SEIFERT FIBER SPACES WITH SINGULAR SURFACES

Abstract

Seifert fiber spaces are compact 3-dimensional manifolds that are foliated by circles. Seifert fiber spaces with isolated singular fibers have been well-studied. We focus on Seifert fiber spaces which have singular surfaces and extend known results to such manifolds. Two-sided incompressible surfaces in Seifert fiber spaces with isolated singular fibers are either horizontal or vertical. Frohman and Rannard have shown that one-sided incompressible surfaces in such manifolds are either pseudo-horizontal or pseudo-vertical. We extend their result to characterise essential surfaces in Seifert fiber spaces which may contain singular surfaces. We also give a complete criterion for the existence of horizontal surfaces in Seifert fiber spaces which may have singular surfaces. We introduce prism complexes as an analogue of simplicial complexes. And show that while every compact 3-dimensional manifold admits a prism complex structure, it admits a special prism complex structure if and only if it is a Seifert fiber space which has either non-empty boundary or singular surfaces or it is a closed Seifert fiber space with Euler number zero. In particular, a compact 3-dimensional manifold with boundary is a Seifert fiber space if and only if it admits a special prism complex structure. We will also briefly discuss our future work towards finding families of manifolds that support the L-space Conjecture.