sum of twenty copies ofS2× S2

Definitions and mathematical statement

Let $S$ be a Riemannian manifold homeomorphic to \[ \#_{i=1}^{20}(S^2\times S^2). \] In particular it is closed and orientable; choose an orientation. Let $B^4(R)\subset{\mathbb{R}}^4$ be the Euclidean ball of radius $R$, and give $S$ its specified Riemannian distance. A map $f_R:B^4(R)\to S$ is 1-Lipschitz if $d_S(f_R(x),f_R(y))\leq|x-y|$ for all $x,y$.

Choose a cocycle $h$ representing the top dimensional cohomology orientation class $[S]\in H^4(S;{\mathbb{R}})$. In the source's differential-form version, take a smooth closed 4-form $h$ normalized by $\int_S h=1$, and interpret \[ h(f_R)=\int_{B^4(R)}f_R^*h, \] using the almost everywhere differential of a Lipschitz map. The question is whether one can find such maps for radii tending to infinity, with a positive constant $c=c(S)$ independent of $R$, satisfying \[ \int_{B^4(R)}f_R^*h\ \geq\ cR^4. \] The integral is signed; no absolute value is inserted. The form is fixed independently of $R$, and there is no stated boundary condition on $f_R$.

The source does not define $h(f_R)$ for an arbitrary cocycle. Since the domain ball has boundary, this is not automatically a pairing of an absolute fundamental cycle with the cohomology class; a cocycle evaluation convention or the differential-form interpretation is needed. The notation $R\to\infty$ also leaves open whether maps are required for every sufficiently large radius or merely along an unbounded sequence. Both readings retain a single uniform positive constant.

Short English statement

Can large Euclidean four-balls map 1-Lipschitzly to the connected sum of twenty copies of $S^2\times S^2$, with a fixed top dimensional cocycle evaluating at least proportionally to $R^4$? The differential-form version uses a signed pullback integral and imposes no boundary condition.

Sources

• Local UnsolvedMath registry, permanent ID 6700069, source code catalogue entry; Misha Gromov, 2017; update 2026-08-06.
• Gromov, 101 Questions, Problems and Conjectures around Scalar Curvature (2017), printed tag [?73], occurrence 72, PDF page 72, subquestion [c]: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf. Registry access date 2026-07-29; normalized PDF extraction; public author-hosted incomplete PDF; present status and reuse terms require review.
• The supplied summary describes a Lipschitz map and cohomological growth question but supplies no precise further reference or resolution.