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

TL;DR

This survey explores the structure of the monoid of homology cobordisms of surfaces and their relationship with the mapping class group, highlighting recent developments and focusing on cases with empty or connected boundary.

Contribution

It provides a comprehensive overview of recent research on homology cobordisms and their connections to the mapping class group, emphasizing new structural insights.

Findings

Identification of the monoid structure of homology cobordisms

Relations between homology cobordisms and the mapping class group

Focus on surfaces with empty or connected boundary

Abstract

Let S be a compact oriented surface. A homology cobordism of S is a cobordism C between two copies of S, such that both the "top" inclusion and the "bottom" inclusion of S in C induce isomorphisms in homology. Homology cobordisms of S form a monoid, into which the mapping class group of S embeds by the mapping cylinder construction. In this paper, 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 the boundary of S is empty or connected.