MATH6071A - Topics in Topology I - 2015/16

Announcement

• There is no class on Sept 8 (Sept 7's lecture had extra 45mins)

General Information

Lecturer

• Yi-Jen LEE
  • Office: AB1 412
  • Email:

Time and Venue

• Lecture: Mondays 7:30-9:00 pm; Tuesdays 4:45-5:30pm at AB1 502a

Course Description

Selected topics in low dimensional topology.

References

• Schultens--Introduction to 3-manifolds
• Casson & Bleiler--Casson & Bleiler--Automorphisms of surfaces after Nielsen and Thurston
• Buoncristiano--Fragments of geometric topology from the sixties, part I. (http://www.emis.de/journals/GT/gtmcontents6.html)
• Hempel--3-manifolds
• Rolfsen--Knot and Links
• Gordon--Lectures on normal surface theory. (https://www.math.wisc.edu/~rkent/normal.pdf)
• Lackenby--Lectures on 3-manifolds. (https://homepages.warwick.ac.uk/~masgar/Articles/Lackenby/thrmn812.pdf)
• Jaco--Lectures on 3-manifold topology.
• Hatcher--Notes on basic 3-manifold topology. (https://www.math.cornell.edu/~hatcher/3M/3M.pdf)
• Aschenbrenner & Friedel & Wilton--3-manifold groups
• Jenkins & Neumann--Lecture notes on Seifert manifolds
• Caligari--Notes on 3-manifolds (http://math.uchicago.edu/~dannyc/courses/3manifolds_2014/3_manifolds_notes.pdf)
• Thurston--Unpublished lecture notes (http://library.msri.org/books/gt3m/PDF/1.pdf)

Pre-class Notes

• W1 &2: Buoncristiano; Schultens Ch.1
• W3: Buoncristiano; Schultens Ch. 3.
• W4 &5: Schultens Ch.3 &5; Rolfsen Ch. 9B; Hempel Ch. 13.
• W6-8: Schultens Ch.3&5; Gordon's and Lackenby's notes.
• W8: Jaco Ch.6.
• W10: Hatcher Section 1.2.
• W11& 12: Jankins-Neumann;
• W13: Aschenbrenner-Friedel-Wilton Ch. 1; Caligari Ch.5; Thurston Ch.4; Casson & Biler

Lecture Notes

• Week 1& 2: the three categories of manifolds (TOP, PL, DIFF); summary and current status of essential basic results/conjectues.
• Week 3: (TOP/PL/DIFF) R^n and S^n; their automorphisms. Connected sums. Knot equivalences and isotopy extension theorems. (3-manifolds): Prime decomposition, Irreducible manifolds, essential surfaces, The sphere theorem.
• Week 4& 5: Dehn's Lemma; loop theorem; Papakyriakopoulos's tower construction; criterion for existence of incompressible surfaces; Hierarchies and Haken's theorem; Waldhausen's theorem; 3-manifolds that are homotopic but not homeomorphic; Lens spaces.
• Week 6: Whitehead manifolds; normal surface theory.
• Week 7: Haken finiteness; Diophantine equalities. "Topological rigidity" for surfaces.
• Week 8: Topological rigidity for Haken manifolds; Algorithm to detect the unknot; Seifert fibered spaces;
• Week 9: East Asian Symplectic Conference
• Week 10: Surfaces in Seifert fibered spaces; the JSJ decomposition.
• Week 11: Seifert invariants, uniqueness of SFS structures; SFS and \pi_1; Brieskorn spheres; manifolds;
• Week 12: "Geometrization" of 2-d orbifolds and SFS
• Week 13: Torus theorem and Seifert fibered theorem; Thurston's geometrization of 3-manifolds; Sol and Nil manifolds; Thurston's classification of surface automorphism and geometry of mapping tori; examples of hyperbolic and non-Haken manifolds from knot surgery

Assignments

• Week 2: Read Ch. 1& 2 of Schultens