||A Cyclic Extension of Heegaard Splittings
In topology, a Heegaard Splitting is the decomposition of a 3-Manifold into simpler topological objects called handlebodies. There exist functions called Morse functions which, for example, map a 3-manifold to the real line. The inverse image of a non-critical value r produces what is known as a Heegaard surface in the 3-Manifold, giving us the boundary for a Heegaard splitting. In stead of dealing with Morse functions which map a 3-manifold to the real line, I will be dealing with functions that map a 3-manifold to the unit-circle and trying to obatin uniquness results possibly simimlar to those already known for usual splittings.
- Week 1:
- During the first week of the REU, I spent the majority of my time doing necessary background reading in order to familiarize myself with material with which I will be working. At the end of the week, I used these references to make a presentation that serveda as an inroduction to my project.
- Week 2:
- This week I continued reading background information relating to my project; specifically, I read papers that developed the idea of Morse 2-functions which are the class of functions that I will be considering in my research. In particular, I need to investigate such functions that admit no local extrema in addition to having connected fibers as inverse images. These funcitons are of interest to me because when mapping a 3-manifold to the unit-circle, they induce Cyclic Heegaard splittings and have no index 0 or index 3 critical points. Aside from research, this week I began attending a lecture series on Simplectic Geometry which is proving to be quite interesting.
- Week 3:
- Continuing with my research, I am looking into conditions on when Morse 2-functions will produce connected fibers as inverse images, and thus induce a cyclic Heegaard splitting on a 3-manifold. However, I am particularly interested in those functions which also have no admit no minimum of maximum value, i.e. no critical points of index 0 or 3, because these are precicely the functions that map a 3-manifold to S1. For this area of my research, I have found the paper 'Fiber-connected, indefinite Morse 2-functions on connected n-manifolds' by Gay and Kirby very useful and informative.
- Week 4:
- One main focus of my research is to identify some sort of uniqueness of Cyclic Heegaard Splittings, in particular to address the questions, 'Given two Cyclic Heegaard splittings of the same 3-manifold, can we get from one to another with a sequence of well-defined moves?'. I began to address this question by considering the cycling of critical points within a 3-manifold. Essentially, this would be a new type of move whereby a critical point is revolved inside of the manifold and, in doing so, changes the genus of inverse images of a Morse 2-function (effectively changing the genus of a Heegaard surface). The issue with this sort of move is that I am not sure what happens when we move a critical point past a genus 0 fiber in such a way that it would decrease the genus. I think that there are two possibilitie; either it would do nothing or it would disconnect the manifold. Based purely on intuition, I think that such a situation would result in nothing happening but I intend to investigare this further. In addition, it may be possible that we can represent the cycling of critical points as a sequence of stabilizations and this is also something I plan to look at.
- Week 5:
- In order to give a proof of what would essentially be an 'S1 valued analogue' of the Reidemeister-Singer Theorem, I read multiple Morse-Theoretic proofs of this theorem and began looking at where certain assumtions break down when considering circle valued Morse functions as opposed to usual ones. In doing so, I prepared and gave a talk on the usual version of the theorem, which also addressed some of the issues that could possibly arise in my attempt to generalize it to circle valued Morse-functions. Also this week I investigated the importance of the connectedness of fibers. A good result is that, if one fiber that is the inverse image of a circle-valued morse function has k connected components, then all such fibers will and this lets us characterize such functions. Something I am looking into is whether or not we can, given a Morse function whose inverse image has k connected components, come up with a Morse-funtion which can be written as the kth root of our original function. In doing so, we will have a Morse-function that only wraps one time around the circle, hopefully making some arguments easier and allowing for simpler comparison between two functions.
- Week 6:
- This week I worked on proving the idea from last week where, given a Morse function whose inverse image has k connected components, we can write associate to it a simpler function. I have made good progress in proving this by using a gradient-flow argument and keeping track of how many times the image has wrapped around S1. I hope to have a detailed proof finished soon. I have also started looking at homotopy classes of maps from S1 to itself in order to hopefully apply known results to homotopies between these circle valued Morse-functions.
- Week 7:
- The majority of this week was spent compiling my research into my final presentation, which can be seen by following the link below. In addition, I also began examining lower dimensional cases of critical point winding in order to hopefully get a better understanding of winding in the three-dimensional case. So far I have worked out the one-dimensional case and will continue working on the two-dimensional case.
- Week 8:
- During the final week of the REU I went over all of the research that I had done in order to write my final report. I am also currently preparing a talk on fiber connected indefinite Morse 2-functions based on a paper by Gay and Kirby to be given next week, as I continue my research through the rest of the Summer and into the school year.