MTG 6346-7 Topology

The topics are divided into two modules:

  1. General Topology
    • Topological spaces, open and closed sets, continuous maps, homeomorphism.
    • Connected, path connected, and locally connected spaces. Connected subspaces of \(\mathbb{R}\).
    • Compact and locally compact spaces. Compact subspaces of \(\mathbb{R}^n\). Compactness in metric spaces.
    • Countability axioms.
    • Separation axioms. The Urysohn Lemma.
    • The Urysohn Metrization Theorem.
    • The Tychonoff Theorem.
    • The Stone-Cech compactification.
    • Complete metric spaces. Contraction theorem.
    • Functional spaces and their topologies. The Arzela-Ascolli Theorem.
    • The Baire property and the Baire Theorem.
  2. Algebraic Topology
    • Homotopy and homotopy type. Deformation retracts and deformation retractions. Contractible spaces.
    • The fundamental group. The fundamental groups of the circle. A topological proof of the Fundamental Theorem of Algebra.
    • The Seifert-van Kampen Theorem.
    • Surfaces and their fundamental groups.
    • Covering spaces.
    • Exact sequences, 5-lemma. Homology groups, homology exact sequence. Homotopy invariance of homology groups. Excision.
    • Homology groups of \(n\)-dimensional sphere. The Brouwer Fixed Point Theorem. The degree of a map \(S^n \to S^n\). Vector fields on \(S^n\).
    • The Euler characteristic. Lefschetz Fixed Point Theorem
    • Cohomology. The cup product and the cap product. The Borsuk-Ulam Theorem.
    • The Universal Coefficient theorem.
    • Universal Coefficient Formula, The Künneth Formula.
    • Manifolds. Orientation. Homology and cohomology of manifolds. Poincaré Duality. The degree of a map.

Literature:

  1. J. Munkres, Topology, 2nd edition, Prentice Hall
  2. A. Hatcher, Algebraic Topology, Cambridge University Press;

Additional Resources:

(posted March 19, 2014)