Constructions of exotic actions on product manifolds with an asymmetric factor

We explore transformation groups of manifolds the form M×Sn, where M is an asymmetric manifold, that is, a manifold which does not admit any nontrivial action finite group. In particular, we prove for n=2, there exists infinite family distinct nondiagonal effective circle actions on such products. A similar result holds cyclic prime order. also discuss free M×S1, belongs to class “almost asymmetric” considered previously by Puppe and Kreck.