# The surface complex of Seifert fibered spaces

## TL;DR

This paper introduces the surface complex for 3-manifolds and explores its properties in Seifert fibered spaces, highlighting the relationship with the curve complex of the base orbifold.

## Contribution

It defines the surface complex for 3-manifolds and investigates its structure specifically within Seifert fibered spaces, establishing a connection to the base orbifold's curve complex.

## Key findings

- Surface complex always contains a subcomplex isomorphic to the base orbifold's curve complex.
- Provides a case study in Seifert fibered spaces.
- Highlights the topological relationship between the surface complex and the base orbifold.

## Abstract

We define the surface complex for $3$-manifolds and embark on a case study in the arena of Seifert fibered spaces. The base orbifold of a Seifert fibered space captures some of the topology of the Seifert fibered space, so, not surprisingly, the surface complex of a Seifert fibered space always contains a subcomplex isomorphic to the curve complex of the base orbifold.