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)