The symmetric group is a classic example in group theory and combinatorics, with many applications in areas such as geometry and topology. While the closely related braid group also has frequent appearances in the same subjects and is generally well understood,there are still many unresolved questions of a geometric nature. The focus of this dissertation is on the dual braid complex, a simplicial complex introduced by Tom Brady in 2001 with natural connections to the braid group. In particular, many subcomplexes and quotients of the dual braid complex have interesting properties, including applications to the “intrinsic geometry” for the braid group. At the root of these analyses is the theory of noncrossing partitions, a modern combinatorial theory with close connections to the symmetric group. In this dissertation, we survey the connections between these areas and prove new results on the geometry and topology of the dual braid complex
