The compact quotient–local fibration equivalence conjecture

Let $G/H$ be a reductive homogeneous space. A local geometric fibration is a smooth fiber bundle whose total space is an open subset of $G/H$ and whose fibers are translates of the maximal compact subspace $X=K/K_H$.

Compact quotient–local fibration conjecture. The following are equivalent:

1. $G/H$ admits compact quotients.
2. $G/H$ admits local geometric fibrations.

The forward implication is motivated by the geometric fibration conjecture and is known to be necessary in the results discussed in the paper. The reverse implication is described as more audacious and remains open.