Homology cylinders of higher-order

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.

Keywords

homology cylinder, homology cobordism, Reidemeister torsion, derived series

Mathematical Subject Classification 2000

Primary: 57M27

Secondary: 57Q10

References

Publication

Received: 8 September 2011

Revised: 18 April 2012

Accepted: 15 May 2012

Published: 16 July 2012

Authors

Takahiro Kitayama
Research Institute for Mathematical Sciences Kyoto University Kyoto 606–8502 Japan
http://www.kurims.kyoto-u.ac.jp/~kitayama/

Volume 12, issue 3 (2012)

Takahiro Kitayama

Abstract

Keywords

Mathematical Subject Classification 2000

References

Publication

Authors