MTH 532 General Topology
MTH 532 General Topology and Fundamental Groups
MTH 676 Topics in Topology CRN 39583
Winter 2017

The primary objective in algebraic topology is to study the shape of geometric spaces by associating algebraic objects to them that remain unchanged under homeomorphisms and continuous deformations. The fundamental group, higher homotopy groups, homology and cohomology groups of a topological space provide the principal examples of such invariants.

In the past century algebraic topology, originally known as combinatorial topology, has evolved into an indispensable tool in topology and geometry, and it bears deeply on various other areas of mathematics, including global analysis, group theory, homological algebra, and number theory. Algebraic topology also affords novel and striking applications to physics, computer science, economics, and biology as a tool for uncovering hidden structures and for identifying obstructions to the equivalence of two objects.

MTH 532 will be taught as an introductory course to the fundamental group of a topological space and its applications. Emphasis will be placed on developing computational techniques required in the analysis of this key invariant. We will use the book Topology, 2nd Ed., by James Munkres as the course text.

Instructor: Juha Pohjanpelto
Office/Phone: Kidder Hall 282B, (541) 737-5156
Office hours: Wednesdays 1:00–1:50 p.m., or by appointment
Email:
Homepage:

Prerequisites: The course will by and large be self-contained but a working knowledge of the rudiments of abstract algebra and topology will be assumed.

 Topics: • Homotopy of paths and the definition of the fundamental group • Covering spaces and their classification • Separation theorems in the plane • The Seifert-van Kampen theorem

 Learning objectives: Upon successfully completing this course a student • appreciates the general notion of the fundamental group. •  recognizes the essential computational techniques for determining the fundamental group and capably applies these in standard examples from topology. • is acquainted with the key applications of the fundamental group. • comprehends the role of the fundamental group in the classification of covering spaces.