7 problems

Positive-injectivity-radius conjecture. All Riemannian manifolds with Lie structures at infinity have positive injectivity radius:

Uniform geometric K-homology conjecture. The natural map

Bounded-geometry positive scalar curvature exact-sequence conjecture. There is a commutative diagram

Bounded-geometry to $\ell^\infty$-coefficient analysis conjecture. This setup results in a framework mapping bounded geometry to analysis with $\ell^\infty$-coefficients.

Finiteness conjecture. The class of all such submetries contains only finitely many foliated homeomorphism types.

Bounded-geometry conjecture. The manifold $({\mathcal{M}},g)$ is of bounded geometry if and only if the five gradients inequality holds.

Let $M$ be a closed manifold with a Riemannian metric whose geometry is bounded, meaning that its dimension, injectivity radius, and sectional curvature satisfy fixed bounds. Suppo…