$\textrm {PL}$ involutions of some $3$-manifolds

Abstract:

Let ${h_1}$ and ${h_2}$ be PL involutions of connected, oriented, closed, irreducible 3-manifolds ${M_1}$ and ${M_2}$, respectively. Let ${a_i},i = 1,2$, be a fixed point of ${h_i}$ such that near ${a_i}$ the fixed point sets of ${h_i}$ are of the same dimension. Then we obtain a PL involution ${h_1}\# {h_2}$ on ${M_1}\# {M_2}$ induced by ${h_i}$ by taking the connected sum of ${M_1}$ and ${M_2}$ along neighborhoods of ${a_i}$. In this paper, we study the possibility for a PL involution h on ${M_1}\# {M_2}$ having a 2-dimensional fixed point set ${F_0}$ to be of the form ${h_1}\# {h_2}$, where ${M_i}$ are lens spaces. It is shown that: (1) if ${F_0}$ is orientable, then ${M_1} = - {M_2}$ and h is the obvious involution, (2) if the fixed point set F contains a projective plane, then ${M_1} = {M_2} = {\text {a}}$ projective 3-space, and in this case, F is the disjoint union of two projective planes and h is unique up to PL equivalences, (3) if F contains a Klein bottle K, then F is the disjoint union of a Klein bottle and two points.