The mapping class group is an important concept in the field of algebraic topology, particularly in the study of surfaces and their automorphisms. Specifically, it is the group of isotopy classes of orientation-preserving diffeomorphisms of a surface. Here's a more detailed explanation: 1. **Surface**: A surface is a two-dimensional manifold, which can be either compact (like a sphere, torus, or more complex shapes) or non-compact.
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The mapping class group of an oriented surface is the group of orientation-preserving self-homeomorphisms of modulo isotopy. It acts on isotopy classes of curves and arcs.