Handbook of Teichmüller Theory, Volume III
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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