SIPS-GT is an online seminar (via Zoom) dedicated to looking into interesting problems in Group Theory and Topology during the summer months. In Summer 2021, we will be focusing on the Generalized Property R and Andrews-Curtis conjectures.

Each week, someone will give a talk by Zoom that lasts about a half hour, with discussion to follow. Our goal is not necessarily to solve these problems, but to make strides in understanding their current status. Most talks will be of an expository and informal nature, but if you wish to give a talk on personal research into these problems, or wish to give a more “formal” talk in preparation for a conference or other speaking engagement, please feel free to do so!

The organizer of this seminar is Nick Meyer You can email him if you have any questions.

If you are interested in getting reminder emails with the Zoom info (and are not already receiving them), please sign up at this Google form

Seminar Google Drive folder: http://bit.ly/sips-gt-lit

Theme: Generalized Property R and Andrews-Curtis Conjectures

In this seminar, we will investigate the status of various open problems in low-dimensional topology and combinatorial group theory. Many of these open problems are intimately connected, and understanding these connections gives a fantastic glimpse into what is still unknown. We will focus on the following conjectures/problems:

• The Generalized Property R Conjecture (GPRC) posits that if an n-component link has 0-framed Dehn surgery to $\#^n \left(S^1 \times S^2\right)$, then that link is handleslide equivalent to the an n-component unlink. This is usually attacked 4-dimensionally by first creating a homotopy 4-sphere X from our 0-framed link, and then modifying its handle structure in a way that changes our initial link into an unlink.
• The Andrews-Curtis (AC) conjecture posits that every balanced presentation of the trivial group can be transformed to a balanced trivial presentation using only AC moves (which resemble Tietze transformations but are slightly more restrictive).

It turns out that AC and GPRC are related in the follow way: Suppose you’re given an n-component link with 0-framed surgery to $\#^n \left(S^1 \times S^2\right)$, from this, build a homotopy 4-sphere $X$ with 1 0-handle, n 2-handles n 3-handles, and 1 4-handle. Taking the “upside down” handle structure on $X$ gives us a balanced presentation for $1 = \pi_1(X)$ by 1- and 2-handles.

In this way, possible counterexamples to AC have been presented by Gompf, Akbulut, and others — many of which have turned out to be AC-trivial. However, there are several infinite families of counterexamples whose AC-triviality is unknown.

The format of this seminar will be one weekly 30-ish minute talk, held on Zoom, followed by informal discussion. A collection of reference literature is included below. Our aim is to have a planning meeting the week of June 7th, and the first talk will be the week of June 14th. A WhenIsGood poll is linked below to determine the date and time of our first meeting. At our first meeting, we will determine when regular seminar meetings will be held, the order of speakers, and other procedural tasks.

If you would like to participate, please email Nick Meyer at nicholas.meyer2@huskers.unl.edu indicate your desire to participate and fill out the WhenIsGood poll. Both of these items should be completed no later than Friday, June 4th.

WhenIsGood Poll: https://whenisgood.net/sips-gt-1

Suggested Literature: http://bit.ly/sips-gt-lit