MTH 532 General Topology and Fundamental Groups

MTH 676 Topics in Topology CRN 39583

Winter 2017

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 PohjanpeltoOffice/Phone: Kidder Hall 282B, (541) 737-5156

Office hours: Wednesdays 1:00–1:50 p.m., or by appointment

Email: juha@math.oregonstate.edu

Homepage: people.oregonstate.edu/~juha

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. |

Additional resources:

The Geometry Center web site, University of Minnesota Science and Technology Center. | |

R. Bott, L. Tu, Differential Forms in Algebraic Topology, Springer-Verlag, 1982. | |

M.J. Greenberg, J.R. Harper, Algebraic Topology, a First Course, Benjamin/Cummings, 1981. | |

J. P. May, A Concise Course to Algebraic Topology. | |

E. L. Lima, Fundamental Groups and Covering Spaces, CRC Press, 2003. | |

J. Rotman, An Introduction to Algebraic Topology, Springer, 1998. |

Grading: Your course grade will be based on a midterm exam, a take-home final exam, and four homework assignments. The midterm and final exams count 30% each toward your grade and the homework assignments 40%. The midterm is scheduled for Friday, 2/17, and the homework assignments will be due on 1/27, 2/10, 3/3, and 3/17. The due date for the final exam will be determined at a late time

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Last Changed: January 10, 2017