Cut and Paste Methods in Low Dimensional Topology

Topology is a branch of mathematics that studies spaces up to continuous deformation. This means, that topologists consider two spaces the same if one can be deformed into the other by stretching, bending, but no cutting or gluing. So distances and areas are allowed to change during these deformations. For example from a topologists viewpoint all circles, ellipses and even squares are the same (as they have only one hole), but a figure eight is dierent (as it has 2 holes). A dimension of a space is the number of coordinates one needs to describe locations of objects. For example any point on the surface of the Earth may be described by its latitude and longitude. Thus it is 2-dimensional. In the Space however one also needs to keep track of another local coordinate; height. Which means it is 3-dimensional. With a bit of abstraction, one can talk about higher dimensional spaces, the fourth dimension is often represented by time, but in general one does not need to give names to these extra dimensions. Dimension in itself is not enough to describe a space. For example the line and the circle are both one dimensional, but they are topologically dierent; one needs to cut open the circle to make it into a line. Similarly there are several 2-dimensional spaces: the surface of a sphere, a mug, or a mug with two or more handles. Once again, these are all dierent 2-dimensional spaces; one needs to cut to change the number of handles. While we have a complete understanding of 1- and 2-dimensional spaces, and higher dimensional spaces turn out to be easier to work with due to the flexibility we gain in the extra dimensions, 3- and 4-dimensional spaces are the most complicated to study. Low dimensional topology is the branch of mathematics dealing exactly with these spaces in the borderline. It has long been a fertile area for the interaction of many dierent disciplines of mathematics, including dierential geometry, hyperbolic geometry, combinatorics, representation theory, global analysis, classical mechanics, and theoretical physics. This project concentrates on questions in low dimensional topology, that can be settled using cut and paste techniques in various forms. This means, that we cut spaces into elementary pieces that are each easier to study, and derive results for the the original space by gluing them back together. The diculty in these approaches lies in developing structures that are complicated enough to remember how the spaces are built up from the elementary pieces, but are still simple enough to be studied successfully.

Topology is a branch of mathematics that studies spaces up to continuous deformation. This means that topologists consider two spaces the same if one can be deformed into the other by stretching and bending, but not cutting or gluing. For example, from a topologist's viewpoint, all circles, ellipses and even squares are the same (as they have only one hole), but a figure eight is different (as it has 2 holes). A dimension of a space is the number of coordinates one needs to describe the locations of objects. For example, any point on the surface of the Earth may be described by its latitude and longitude. Thus, it is 2-dimensional. In space, one also needs to keep track of height, which means it is 3-dimensional. With a bit of abstraction, one can talk about higher-dimensional spaces; the fourth dimension is often represented by time, but in general, one does not need to give names to these extra dimensions. Dimension in itself is not enough to describe a space. For example, the line and the circle are both one-dimensional, but they are topologically different; one needs to cut open the circle to make it into a line. Similarly, there are several 2-dimensional spaces: the surface of a sphere, a mug, or a mug with two or more handles. These are different 2-dimensional spaces; one needs to cut to change the number of handles. While we have a complete understanding of 1- and 2-dimensional spaces, and higher-dimensional spaces turn out to be easier to work with due to the flexibility we gain in the extra dimensions, 3- and 4-dimensional spaces are the most complicated to study. Low-dimensional topology is the branch of mathematics dealing with these borderline cases. It has long been a fertile area for the interaction of many different disciplines of mathematics, including differential geometry, hyperbolic geometry, combinatorics, representation theory, global analysis, classical mechanics, and theoretical physics. This project concentrated on questions in low-dimensional topology that can be addressed using cut-and-paste techniques in various forms. In this approach, spaces are decomposed into simpler pieces that are easier to study, and results are obtained for the original space by understanding how these pieces fit together. The project developed new structures that successfully encode how such spaces are built from elementary components. These methods made it possible to compare different decompositions of the same space and to understand when they describe the same underlying geometry. As a major outcome, the project established a complete proof of a fundamental correspondence in three dimensions, providing a precise understanding of how different combinatorial descriptions of these structures are related.