The cobordism group of homology cylinders

被引：10

作者：

Cha, Jae Choon ^{[1
,2
]}

Friedl, Stefan ^{[3
]}

Kim, Taehee ^{[4
]}

机构：

[1] Pohang Univ Sci & Technol, Dept Math, Pohang Gyungbuk 790784, South Korea

[2] Pohang Univ Sci & Technol, Pohang Math Inst, Pohang Gyungbuk 790784, South Korea

[3] Univ Warwick, Math Inst, Coventry CV4 7AL, W Midlands, England

[4] Konkuk Univ, Dept Math, Seoul 143701, South Korea

来源：

COMPOSITIO MATHEMATICA | 2011年 / 147卷 / 03期

关键词：

torsion invariant; homology cylinder; homology cobordism; FINITE-TYPE INVARIANTS; REIDEMEISTER TORSION; TORELLI GROUP; ALEXANDER INVARIANTS; KNOT COBORDISM; MAHLER MEASURE; LINKS; REPRESENTATION; 3-MANIFOLDS; SURFACES;

D O I：

10.1112/S0010437X10004975

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

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.

引用

页码：914 / 942

页数：29

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