MAT 540 Advanced Topology, Geometry I, Fall 2023.

  • Course description: CW-complexes, homotopy, fundamental group and higher homotopy groups, fiber bundles and covering, homotopy exact sequences of pairs, fiber bundles, classification of coverings, homotopy excision, suspension, topological manifolds, topological classification of 1- and 2-manifolds.

    This course is intended for students with a strong background in point-set topology and some familiarity with the fundamental group and covering spaces. For students with weaker background, MAT 530 is more appropriate. Both courses include the same core material on the fundamental group and covering spaces, but MAT 540 discusses these concepts in the context of homotopy groups and fibrations. MAT 530 builds a solid foundation in point-set topology while MAT 540 only revisits point-set topology as necessary (for example, to establish properties of CW-spaces); students will be expected to work with point-set-topological concepts confidently and correctly. The emphasis of the course will be on the basics of homotopy theory (typically not included in introductory topology courses).

    MAT 530 and MAT 540 satisfy the same requirements for the PhD and master's programs in Mathematics. MAT 540 cannot be taken for credit if you already completed MAT 530.