Math 3330, Elements of Topology.

Instructor: Ovidiu Munteanu
Office: MONT 434, Phone: (860) 486-4003
Office Hours: MONT 434, WF 11:00 am - 12:00 pm
Class Meets: MONT 414, MWF 10:10 - 11:00 am

Textbook:  Introduction to Topology, Second Edition, by Theodore Gamelin and Robert Greene, Dover 1999
 
Course description: Topology is the study of the shape of objects. It considers properties that do not change under stretching and bending, and is fundamental to many areas of mathematics: geometry, analysis, and even an area as apparently non-geometric as number theory. Topology also has applications in computer science (distributed computing) and biology (tangled DNA). Some objectives of this course will be to understand metric spaces and topological spaces, particularly properties these spaces may have such as compactness, connectedness and separability. We will also look at techniques for obtaining new spaces from old ones, such as product spaces and quotient spaces. Carefully selected examples will develop our intuition for the subject and motivate the results we find. At the end of the course we will learn how to distinguish topological spaces from one another using topological invariants related to combinatorics and algebra, such as the Euler characteristic, the fundamental group, and homology groups. 

Other useful texts:
1. Topology, Second Edition,  by James Munkres,  Pearson 1974.
2. Introduction to topological manifolds, by John Lee, Springer 2011.
3. Introduction to Topology, Third Edition,  by Bert Mendelson, Dover 1990.
4. Online notes on Allen Hatcher's homepage.


Homework: There will be several homework assignments (graded). The homework score makes up for 40% of your total grade in this class. Please make sure to hand in the homework before its due date, as I do not accept late homework (for any reason). You may discuss homework with other students, or look for other resources, but you are expected to write the solutions on your own.  

Midterm: There will be an in class midterm, its projected date is Wednesday, March 7. This will be 30% of the total grade.

The final exam is also in class, on the date scheduled by the registrar. The final exam is 30% of the total grade.

Syllabus, subject to change. Depending on need or interest, we may spend more time on a certain topic.
Week1 Set theory, metric spaces.
Week2 Metric spaces, topological spaces, subspaces.
Week3 Continuous functions, base of topology.
Week4 Separation axioms, compactness.
Week5 Compactness, local compactness.
Week6 Connectedness, path connectedness.
Week7 Product spaces, group actions, quotient spaces.
Week8 Quotient spaces, Midterm  covers material from weeks 1-6.
Week9 Topological surfaces: examples, polygonal representation.
Week10 Euler characteristic, classification of compact surfaces.
Week11 Homotopy theory. 
Week12 The fundamental group and examples. 
Week13 Computations for surfaces.
Week14 Covering maps, review.