Let Zm be the cyclic group of
order m. Denote by ρ a free PL (orientation preserving) action of Zm on the sphere
S2k+1, k ≥ 3. In this paper, we study submanifolds, K2k−1, of the sphere which are
left invariant by the free action ρ. In particular, the submanifolds considered are
highly conneced, that is, πi(K) = 0, i < k − 1. We apply the techniques of B-surgery
theory as developed in [Szczepanski, 1983] and some new results in this homology
surgery theory which we develop herein. We obtain a classification (up to cobordism)
of invariant submanifolds with given restriction τ = ρ|K and a relationship
between invariant highly connected submanifolds and invariant (homotopy)
spheres.
