Math 60440 (Basic Topology II), Spring Semester 2016
General Information
Goals: The goal of this course is to learn the basic tools of algebraic topology, namely homology and cohomology.
Text: The course will be loosely based on the book ``Algebraic Topology" by Allen Hatcher. This book is recommended, but not required. It is also available online
for free downloads.
Syllabus: We will roughly cover Chapters II and III of Hatchers book; i.e., we will define singular homology groups of topological spaces, discuss their properties and develop tools to calculate homology groups (in particular for CW-complexes). Then we will define cohomology groups and the cup product, and finally we will deal with Poincare duality which relates homology and cohomology groups for closed orientable manifolds.
Grades: Grades will be based on a total of 450 points, distributed as follows:
Midterm Exam - 100 points;
Final Exam - 150 points;
Quizzes - 100 points;
Homework - 100 points;
Exams: The midterm as well as the final exam will both consist of two parts: a take-home and an in-class part, each worth half of the total points. The in-class part, similar to the quizzes, is about definitions and theorems, possibly requiring some basic calculations, but involving no proofs. The take-home part, similar to homework problems, will involve more extensive calculations and proofs.
Homework: Homework is an integral part of the course. You are permitted,
in fact encouraged, to work together and help one another with homework, although what you turn in should be written by you. Providing detailed arguments in your homework is important, since learning how to write mathematics in a rigorous and yet concise and readable way is an essential part of graduate school in mathematics. Homework problems will be posted on the course web-page and will be due on a to be determined day each week. After the due date I will provide a detailed solution to each homework set online.
Pre/post lecture preparation: As with most math classes, the material in one class
often depends heavily on stuff covered in previous classes.
Hence it is a good idea to look over your notes before the next class. I'm very happy to discuss questions concerning the material of previous classes at the beginning of each class.
Quizzes: There will be circa six 10 minute in-class quizzes during the course of the semester on a to be determined day of the week testing basic understanding of definitions and theorems.
Course website:
www.nd.edu/~stolz/Math60440(S2016)/