MATH 7512: Topology II

# MATH 7512 Topology II

## Course Information

 Course: MATH 7512 Topology II Time and Place: Tuesday & Thursday, 12:10 - 1:30 PM, in 280 Lockett Instructor: Dan Cohen Office Hours: Tuesday 1:30 - 3:00, Wednesday 11:00 - 1:00 in 372 Lockett, and by appointment 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. Text: Algebraic Topology, An Introduction, by W.S. Massey, Springer-Verlag, GTM 56† We will cover portions of Chapters II--V, and topics from Chapters VI, VII, and the Appendices. This book is on reserve at Middleton Library. The call number is QA612.M37. Other sources for material covered in this course include Chapter 8 of Topology, A First Course, by J. Munkres This book is also on reserve at Middleton Library. The call number is QA611.M82. Chapter 1 of Algebraic Topology I, by A. Hatcher, available his website

## 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.

Students who anticipate taking Algebraic Topology (MATH 7520) next Fall might consider purchasing the text A Basic Course in Algebraic Topology, Springer-Verlag, GTM 127 (also by W.S. Massey). This book contains all the material in the text book noted above, as well as much (if not all) of the material typically covered in MATH 7520. As of January 19, 2000, this text was on sale for \$38.50 as part of the Springer-Verlag Yellow Sale. Click on the title of the book for further details.

Department of Mathematics
Louisiana State University
Baton Rouge, LA, 70803
Office: 372 Lockett
Phone: (225) 388-1576
Email: cohen@math.lsu.edu