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The cobordism group of homology cylinders

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Cha, Jae Choon, 1971-, Friedl, Stefan and Kim, Taehee. (2011) The cobordism group of homology cylinders. Compositio Mathematica, Vol.147 (No.3). pp. 914-942. ISSN 0010-437X

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Official URL: http://dx.doi.org/10.1112/S0010437X10004975

Abstract

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.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Homology theory, Cobordism theory, Abelian groups
Journal or Publication Title:	Compositio Mathematica
Publisher:	Cambridge University Press
ISSN:	0010-437X
Date:	2011
Volume:	Vol.147
Number:	No.3
Page Range:	pp. 914-942
Identification Number:	10.1112/S0010437X10004975
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	National Research Foundation of Korea (NRF), Korea (South). Kyoyuk Kwahak Kisulbu
Grant number:	2007-0054656 (NRF), 2009-0094069 (NRF), 2009-0068877 (NRF), 2009-0086441 (NRF)
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URI:	http://wrap.warwick.ac.uk/id/eprint/39993