ETD

• cw-models for Milnor fiber
• hyperplane arrangements
• Milnor fiber
• reflection arrangements

In this thesis, we study the Milnor fiber of a complexified real hyperplane arrangement. The topology of the Milnor fiber is closely related to the combinatorics of the arrangement, but its cohomology is not fully understood in general, and it is not known whether it depends only on the combinatorial structure.
We describe two recent combinatorial approaches to constructing CW-models for the Milnor fiber. First, for reflection arrangements, we study the construction of Brady, Falk, and Watt, which uses noncrossing partitions associated with a finite reflection group to construct a cell complex having the homotopy type of the Milnor fiber. We then discuss the work of Mücksch and Yoshinaga, which provides a combinatorial cellular model for the Milnor fiber of a general real hyperplane arrangement using its associated oriented matroid. In particular, their construction shows that the homotopy type of the Milnor fiber depends only on the oriented matroid.
Finally, we study several explicit examples, mainly reflection arrangements, and use the combinatorial models to compute their homology and the homology of quotients of the Milnor fiber by the action of the corresponding Coxeter group.