𝑃𝐿 involutions of some 3-manifolds

Abstract

Let h 1 {h_1} and h 2 {h_2} be PL involutions of connected, oriented, closed, irreducible 3-manifolds M 1 {M_1} and M 2 {M_2} , respectively. Let a i , i = 1 , 2 {a_i},i = 1,2 , be a fixed point of h i {h_i} such that near a i {a_i} the fixed point sets of h i {h_i} are of the same dimension. Then we obtain a PL involution h 1 # h 2 {h_1}\# {h_2} on M 1 # M 2 {M_1}\# {M_2} induced by h i {h_i} by taking the connected sum of M 1 {M_1} and M 2 {M_2} along neighborhoods of a i {a_i} . In this paper, we study the possibility for a PL involution h on M 1 # M 2 {M_1}\# {M_2} having a 2-dimensional fixed point set F 0 {F_0} to be of the form h 1 # h 2 {h_1}\# {h_2} , where M i {M_i} are lens spaces. It is shown that: (1) if F 0 {F_0} is orientable, then M 1 = − M 2 {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 = a {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.