 The case "n" = 1 and 2 has long been known, by classification of manifolds in those dimensions.For a "PL" or "smooth" homotopy n-sphere, in 1960 Stephen Smale proved for formula_1 that it was homeomorphic to the "n"-sphere and subsequently extended his proof to formula_2; he received a Fields Medal for his work in 1966.Shortly after Smale's announcement of a proof, John Stallings gave a different proof for dimensions at least 7 that a PL homotopy "n"-sphere was homeomorphic to the "n"-sphere using the notion of "engulfing".