4 problems

Gromov's rational inessentiality conjecture. Then $M$ is not rationally essential.

Gromov's rational essentialness conjecture. A closed $\mathbb{Q}$-essential $n$-manifold does not admit any Riemannian metric with positive scalar curvature.

Gromov's systolic conjecture. There exists an absolute constant $C$ such that, for every essential closed Riemannian manifold $M^n$,

Essentialness conjecture. An essential manifold does not admit a metric of positive scalar curvature. The paper explains that essential spin$^c$ manifolds are K-theoretically essen…