Triangulations and Nonorientable Incompressible Surfaces

In this paper, we classify all nonorientable incompressible surfaces, up to isotopy, in the generalized quaternion spaces S-3/Q(4k), which are M-k = (S-2 : (2,1), (2,1), (k, -k + 1)), k >= 1. The techniques used can also be expanded to give the classification of nonorientable incompressible surfaces in the minimal layered chain pair triangulations of Seifert fibered spaces M-r,M-s = (S-2 : (2, -1), (r + 1,1), (s + 1, 1)), r,s >= 1.