Applications of Noncommutative Algebra to Low-Dimensional Topology and Geometry

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The principal investigator proposes to continue the investigation of topology and geometry of low-dimensional manifolds using invariants that arise from non-commutative algebra and von Neumann Algebras. These reflect the highly non-commutative nature of the fundamental group. In previous work, the PI showed that this type of invariant gives estimates for the Thurston norm of a 3-manifold, obstruct a 3-manifold fibering over the circle, obstruct the existence of a symplectic structure on certain 4-manifolds, give new information about the structure of the link concordance group, and obstruct a group being the fundamental group of a 3-manifold or having positive deficiency. The PI proposes to find new interesting non-commutative algebraic invariants and to apply these invariants to questions in low-dimensional topology; for example, homology cobordism of 3-manifolds (and link concordance), symplectic structures of 4-manifolds, genera of contact structures of 3-manifolds, Betti numbers of finite covers of 3-manifolds, and depth of foliations of 3-manifolds. The PI also proposes to find a specific relationship between her invariants of a three manifold and the Heegard Floer Homology of a 3-manifold.Topology is the study of the continuous change of space (by stretching or twisting but not tearing). In this project, the PI will focus on spaces that are locally modelled on 3-dimensional space (the space that we live in) and 4-dimensional space (3-dimensional space along with a time dimension). These are called 3 and 4-dimensional manifolds respectively.One of the ways that we can better understand these spaces is via their "fundamental group." The fundamental group isan algebraic object associated to any topological space which measures the number of holes in a the space. It is defined as the set of loops starting and ending at a point. We can formally multiply loops in the fundamental group as follows. If A is a loop and B is another loop, we define "A times B", denoted AB, to be the loop obtained by first traversing A and then traversing B. The multiplication of loops is non-commutative as in matrix multiplication. That is, AB is not the same as BA, since traversing A then B is not the same as traversing B then A. Unfortunately,the fundamental group of a space is quite difficult to understand. In this project, the PI will use noncommutative algebraic techniques to better understand the fundamental group and hence better understand the 3 and 4-dimensional manifolds themselves. For example, one of the non-commutative techniques the PI will use involves matrices which are no longer finite but have an infinite number of rows and columns.