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

  • Kazuo Habiro

    Kyoto University, Japan
  • Gwénaël Massuyeau

    Université de Strasbourg, France
From mapping class groups to monoids of homology cobordisms: a survey cover
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Abstract

Let Σ\Sigma be a compact oriented surface. A homology cobordism of Σ\Sigma is a cobordism CC between two copies of Σ\Sigma, such that both the “top” inclusion and the “bottom” inclusion ΣC\Sigma \subset C induce isomorphisms in homology. Homology cobordisms of Σ\Sigma form a monoid, into which the mapping class group of Σ\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 Σ\partial \Sigma is empty or connected.