MATH 7512: Topology II

MATH 7512 Topology II

Spring 2005

Course Information

Course: MATH 7512 Topology II
Time and Place: Tuesday & Thursday, 9:10 - 10:30 AM, in 282 Lockett
Instructor: Dan Cohen
Office Hours: Tuesday & Thursday, 10:30 - 11:30 AM in 372 Lockett, and by appointment.
G. Pruidze, the teaching assistant for this course, will also hold an office hour
Monday, 2:30 - 3:30 PM in 379 Lockett.
Prerequisites: MATH 7510 Topology I, and basic group theory.
Grade: Based on homework (~80%) and a final exam (~20%). Homework will be announced in class, and posted here.
The final will probably be an in-class exam.
Topology, by J. Munkres (second edition). We will cover portions of Part II, Chapters 9 - 14.
Other sources for material covered in this course include:
Algebraic Topology, by A. Hatcher, Chapters 0 & 1.
A Basic Course in Algebraic Topology, by W. Massey, Chapters I - V.

Course Description

This course provides an introduction to algebraic topology. The basic idea of this subject is to associate to a topological space an algebraic object (a polynomial, a group, a ring, etc.) in such a way that topologically equivalent spaces get assigned equivalent objects (e.g., the same polynomial, isomorphic groups). Such an algebraic object is an invariant of the space, and provides a means for distinguishing between topological spaces: if two spaces have inequivalent invariants, they cannot be topologically equivalent.

The focus of MATH 7512 is on one such algebraic invariant, the fundamental group (consisting, loosely speaking, of unshrinkable loops in the topological space in question). Using this tool, we can (attempt to) reduce topological problems about spaces to purely algebraic problems about groups. For instance, with the fundamental group, we will be able to distinguish between the surface of a donut and the surface of a sphere, despite the fact that these surfaces appear the same on a small scale.

We will also pursue a number of topics and applications related to the fundamental group, including covering space theory, the Ham Sandwich Theorem, and even the Fundamental Theorem of Algebra. Time permitting, we may also pursue topics such as Kurosh's theorem on subgroups of free groups, methods for describing the fundamental group of the complement of a knot, and braids.

Department of Mathematics
Louisiana State University
Baton Rouge, LA, 70803
Office: 372 Lockett
Phone: (225) 578-1576

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