Homology cylinders: an enlargement of the mapping class group
The mapping class group \(\Gamma_g\) is the group of diffeotopy classes of orientation preserving diffeomorphisms of the closed oriented surface \(\Sigma_g\) of genus \(g\). One reason for interest in \(\Gamma_g\) is that it is related in an obvious way to the structure of 3-manifolds via the Heegaard construction. From this viewpoint, the subgroup \(\Gamma_g^B\) consisting of diffeomorphisms which extend over the handlebody \(T_g\) of genus \(g\), and the coset space \(\Gamma_g/ \Gamma_g^B\) are of obvious interest.NEWLINENEWLINENEWLINEIn this note, the author proposes a ``homological'' generalization of these groups, defined by homology cylinders over a closed oriented surface (up to homology cobordism). In this context, the group \(\Gamma_g\) is replaced by the group \({\mathcal H}_g\) of homology bordism classes of homology cylinders over \(\Sigma_g\). Homology cylinders have appeared and have been studied in recent work of \textit{M. Goussarov} [C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 6, 517-522 (1999; Zbl 0938.57013)] and \textit{K. Habiro} [Geom. Topol. 4, 1-83 (2000; Zbl 0941.57015)], as important model objects for their new theory of finite-type invariants of general 3-manifolds. The notion of Heegaard decomposition translates to the more general context of homology cylinders -- the relevant subgroup is now \({\mathcal H}_G^B\) consisting of those homology cylinders which extend to homology cylinders over \(T_g\). The structure of the group \({\mathcal H}_g\) is studied from several directions: the relative weight filtration of Dennis Johnson, the finite-type filtration of Goussarov-Habiro, and the relation to string link concordance.
scientific article; zbMATH DE number 1605726
scientific article