Honors BS

honors thesisWe will prove the Birman exact sequence as motivated by the study of mapping class groups. In order to do so we introduce basic concepts and theorems about surfaces, exact sequences, mapping class groups, higher dimensional homotopy groups, and fiber bundles. In particular, the proof of the Birman exact sequence depends heavily on the long exact sequence of homotopy groups of a fiber bundle, and a majority of the thesis is devoted to establishing this sequence