We study algebraic structures of certain submonoids of the monoid of homology
cylinders over a surface and the homology cobordism groups, using Reidemeister
torsion with non-commutative coefficients. The submonoids consist of ones whose
natural inclusion maps from the boundary surfaces induce isomorphisms on higher
solvable quotients of the fundamental groups. We show that for a surface whose first
Betti number is positive, the homology cobordism groups are other enlargements
of the mapping class group of the surface than that of ordinary homology
cylinders. Furthermore we show that for a surface with boundary whose first
Betti number is positive, the submonoids consisting of irreducible ones as
–manifolds
trivially acting on the solvable quotients of the surface group are not finitely
generated.
Keywords
homology cylinder, homology cobordism, Reidemeister
torsion, derived series
