A cooriented foliation of an oriented –manifold is taut if and only if there is a map from to the –sphere whose restriction to every leaf is a branched cover.
Danny Calegari. "Taut foliations leafwise branch cover ." Algebr. Geom. Topol. 21 (5) 2523 - 2541, 2021. https://doi.org/10.2140/agt.2021.21.2523