Abstract
For the symmetric groups S( n) with n ≥ 6 we construct representations which satisfy a strong gap condition for the dimensions of various fixed point sets, as desired in equivariant surgery, and a condition on the isotropy groups, as used in the construction of manifolds with interesting fixed point sets. This allows us to construct smooth actions of S( n) on sphere such that the fixed point set is exactly one point, or any given closed, stably parallelizable manifold, all of whose components have the same dimension.
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