Mapping Class Group - Motivation

Motivation

Consider a topological space, that is, a space with some notion of closeness between points in the space. We can consider the set of homeomorphisms from the space into itself, that is, continuous functions with continuous inverses: functions which stretch and deform the space continuously without puncturing or breaking the space. This set of homeomorphisms can be thought of as a space itself. It can be seen fairly easily that this space forms a group under functional composition. We can also define a topology on this new space of homeomorphisms. The open sets of this new function space will be made up of sets of functions that map compact subsets K into open subsets U as K and U range throughout our original topological space, completed with their finite intersections (which must be open by definition of topology) and arbitrary unions (again which must be open). This gives a notion of continuity on the space of functions, so that we can consider continuous deformation of the homeomorphisms themselves: called homotopies. We define the mapping class group by taking homotopy classes of homeomorphisms, and inducing the group structure from the functional composition group structure already present on the space of automorphisms.