Some Cyclic Group Actions on Homotopy Spheres

In [4J Orlik defined a free cyclic group action on a homotopy sphere constructed as a Brieskorn manifold and proved the following theorem: THEOREM. Every odd-dimensional homotopy sphere that bounds a para-llelizable manifold admits a free Zp-action for each prime p. On the other hand, it was shown ([3J) that there exists a free Zp-action on a 2n-1 dimensional homotopy sphere so that its orbit space is stably parallelizable if and only if n-::;'p. Naturally, one can ask: QUESTION. Does each odd dimensional homotopy sphere ~2n-I, n>2 that bounds a parallelizable manifold admit a free Zp-action so that its orbit space is stably parallelizable whenever n-::;'p? As a partial answer of the question and as a generalization of Orlik's theorem for 4n-3, n 2 2, dimensional homotopy spheres, we will show: MAIN THEOREM. For n23 odd, p2n, any homotopy sphere of dimension 2n-1 that bounds a parallelizable manifold admits a free Zp-actio1Z so that its orbit space is stably parallelizable. To prove it, we need a generalized Brieskorn manifold. Let n+m fi (Zl> "', zn+m) = ~aijZjajj, j=l be polynomials having only one critical point at the origin, where aij are integers greater than 1, aij are real numbers, and n>2. Set