The cobordism group of homology cylinders

Garoufalidis and Levine introduced the homology cobordism group of homology cylinders over a surface. This group can be regarded as a generalization 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-Levine and Goda-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 hold for the homology cylinder analogue of the Torelli group as well.

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