| Date |
Lecture Content |
| 8/19/25 |
Introduction. Dehn Surgery on knots. |
Saveliev, Lectures on the Topology of 3-Manifolds: Lecture 2 §2.1-2.3 (see also Lecture 1 for some relevant background)
Further reading: Gordon, Dehn surgery and 3-Manifolds (2006). Many examples and exercises. (Not up to date, though.)
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| 8/21/25 |
Wirtinger presentations. Fundamental groups of surgeries. |
|
Rolfsen, Knots and Links: The Wirtinger Presentation §3.D
|
| 8/26/25 |
SU(2) and quaternions. Representations spaces and conjugation action. |
Saveliev, Lectures on the Topology of 3-Manifolds: Lecture 13
house notes: SU(2) as unit quaternions
|
| 8/28/25 |
Stabilizer types of representations. The reducible part of the character variety. |
| house notes: Stablizers and orbits, and the reducible characters
|
| 9/2/25 |
SU(2) character varieties of knots. Example computations. The pillowcase. |
|
Kirk, SU(2) Representation Varieties of 3-manifolds, Gauge Theory Invariants, and Surgery on Knots: pdf Ch.1
|
| 9/4/25 |
Torus knots. Herald's Theorem on the structure of the SU(2) character variety (intro). |
|
Klassen, Representations of knot groups in SU(2): pdf IA, IB
|
| 9/9/25 |
Perturbations, Herald's Theorem. Character varieties, amalgamated groups, and Dehn surgery. |
Herald, Legendrian cobordism and Chern-Simons theory on 3-manifolds: link
Herald, Kirk, Holonomy perturbations in a cylinder... link, see especially §2
|
| 9/11/25 |
Understanding Dehn surgery in the pillowcase. Proving first cases of Thm 1 from from [KM] |
|
Kirk paper above, KM paper above
|
| 9/16/25 |
Proof of Thm 1 from [KM]. Perturbations in a thickened torus. |
|
KM paper above; Herald-Kirk paper above does more involved perturbations
|
| 9/18/25 |
Local structure of representation spaces. Zariski tangent spaces. Group cohomology. |
Saveliev, Lectures on the Topology of 3-Manifolds: Lecture 15
Weil, Remarks on the cohomology of groups: pdf
|
| 9/23/25 |
Homology with local coefficients. Zariski tangent space computations for the pillowcase. |
|
Kirk paper above, Hatcher Algebraic Topology § 3.H
|
| 9/25/25 |
Local structure at a reducible, and relation to the Alexander polynomial. |
Lickorish, An introduction to Knot Theory Ch 6
Milnor, Infinite cyclic coverings §1-2
|
| 9/30/25 |
More on Alexander polynomial, definition of Levine-Tristram signature. |
|
See Milnor paper above
|
| 10/2/25 |
Herald's theorem on Levine-Tristram signature and SU(2) representations. |
|
Herald, Flat connections, the Alexander invariant, and Casson's invariant: link
|
| 10/7/25 |
Definition of Floer's instanton homology for admissible pairs. |
Floer, Instanton homology and Dehn surgery: link
Kronheimer and Mrowka, Knots, three-manifolds and instantons ICM notes: pdf
|
| 10/9/25 |
More instanton homology. Kronheimer and Mrowka's non-vanishing result and implications. |
|
Kronheimer and Mrowka ICM paper, also their Knots, sutures, and excision: link
|
| 10/14/25 |
No class (Fall break) |
|
|
| 10/16/25 |
Proof sketch of Kronheimer and Mrowka's non-vanishing result. Smith Conjecture. |
Kronheimer and Mrowka, Witten's conjecture and Property P: link
See also the references from the previous lecture
|
| 10/21/25 |
Surfaces in 3-manifolds and trees: from surfaces to trees. Examples. |
|
Shalen, Representations of 3-manifold groups, § 1: link
|
| 10/23/25 |
From surfaces to trees. (Zoom lecture) |
|
Shalen § 2
|
| 10/28/25 |
Nontrivial actions on trees and essential surfaces in 3-manifolds. |
|
Shalen § 2
|
| 10/30/25 |
From discrete valuations to trees. |
|
Shalen § 3; Boyer, Applications of character variety methods to Dehn surgery, § 3.1-3.2: link
|
| 11/4/25 |
Properties of the SL_2 action on the tree of a discrete valuation. |
|
Shalen § 3; Boyer, § 3.3
|
| 11/6/25 |
SL(2,C) representation and character varieties. |
|
Shalen § 4; Boyer § 4-5
|
| 11/11/25 |
The relation between ideal points and essential surfaces. |
|
Shalen § 5; Boyer § 6
|
| 11/13/25 |
Boundary slopes. Separating essential surfaces in knot complements. Culler-Shalen seminorms. |
|
Shalen § 5.6, § 8; Boyer § 7
|
| 11/18/25 |
The cyclic surgery theorem. |
|
Shalen § 9.1; Boyer § 8
|
| 11/20/25 |
... |
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|
| 11/25/25 |
Thanksgiving break. |
|
|
| 11/27/25 |
Thanksgiving break. |
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|
| 12/2/25 |
Last class |
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