## MAT 402 Spring 2004

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Important: I shall be absent due to professional meetings on TU FEB 24, TH Feb 26, TU Mar 30, TU Apr 27, TH Apr 29. Please check the web page before those dates in case my trips are canceled. In that case, classes WILL TAKE PLACE.

Important: Schedule of ``make-up" classes. Rooms to be announced. Wednesdays Mar 3 (5-127), April 21, 12:40 - 2:00 (5-127).

Lectures: TU and TH 11:20pm - 12:40 pm, Physics P-129

Instructor: Mark Andrea de Cataldo. Office hours: TU 1-2 MAT P-143 and by appointment.

The topic of study will be Covering Spaces. A covering space X of a space Y is, roughly, one that ``locally" looks like Y but is ``globally" quite different. For example: the line is a covering space for the unit circle: just wrap the line around the circle infinitely many times. Another example: take the unit disk in the plane, take its boundary, the circle of radius one, and identify to each other the pairs of points on the circle of type (a,b) and (-a,-b). You get a surface, called the real projective plane, which admits the sphere as a 2:1 covering space. The relation between the space Y and its covering spaces X is explained precisely via the notion of the fundamental group. In the former example this is the additive group Z, in the latter it is the additive group Z/2Z. The fundamental group is, as the name suggests!, also an intriguing object: it describes the different ways in which one can wrap a rope around spaces. In this seminar, we would start with reviewing basic topology, study the fundamental group, study covering spaces. There will be many concrete and many less concrete examples.

Tentative Syllabus. The map z-->z^n, the exponential map, the logarithm. Winding numbers. Topological spaces. The fundamental group. Examples. Covering spaces. Classification of covering spaces in terms of conjugacy classes of subgroups of the fundamental group. Applications to maps of spehres (Borsuk-Ulam Theorem), to group theory (subgroups of free groups via graphs and trees). Coverings associated with algebraic functions and connection with the Galois theory of fields.

Syllabus. Lecture 1,2: z->z^n, log. Lecture 3: associated surfaces, winding number. Lecture 4: winding number (cont.), composition of paths, inverse paths. Lecture 5: homotopic paths, fundamental group. Lecture 6: reminder of algebra presentation. Lecture 7: reminder of point-set topology presentation. Lecture 8: reminder continued. Lecture 9: the fundamental group is a group. Lecture 10: lifting paths. Lecture 11: lifting homotopies; the fundamental group of a circle. Lecture 12: fundamental group of a product; covering spaces. Lecture 13-20: covering spaces: basic properties and classification. Remaining lectures: presentations.

I will give introductory lectures and all the other participants will lecture on the topics. Preferably in groups of two.

Textbook: W. Fulton, Introduction to Algebraic topology, Springer. I will hand out additional material.

Grade: based on in-class participation, the presentations and homework. The will be no final exam.

PRESENTATIONS.

1) Reminder of basic algebra, Tom Clark and Althea Smith.

2) Reminder of point-set topology, Vincent Beltrami and Samir Shah.

3) Fundamental groups of spheres: Lesley Burnett.

4) Homology: Amy Roberts and Fatima Zarinni.

5) Classification of compact surfaces: Vincent Beltrami, Tom Clark and Althea Smith.

Homework.

1) discuss z->z^3, z->z^n. Due 2/5.

2) prove the continuity lemma. Due 2/19.

Special needs. If you have a physical, psychological, medical, or learning disability that may impact on your ability to carry out assigned course work, you are strongly urged to contact the staff in the Disabled Student Services (DSS) office: Room 133 in the Humanities Building; 632-6748v/TDD. The DSS office will review your concerns and determine, with you, what accommodations are necessary and appropriate. All information and documentation of disability is confidential. Arrangements should be made early in the semester (before the first exam) so that your needs can be accommodated.

• Mark Andrea de Cataldo's homepage.