Orientation reversing involutions on closed 3-manifolds (Q921663)

Let M be an arbitrary connected, closed, orientable 3-manifold admitting an orientation reversing PL involution \(\tau\). Various algebraic and geometric consequences (concerning M and the fixed point set Fix(\(\tau\))) have been derived from these hypotheses. In the present paper the author rounds off results of this kind. She observes that Fix(\(\tau\)) is either a closed 2-manifold separating M into two pieces which are interchanged by \(\tau\) or a nonseparating disjoint union of closed surfaces and isolated points. For either case she lists a complete set of properties of the group \(G=H_ 1(M)\) and the Betti numbers of \(F=Fix(\tau)\) which follow from the stated hypotheses. Completeness is to be understood in the following sense: if G is any abelian group and F is any orientable closed 2-manifold (or a disjoint union of a closed 2-manifold and a finite discrete set) such that G and the Betti numbers of F possess the listed properties, then G and F can be realized as \(H_ 1(M)\) and Fix(\(\tau\)) for some connected, closed, orientable 3-manifold M and an orientation reversing involution \(\tau\) of M.