We give a Dehn–Nielsen type theorem for the homology cobordism group of homology cylinders by considering its action on the acyclic closure, which was defined by Levine, of a free group. Then we construct an additive invariant of those homology cylinders which act on the acyclic closure trivially. We also describe some tools to study the automorphism group of the acyclic closure of a free group generalizing those for the automorphism group of a free group or the homology cobordism group of homology cylinders.
"Homology cylinders and the acyclic closure of a free group." Algebr. Geom. Topol. 6 (2) 603 - 631, 2006. https://doi.org/10.2140/agt.2006.6.603