# Handbook of Teichmüller Theory, Volume III

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#### DOI: 10.4171/103-1/9

From mapping class groups to monoids of homology cobordisms: a survey

Kazuo Habiro[1] and Gwénaël Massuyeau[2]

(1) Research Institute for Mathematical Sciences, Kyoto University, 606-8502, KYOTO, JAPAN
(2) Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS, 7 Rue René Descartes, 67084, STRASBOURG CEDEX, FRANCE

Let $\Sigma$ be a compact oriented surface. A homology cobordism of $\Sigma$ is a cobordism $C$ between two copies of $\Sigma$, such that both the “top” inclusion and the “bottom” inclusion $\Sigma \subset C$ induce isomorphisms in homology. Homology cobordisms of $\Sigma$ form a monoid, into which the mapping class group of $\Sigma$ embeds by the mapping cylinder construction. In this chapter, we survey recent works on the structure of the monoid of homology cobordisms, and we outline their relations with the study of the mapping class group. We are mainly interested in the cases where $\partial \Sigma$ is empty or connected.

Keywords: Mapping class group, Torelli group, 3-manifold, homology cobordism, homology cylinder, Johnson homomorphism, finite-type invariant

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