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We give a general method that may be effectively applied to the question ofwhether two components of a function space have the same homotopy type. Wedescribe certain group-like actions on function spaces. Our basic resultsassert that if two maps are in the same orbit under such an action, then thecomponents of the function space that contain these maps have the same homotopytype.

Criteria for components of a function space to be homotopy equivalent