Abstract
It is shown that there are homotopy ( 4 n + 1 ) (4n + 1) - or ( 4 n + 3 ) (4n + 3) -spheres which admit infinitely many differentiable free actions of S 1 {S^1} or S 3 {S^3} with characteristic homotopy spheres in certain dimensions and without characteristic homotopy spheres in some other dimensions.
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