A Seifert fiber space M is a
compact 3-manifold which decomposes into a collection ℱ of disjoint simple closed
curves, called fibers, such that each fiber has a tubular neighborhood which consists
of fibers and is a “standard fibered solid torus.” We consider the question, given a PL
involution h of M, can the fiber structure ℱ be chosen in such a way that h will
be fiber-preserving? We give an affirmative answer for the case when M is
orientable, irreducible, and either ∂M≠∅ or M contains an incompressible fibered
torus.
