2018 Spring Course Syllabus - Mathematics 9072.001
Course: Mathematics 9072.001.
Course Title: Differential Topology II: The Life and Times of the Poincare Homology Sphere.
Time: 11:00-12:20 TTh.
Place: Wachman 527.
Instructor: Edgar Bering.
Instructor Office: Wachman 1021.
Instructor Email: edgar.bering@temple.edu
Instructor Phone: NA.
Office Hours: 10:00-10:50 TTh or by appointment.
Prerequisites: Math 8061-2.
Textbook: ``Eight Faces of the Poincare Homology 3-Sphere'' Kirby and Scharlemann; and other sources on reserve in the library.
Course Goals: The goal of this course is to survey constructions, classifications, and structure theorems used in three and four dimensional topology, using a rich family of examples with the PoincarÃÆÃ'Ãâ homology sphere as the central example.
Topics Covered: Before stating his famous conjecture Poincare claimed, in the second supplement to Analysis situs, that a 3-manifold has the homology groups of the 3-sphere if and only if it is homeomorphic to the 3-sphere. Later, in the fifth supplement, and after almost decade of correspondence with Heegaard and other early topologists, Poincare discovered a counterexample: his famous homology sphere, a 3-manifold that is not homeomorphic to the 3-sphere but has the same homology groups. This discovery led to the well-known the Poincare conjecture: a 3-manifold with trivial fundamental group is homeomorphic to the 3-sphere. Since then, the Poincare homology sphere has been a driving example in low-dimensional toplogy. In this course we will sample themes (constructions, classification theorems, and structure theorems) from the past century and a quarter of low-dimensional topology, using the Poincare homology sphere as our guide. We will touch on and connect knot theory, mapping class groups of surfaces, crystallographic groups, singularities of algebraic surfaces, and more; meeting a menagerie of mesmerising manifolds in the meanwhile. Time permitting we will discuss more contemporary (related) subjects using the Poincare sphere, these will be chosen by class interest. Some (interrelated) possibilities are geometric structures on 3-manifolds, Ricci flow, the L-space conjecture, and contact topology.
Course Grading: The course grade will be based on writing for the course blog and written exercises, as well as a presentation.
Exam Dates: No examinations.
Attendance Policy: Attendance will not be closely monitored. This is a graduate topics course, my expectation is that as senior graduate students class participants will behave like professionals and be present and engaged.
Any student who has a need for accommodation based on the impact of a disability should contact me privately to discuss the specific situation as soon as possible. Contact Disability Resources and Services at (215) 204-1280, 100 Ritter Annex, to coordinate reasonable accommodations for students with documented disabilities.
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