In this Chapter we shall present proofs of the theorems of R. Myers My, generalising earlier work of R. Livesay Li, and of J. Rubinstein R. Taken together these come close to a topological classification of free actions by groups of order 3s 2t on S3, and provide the major ingredient in the reduction theorem of the next chapter. As before let L3(m,q) be a lens space with cyclic fundamental group of order m.

THEOREM 3.1 Let h be a free involution on M = L3(m,q). The orbit manifold M* = M/h is Seifert fibered, and hence homeomorphic to a manifold of constant positive curvature.

The proof will be broken up into several lemmas.

LEMMA A There is a pℓ Morse function f:M → R and a triangulation K of M, such that

1. (1) f has 4 critical points {x0, x1, x2, x3} such that f(xi) = i = index (xi),

2. (2) A = {x ∈ M: fh(x) = f(x)} is a closed orientable surface containing no critical points,

3. (3) f is linear on the simplexes of K,

4. (4) h is simplicial with respect to K, and

5. (5) A and {xi}, O ≤ i ≤ 3, are triangulated as full subcomplexes of K.

Proof. The description of L(m,q) as a Seifert manifold with one exceptional fibre in the first section shows that M has a handle decomposition with one handle each of index i, O ≤ i ≤ 3.