Mathematics
 

Foundations of Topology 2

Foundations of Topology 2
An introduction to the topology of manifolds with particular emphasis on low dimensions: topological and smooth manifolds; smooth maps; tangent spaces; vector fields and flows; Sard's Theorem; Morse Theory; classification of compact manifolds in dimensions 1 and 2; and theory of 3 dimensional manifolds.
MATH
554
 Hours3.0 Credit, 3.0 Lecture, 0.0 Lab
 PrerequisitesMath 553 or instructor's consent.
 TaughtWinter
 ProgramsContaining MATH 554
Course Outcomes: 

Learning Outcomes

Students should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the concepts below, related to, but not identical to, statements proven by the text or instructor. For more detailed information visit the Math 554 Wiki page.

Overview

  1. The Fundamental Group
  2. The topology of the plane
  3. Jordan Curve Theorem
  4. Seifert-van Kampen Theorem
  5. Classification of Surfaces
  6. Classification of Covering Spaces
  7. Group Theory
  8. Free groups
  9. Free abelian groups
  10. Presentations of groups
  11. Subgroups of free groups