If is a compact set, a topological contraction is a self-embedding such that the intersection of the successive images , , consists of one point. In dimension 3, we prove that there are smooth topological contractions of the handlebodies of genus whose image is essential.
"Essential curves in handlebodies and topological contractions." Geom. Topol. 12 (2) 981 - 985, 2008. https://doi.org/10.2140/gt.2008.12.981