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The cobordism group of homology cylinders. (English) Zbl 1225.57003

Let \(\sum_{g,n}\) be a fixed oriented, connected and compact surface of genus \(g\) with \(n\) boundary components. The authors intend to study an enlargement of the mapping class group, namely the group of homology cobordism classes of homology cylinders. A homology cylinder over \(\sum_{g,n}\) is roughly speaking a cobordism between surfaces equipped with a diffeomorphism to \(\sum_{g,n}\) such that the cobordism is homologically a product. By considering smooth (respectively topological) homology cobordism classes of homology cylinders we obtain a group \({H}^{\text{smooth}}_{g,n}\) (respectively \({H}^{top}_{g,n})\). These groups were introduced by Garoufalidis and Levine. They can be regarded as an enlargement of the mapping class group.
An argument of Garoufalidis and Levine shows that the canonical map \({M}_{g,n} \rightarrow {H}_{g,n}\) is injective. It is a natural question which properties of the mapping class group are carried over to \({H}_{g,n}\). In particular, Goda and Sakasai asked whether \({H}^{\text{smooth}}_{g,1}\) is a perfect group and Garoufalidis and Levine asked whether \({H}^{\text{smooth}}_{g,1}\) is infinitely generated. The following theorem answers both questions.
Theorem 1.2. If \(b_{1}(\sum_{g,n}) > 0,\) then there exists an epimorphism \({H}_{g,n} \rightarrow (\frac{\mathbb Z}{2})^{\infty}\) which splits (i.e. there is a right inverse). In particular, the abelianization of \({H}_{g,n}\) contains a direct summand isomorphic to \((\frac{\mathbb Z}{2})^{\infty}.\)
Furthermore, the authors 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.

MSC:

57M07 Topological methods in group theory
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
57N70 Cobordism and concordance in topological manifolds
14H15 Families, moduli of curves (analytic)
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