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The cobordism group of homology cylinders

Part of: Topological manifolds Low-dimensional topology

Published online by Cambridge University Press:  07 September 2010

Jae Choon Cha ,
Stefan Friedl  and
Taehee Kim
Jae Choon Cha
Affiliation:
Department of Mathematics and Pohang Mathematics Institute, Pohang University of Science and Technology, Pohang Gyungbuk 790–784, Republic of Korea (email: [email protected])
Stefan Friedl
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (email: [email protected])
Taehee Kim
Affiliation:
Department of Mathematics, Konkuk University, Seoul 143–701, Republic of Korea (email: [email protected])
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Abstract

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Garoufalidis and Levine introduced the homology cobordism group of homology cylinders over a surface. This group can be regarded as an enlargement of the mapping class group. Using torsion invariants, we show that the abelianization of this group is infinitely generated provided that the first Betti number of the surface is positive. In particular, this shows that the group is not perfect. This answers questions of Garoufalidis and Levine, and Goda and Sakasai. Furthermore, we show that the abelianization of the group has infinite rank for the case that the surface has more than one boundary component. These results also hold for the homology cylinder analogue of the Torelli group.

Keywords

torsion invarianthomology cylinderhomology cobordism

MSC classification

Secondary: 57M27: Invariants of knots and 3-manifolds 57N10: Topology of general $3$-manifolds
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 3 , May 2011 , pp. 914 - 942
Copyright
Copyright © Foundation Compositio Mathematica 2010

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