Structure of knot and link concordance

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Low dimensional topology is a fascinating subject, concerned with understanding spaces of dimension 3 and 4, much like the universe we inhabit.  I investigate the topology of 3- and 4-dimensional manifolds, in particular knot and link concordance.  This is the study of embeddings of circles in 3-space and embeddings of surfaces in 4-space.  My proposal aims to achieve a greater understanding of the structure of concordance spaces. Knot and link concordance is the key local question in the problem of embedding surfaces in 4-manifolds, which is central to attempts to classify 4-dimensional manifolds. I use principally algebraic and geometric tools, but sometimes analytic methods as well, to attack the questions. The grand goal of the proposal is to classify knot and link concordance. I will approach this from several viewpoints. One is to try to construct and compute an algebraic obstruction group which decides whether two knots are concordant. Another uses the observation that the failure of concordance can be measured by towers of surfaces embedded in 4-space called gropes and Whitney towers. By considering the space of knots as a metric space, with the metric defined in terms of gropes, I would like to understand its purported fractal structure.  The complexity of Whitney towers can be used to filter the link concordance space, and understanding how certain links fit into the filtrations, and indeed how nontrivial the filtrations are, should promote our understanding and inform potential classification frameworks. Approaching from the other side, I aim to achieve positive results, showing that knots and links are slice using specifically topological tools derived from the work of Freedman.  This has relations to the famous surgery conjecture for 4-manifolds. Another goal of the grant is to increase our understanding of related concordance problems.  The double concordance group of high dimensional knots is one such problem.  I also intend to study the concordance of knots in general 3-manifolds, and the concordance of non-spherical knots in high dimensions.  The goal is to find new phenomena which are not exhibited in the classical problem.