Involutions on 𝑆¹×𝑆² and other 3-manifolds

Abstract

This paper exploits the following observation concerning involutions on nonreducible 3-manifolds: If the dimension of the fixed point set of a PL involution is less than or equal to one then there exists a pair of disjoint 2-spheres that do not bound 3-cells and whose union is invariant under the given involution. The classification of all PL involutions of S 1 × S 2 {S^1} \times {S^2} is obtained. In particular, S 1 × S 2 {S^1} \times {S^2} admits exactly thirteen distinct PL involutions (up to conjugation). It follows that there is a unique PL involution of the solid torus S 1 × D 2 {S^1} \times {D^2} with 1-dimensional fixed point set. Furthermore, there are just four fixed point free Z 2 k {Z_{2k}} -actions and just one fixed point free Z 2 k + 1 {Z_{2k + 1}} -action on S 1 × S 2 {S^1} \times {S^2} for each positive integer k (again, up to conjugation). The above observation is also used to obtain a general description of compact, irreducible 3-manifolds that admit two-sided embeddings of the projective plane.