### MAT 530 Geometry/Topology I  Fall 2008 TTh 9:50-11:10 Physics P-123

Prof. Claude LeBrun.
Office hours:
TuTh 11:30 -- 12:30,
Math Tower 3-108.

This course offers an introduction to both point-set and algebraic topology. It is one of the core courses of the mathematics graduate program at Stony Brook.

## Homework Assignments

Textbook: J. Munkres, Topology,
2nd edition, Prentice Hall, 1999.

 Assignment Due Date Section: Problems #1 Tuesday, September 23 §17:   9, 11, 12 §18:   11, 12, 13 §20:   1, 3 #2 Thursday,October 2 §22:   2, 4, Suppl. Ex. 5 §23:   2, 5, 9 §24:   1, 2, 3, 8 #3 Tuesday,October 14 §25:   9, 10 §26:   5, 6, 7, 8 #4 Tuesday,October 28 §27:   2, 5 §28:   4, 5 §30:   3, 4, 5 #5 Tuesday,November 25 §52:    4, 7 §53:    5 §54:    6, 8 #6 Monday,December 15* §58:    2, 4, 5 §59:    2, 3 §73:    1, 2 §79:    1, 2, 3, 4

*Note that our last meeting for the semester will take place on "correction day."

## ☞ Announcement Regarding the Final Exam ☜

The final exam will employ a take-home format. This will allow those needing to leave early for the holidays (e.g. in order to spend Christmas with their families abroad) to turn in their exams well before the "official" due-date of Dec. 23.

However, those of you with teaching and/or grading duties should also consult with your work supervisors to ensure that you will actually be able to fulfill all your responsibilities before any such early departure for the holidays.

Syllabus: We will cover most of the basic core course syllabus given below. I will also discuss appropriate additional topics, as time permits.

1. Basic point-set topology
• Metric Spaces
• Topological spaces and continuous maps
• Comparison of topologies
• Separation axioms and limits
• Countability axioms, the Urysohn metrization theorem
• Compactness and paracompactness, the Tychonoff theorem
• Connectedness
• Product spaces
• Function spaces and their topologies, Ascoli's theorem
2. Introduction to algebraic topology
• Fundamental group
• Fundamental group of Sn; examples of fundamental groups of surfaces
• Seifert-van Kampen theorem
• Classification of covering spaces, universal covering spaces; examples
• Homotopy; essential and inessential maps

Your grade will be based upon your performance on the homework (50%), the mid-term (25%), and the Final Exam (25%). Incompletes will be granted only if documented circumstances beyond your control prevent you from taking the final exam.

Homework assignments and other useful information regarding the course will be posted regularly on this web-page.

You may e-mail Prof. LeBrun by This is the best method for making appointments outside normal office hours.

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DSS advisory. If you have a physical, psychiatric, medical, or learning disability that may affect your ability to carry out the assigned course work, please contact the office of Disabled Student Services (DSS), Humanities Building, room 133, telephone 632-6748/TDD. DSS will review your concerns and determine what accommodations may be necessary and appropriate. All information regarding any disability will be treated as strictly confidential.

Students who might require special evacuation procedures in the event of an emergency are urged to discuss their needs with both the instructor and DSS. For important related information, click here.