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Homology cylinders of higher-order

Takahiro Kitayama

Algebraic & Geometric Topology 12 (2012) 1585–1605

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 3–manifolds trivially acting on the solvable quotients of the surface group are not finitely generated.

homology cylinder, homology cobordism, Reidemeister torsion, derived series
Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 57Q10
Received: 8 September 2011
Revised: 18 April 2012
Accepted: 15 May 2012
Published: 16 July 2012
Takahiro Kitayama
Research Institute for Mathematical Sciences
Kyoto University