# Introduction to Geometry and Topology (5310/4310)

## Prelim

Here is an updated prelim study guide to give you an overview of the content of the prelim. Since the syllabus of 5310 changed a bit, the information on the Past Prelim Exams Wiki page is somewhat outdated; some of the old prelim problems will still be useful, other maybe less so, and they don't cover the Algebraic topology part of the new 5310 syllabus. For the latter, almost all problems in Hatcher, Chapter 1, are useful (you can skip those involving CW complexes, obviously). Also, you may want to look for Prelims given at other Universities; there is a collection of links to prelim collection ath the prelim page of Ohio State University. Universities with a similar topology prelim include LSU, Purdue and Northeastern (Northeastern's exam is based on a more advanced course, with similar topics).

Office: Room MSB 206
Office phone: 486-4237
Office hours: Feel free to drop by any time. My official office hours are Mondays, 11am-12pm and Thursdays after class (2pm-3pm).
Email: arend.bayer@uconn.edu

QUICK SURVEY: Have you taken a topology class before? If you haven't done that yet, please let me know by filling out a 5-second google survey.

## Class information

This class will be an introduction to topology. Topology is about ''things that don't change when you wiggle around''. E.g. you can wiggle (deform) a donut into a coffee cup with handle without every breaking anything, but you can't deform a donut into a pretzel. The goal of topology is to make such a statement (and many more!) precise, rigorous, and useful.

Roughly the first half will be a crash-course through point set topology. In the second half (which, as in most sports, will probably be longer than the first half), we will get started on algebraic topology, in particular to the fundamental group and its applications.

ENROLLMENT CAP: You may currently be unable to sign up for the class because it has reached the enrollment cap as set with the registrar. However, anyone is still welcome to join the class; please e-mail me if you want to sign up and I will send you a permission number.

TEXTBOOK: There is no mandatory textbook for this course, and there is no single book that I will follow closely. The first half of the course is covered in most point-set topology textbooks, so if you already own one that you like, you may want to keep using it. (Feel free to e-mail me if you are wondering whether the book you own would be adequate.) A particular inexpensive one, and also a good reference, is Gamelin, Greene, Introduction to Topology. Another option would be to rely on the notes you take in class, where I will cover all relevant material explicitly.
The second half will cover pretty much the material of Chapter I of Hatcher, Algebraic Topology, which I recommend very much; it is also freely (and legally) available for download.
The material of the full course is covered by Lee, Introduction to Topological Manifolds, which should be available at the bookstore (and possibly cheaper elsewhere). Another book that some people like is Munkres, Topology; it goes into many more details than we will be able to in the class.
If you are interested in getting an idea of topology before the start of the semester, I highly recommend Jaenich, Topology, which is a very nicely written introduction to topology (and is also thin and readable).

Author Title ISBN
Allen Hatcher Algebraic Topology 0-521-79540-0
Theodore W. Gamelin and Robert Everist Greene Introduction to Topology 0486406806
John M. Lee Introduction to Topological Manifolds 0-387-98759-2
Klaus Jänich Topology 0-387-90892-7
James R. Munkres Topology 0131816292

HOMEWORK: You will get homework (with about 4-5 problems) every two weeks. You are encouraged to do the homework in groups. The only rule is that in the end, you have to write the solution in your own words; also, I will ask you to give list of names of everyone else in your group.
Given the size of the class I won't be able to grade every problem. Hence I will also ask you to mark one problem that you definitely want me to grade (e.g. the most difficult problem that you could solve, or the one where you found the nicest solution).

EXAMS: There will be one midterm in class, and a cumulative final. The midterm will be October 28 (tentatively).

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