The Identity For The Quotient Group of Homotopic Loops

Theorem

is an equivalence relation on a setfor the set of loops with the same base pointwithifis a loop inhomotopic to a loopandandhave the same base point

The set of homotopy classes is then

The identity functionhas equivalence classand this is the identity function for

Proof

By definitionand

so thattraces the loop a and then stops athence