Decomposability of Homotopy Lens Spaces and Free Cyclic Group Actions on Homotopy Spheres

Let p be a linear Zn action on C and let p also denote the induced Z„ action on S2p~l x D2q, D2p x S2q~l and S2p~l x S2q~l " 1m_1 where p = [m/2] and q = m — p. A free differentiable Zn action (£ , ju) on a homotopy sphere is p-decomposable if there is an equivariant diffeomorphism of (S2p~l x S2q~l, p) such that (S2m_1, ju) is equivalent to (£(*), ¿(*)) where S(*) = S2p_1 x D2q U^, D2p x S2q~l and A() where 2(4») = Sp x Dp+1 U<¡, Dp+l x Sp such that the inclusions S" x Dp+l —+ 2(d>), Dp+1 x Sp —*■ 2(4>) are equivariant. In [9], Livesay and Thomas have proved the following: Theorem 0.1. Let (22p+1, T) be a free involution on a homotopy sphere 22p+1, then there is an equivariant diffeomorphism i> of(Sp x Sp, A) such that (22p+1, T) is equivalent to (2($), A($)). For n > 2, let p be a linear Zn action on Cm which is free on Cm 0. Let p = [m/2] and q = m p. Then 52m_1, S2p~x x D2q,D2p x S2q~l and S2p~1 x S2q~l are invariant subspaces and the induced actions are free. We Received by the editors January 27, 1975. AMS (MOS) subject classifications (1970). Primary 57E1S, 57E25, 57E30.