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It is less transparent how mapping class groups are related to 4–dimensional topology. By results of Donaldson and Gompf, closed symplectic manifolds (admitting Lefschetz fibration or Lefschetz pencil structures) give rise to various objects in mapping class groups, and therefore the study of these groups has implications to 4–dimensional symplectic topology. There are also converse results; there are 4–dimensional topological theorems that have implications to mapping class group theory. For compact 4–manifolds with nonempty boundary a very similar correspondence can be set up, provided the manifolds admit Stein structures and we consider mapping class groups of surfaces with nonempty boundary.

Symplectic 4-manifolds, Stein domains, Seiberg-Witten theory and mapping class groups