Meetings:
We will follow Nikolai Saveliev's book "Lectures on the Topology of 3-manifolds", covering major topics like Heegaard splittings, Dehn surgery, Kirby calculus, intersection forms of 4-manifolds, invariants of knots and links, and fibered knots.
Here is the publisher's page: [book]
(You should check the library for a copy first.)
On Thursday, January 23rd, I will need to join the meeting on Zoom. I'll send out a Zoom link. You all can organize/choose whether to gather in-person or on Zoom.
The topics/breakdown are subject to change.
There are many good exercises in the book. You should try them to check your understanding of each chapter.
| Date | Speaker | Proposed Topic (subject to change) |
|---|---|---|
| 1/16 | Melissa Zhang | Ch 1, Heegaard splittings and diagrams:
gluing construction, boundary orientation, smooth/top concerns; handlebodies, triangulations; Heegaard diagrams, preliminary examples (S3, S1xS2, T3, maybe RP3); crash course in Morse theory and handlebody theory in 3D (not in the book) |
| 1/23 | Annette Belleman | (Melissa will be on Zoom)
Ch 1, Examples: genus 0, Alexander trick; genus 1, framing on torus, setting conventions (in preparation for Dehn surgery), computing fundamental group; |
| 1/30 | Wren Burrill | Ch 2, Dehn surgery:
surgery on links, Lickorish-Wallace theorem; lens space and Seifert fibered space examples; surgery and 4-manifolds |
| 2/6 | Ella Curtiss | Ch 3, Kirby calculus:
linking number, Kirby moves; linking matrix (for use later in comparison with intersection form) There are a lot of tricks here to get used to, so any extra time should be spent on working useful examples. |
| 2/13 | TBD | Ch 5, 4-manifolds:
intersection form, properties that relate to topology (e.g. unimodularity) and Kirby calculus; don't get bogged down on deep algebraic topology / gauge theory and stick to the topology + linear algebra |
| 2/20 | TBD | Ch 4, Ch 6, 4-manifolds with boundary:
intersection form vs. linking matrix; even surgeries, characteristic sublink; Seifert homology sphere examples; (No need to define the Rohlin invariant if there's no time; can return when we get to Chapter 11.) |
| 2/27 | Chris Luevano | Ch 7, Seifert surfaces:
Seifert surfaces, Seifert matrix, S-equivalence, intersection form; Alexander polynomial / module, briefly; There's a lot that can be expanded on here; we can discuss based on interest. |
| 3/6 | TBD | Ch 8, fibered knots:
fibered knots in various 3-manifolds, monodromy, examples; open-book decompositions |
| 3/13 | TBD | Ch 9, 10, 11, Arf invariant and Rohlin's invariant:
This is a lot; the speaker will have to pick and choose. We also might not get to this lecture if a previous lecture needs to be split into more. |