Dehn Surgery, Four-Manifolds, and Symplectic Topology

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Topology is an area of mathematics that studies the intrinsic shapes of objects, such as a hula hoop, a massive data set, our universe, or a strand of DNA. For example, topology can be used to measure how biological processes change the shape of our DNA. Surprisingly, these shape changes can be analyzed through three- and four-dimensional manifolds, fundamental shapes in topology that appear throughout mathematics and physics. This leads to the fundamental problem of trying to completely understand and classify these three- and four-dimensional manifolds. While we understand many aspects of three-dimensional objects, in dimension four we are still mostly in the dark. One effective tool for studying these objects is called Floer homology, which comes from solving powerful equations from physics. In this project, the PI will use Floer homology to study the complexity of these shapes by analyzing the configurations of knotted loops and surfaces inside of them. The potential outcome of this project will be to strengthen connections between knots and three-/four-dimensional manifolds while discovering new structure in the topology of four-dimensional manifolds. This project will then be disseminated to the public through a collaboration with Black Box Dance Theater, a North Carolina non-profit dance company. By way of dance performances and public workshops, this will present to the public the fundamental notions of topology used in the proposed project. This joint project will also work to improve public perception of mathematics and enhance connections between STEM and the arts.Knots can be used to produce new three- and four-manifolds by an operation called Dehn surgery, where one removes a tubular neighborhood and reglues via a homeomorphism. Many invariants and tools in low-dimensional topology, especially Floer homology, are particularly well-behaved under Dehn surgery, and the PI will use these tools to improve our understanding of three- and four-manifolds and further their connections with knot theory. Three major goals of this project are to: 1) construct homology three-spheres which cannot be obtained by Dehn surgery on a two-component link, 2) find new constraints on the algebraic topology of four-manifolds admitting symplectic structures, and 3) further explore the algebraic structure of the concordance group of knots in homology spheres modulo concordance in homology cobordisms. Two potential outcomes would be a better understanding of the complexity of four-manifolds with boundary measured in terms of their handlebody structures and a stronger connection between properties of a knot and the topology of its Dehn surgeries.