The Generalized Smith Conjecture

Abstract

formation of the 3-sphere [5]. The Smith Conjecture may be generalized to higher dimensions by asking whether a nice knotted (n -2)-sphere (one which does not bound a tame disk) can be the fixed point set of a nice nontrivial periodic transformation of the n-sphere. This paper shows that, even with the usual smoothness restriction, the generalized Smith Conjecture is false in dimensions greater than three. This result requires the generalized Poincare Conjecture in dimensions greater than four. Employing special techniques, we settle with the Smith Conjecture for dimension four in the negative for semilinear odd periodic transformations. If the generalized Poincarle Conjecture in dimension four is true, then the generalized Smith Conjecture in dimension four is also false for even periodic transformations. We do not know whether this is the best possible result in dimension four. There is another generalization of the Smith Conjecture which says that