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In this paper the author proves Generalized Homotopy Lemmas. These type of results play an important role in the classification theory of $*$-homomorphisms up to asymptotic unitary equivalence. Table of Contents: Prelude; The basic homotopy lemma for higher dimensional spaces; Purely infinite simple $C^*$-algebras; Approximate homotopy; Super homotopy; Postlude; Bibliography. (MEMO/205/963)