Analogy With Mapping Class Groups
Because Fn is the fundamental group of a bouquet of circles, Out(Fn) can be thought of as the mapping class group of a bouquet of n circles. (The mapping class group of a closed surface is the outer automorphism group of the fundamental group of that surface.) By analogy, Out(Fn) can be described as the quotient G/H, where G is the group of all self-homotopy equivalences of the bouquet of circles, and H is the subgroup of G consisting of homotopy equivalences that are homotopic to the identity map.
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