## MATH 527, Spring 2021

**Course Title:**Topology**Instructor:**Nigel Higson**Meeting Times:**Tuesdays and Thursdays, 12:05-1:20 via Zoom (contact me for coordinates). Because some students will be joining from distant time zones, I shall record the lectures and make the recordings available to the class. I shall share details about that when I’ve figured them out.**Office Hours:**I shall arrange weekly “tea times” during which anyone is welcome to drop in and discuss topology. These will be: Mondays, 4:00-5:00pm on January 25, February 8, 22, March 8, 22 and April 5, 19; and Tuesdays, 4:00-5:00pm on February 2, 16, March 2, 16, 30 and April 13, 27. All teas will be held via Zoom, using the class Zoom address. The teas will be open to all, simultaneously, but if you want to meet with me privately, then you can request an appointment (contact me in class or by email).**Prerequisites:**The main requirements will be familiarity with the basic concepts of general topology (topological spaces, continuity, campactness, connectedness and so on) as well as familiarity with basic abstract algebra (groups, homomorphisms, quotient groups) and linear algebra. Some experience with manifolds will be helpful, but not required.**Overview:**The main topic of the course will be the homology and cohomology of topological spaces: I’ll give definitions, properties and applications to manifolds and fixed point theory. But I’ll begin with a review of general topology and a discussion of covering spaces and the fundamental group.**Texts:**There are plenty of decent books out there, but I’ll mostly use Rotman - An introduction to algebraic topology. If you’re a Penn State student you should be able to follow the link and download the book at no charge for class use. Here are some other standard books, also available at no charge: Fulton - Algebraic topology, a first course, Hatcher - Algebraic topology, Vick - Homology theory, Weintraub - The fundamentals of algebraic topology. In addition, I’ll probably recommend some other sources during the lectures.**Homework**: I’ll hand out occasional homework problems to help you deepen your understanding of the material that we’ll cover. I’ll ask you to hand in some, but not all of the homework, so that I can monitor your progress in presenting your arguments clearly. I’ll try to devote class periods once in a while to a discussion of problems.**Exams:**There will be none. But many of you will be preparing for the qualifying exam in topology, and I will perhaps create one or two practice exams based on previous qualifying exams and other sources.**Academic Integrity:**Students must meet University and the College standards of academic integrity. The University defines academic integrity as "the pursuit of scholarly activity in an open, honest, and responsible manner." It goes on to say that "academic integrity includes a commitment not to engage in or tolerate acts of falsification, misrepresentation or deception. Such acts of dishonesty violate the fundamental ethical principles of the University community and compromise the worth of work completed by others." See this page.**Disability Statement:**Penn State welcomes students with disabilities into the University's educational programs. If you have a disability-related need for reasonable academic adjustments in this course, contact the Office for Disability Services (ODS) at 814-863-1807 (V/TTY). For further information regarding ODS, please visit the Office for Disability Services Web site at http://equity.psu.edu/ods/. In order to receive consideration for course accommodations, you must contact ODS and provide documentation (see the documentation guidelines at http://equity.psu.edu/ods/guidelines/documentation-guidelines). If the documentation supports the need for academic adjustments, ODS will provide a letter identifying appropriate academic adjustments. Please share this letter and discuss the adjustments with your instructor as early in the course as possible. You must contact ODS and request academic adjustment letters at the beginning of each semester.