The set of fixed points which do not move as a knot is transformed into itself is not a knot. The conjecture was proved in 1978 (Morgan and Bass 1984). According to Morgan and Bass (1984), the Smith conjecture stands in the first rank of mathematical problems when measured by the amount and depth of new mathematics required to solve it.
The generalized Smith conjecture considers to be a piecewise linear -dimensional hypersphere in , and the -fold cyclic covering of branched along , and asks if is unknotted if is an (Hartley 1983). This conjecture is true for , and false for , with counterexamples in the latter case provided by Giffen (1966), Gordon (1974), and Sumners (1975).