In the 1960s, Milnor proved that certain intersections of balls with complex hypersurfaces can be expressed as fiber bundles. In particular, the complement of a central (non-affine) arrangement of complex hyperplanes is a fiber bundle over the circle. However, in even the "nicest" such setting, the braid arrangement, the homology of the fiber is unknown in general. In this talk we will discuss a way of leveraging the combinatorial structure of noncrossing partitions to create a geometrically appealing simplicial complex for the Milnor fiber. Knowledge of fundamental groups and covering spaces will be useful, although the content is otherwise introductory.