MATH 7512 Topology II

Course Information

 Course: MATH 7512 Topology II Time/Place: Monday, Wednesday, Friday, 11:40 am - 12:30 pm, in 132 Lockett Instructor: Dan Cohen Office Hours: Monday, Wednesday, Friday, 2:30 - 3:30 pm, and by appointment in 372 Lockett Prerequisites: MATH 7510 (and MATH 7210), or equivalent. Grade: Based on in-class quizzes (25%) and homework (75%), including a final homework set which will be due during finals week. Quizzes will be used to test familiarity with various definitions, terms, and examples encountered in class. Homework problems will be assigned and collected regularly in class throughout the semester. The purpose of these exercises is to help the student engage with the material, and to develop mastery of the techniques and concepts introduced in the course. Homework will be posted here. Text: A. Hatcher, Algebraic Topology, Cambridge University Press. Some other potential sources for material covered in this course may be found here.

Course Description

A fundamental problem in topology is that of determining whether or not two spaces are topologically equivalent. The basic idea of algebraic topology is to associate algebraic objects (groups, rings, etc.) to a topological space in such a way that topologically equivalent spaces get assigned isomorphic objects. Such algebraic objects are invariants of the space, and provide a means for distinguishing between topological spaces. Two spaces with inequivalent invariants cannot be topologically equivalent.

The focus of this course will be on homology and cohomology. To a topological space, we will associate sequences of abelian groups, the homology and cohomology groups. Topics (from chapters 2 and 3 of Hatcher's book) include simplicial, singular, and cellular homology, Mayer-Vietoris sequences, and universal coefficient theorems. Geometric examples, including surfaces and projective spaces, will be used to illustrate the techniques. Discussion of cohomology theory, including products and duality, will continue in MATH 7520 Algebraic Topology.

Department of Mathematics
Louisiana State University
Baton Rouge, LA, 70803
Office: 372 Lockett
Phone: (225) 578-1576
Email:
URL:   www.math.lsu.edu/~cohen