Topology

Instructor

Prof. Paul Gunnells, LGRT 1115L, 413.545.6009, gunnells at math dot umass dot edu.

Office Hours

Tuesdays and Thursdays, 8:30-9:30, and by appointment.

Overview

This course covers the basics of point set topology and the fundamental group. Topics include product and quotient topologies, continuity, compactness, connectedness, and complete metric spaces. Grade will be based on homework problems and exams. This course is the basis of the Topology Basic Exam for PhD students in the Department of Mathematics and Statistics.

Textbook

• Topology, James Munkres - 2nd Edition (Required) Algebraic Topology
• Algebraic Topology, Allen Hatcher (recommended), available free online at Hatcher's website (Cornell Math Department).

Grading

Problem sets will count 30%. Late problem sets will not be accepted. There will be a midterm exam worth 30% and a final exam worth 40%.

Course material

The following is tentative and may change as the course progresses.

• Topologies and continuity (Sections 12-22 of Munkres).
• Connectedness and compactness (Sections 23-29 of Munkres).
• Fundamental Group (Sections 1.1, 1.2 of Hatcher).
• Completeness (Sections 43, 45, 46 of Munkres).
• Countability and Separation axioms (Sections 30-33 of Munkres (hopefully)).
• Covering spaces (Section 1.3 of Hatcher (hopefully)).
• Schedule

Exam Dates

All exams will be held in-class.

• Midterm, 22 October (Thursday).
• Final exam, 10 December (Thursday). This is the last regular class meeting.

Problem Sets

All exercises are in Munkres unless otherwise indicated.

• I: Due Thursday, 24 Sept. p.83: 1, 3, 8. p.91: 1, 4, 6. p.100: 2, 8, 9, 13, 14.
• II: Due Thursday 1 Oct. p.111: 2, 9, 13. p.118: 2, 6, 8.
• III: Due Tuesday 20 Oct. p.127: 3, 4, 9, 10. p.133: 7, 9, 11. p.144: 2, 4.
• IV: Due Thursday 5 Nov. p.152: 2, 4, 5, 10. p.157: 1, 2, 3. p.162: 1. p.170: 4, 6, 7, 8.
• V: Due Tuesday 1 Dec. p.177: 2, 4. p.181: 6, 7. p. 186: 1, 8. Additional problems on Pi_1