Classical Knot Concordance and Problems in Four-Manifold Theory

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Two knots are defined to be concordant if the connected sum of one with the mirror of the other is slice: that is, if the connected sum bounds an embedded disk in the four-ball. The set of equivalence classes forms what is called the concordance group of knots, first studied in the early 1960s. It is known that this group is countable, abelian, and that it maps onto Levine's algebraic concordance group, which is isomorphic to an infinite direct sum of cyclic groups of infinite order and of order 2 and 4. The kernel of this map is infinitely generated. A main goal of the proposal is the continued study of the concordance group and relationships between three-dimensional properties of knots, such as symmetry and genus, and their properties in the concordance group. Specific topics include understanding the algebraic structure of the kernel of Levine's map (in particular the part of the kernel generated by knots of Alexander polynomial one), torsion in the concordance group, and the splitting of Levine's homomorphism. To pursue this study the principal investigator will develop and apply methods coming from the theory of Heegaard Floer homology as defined by Oszvath and Szabo. Other topics of study under the proposal relate to the interplay between the fundamental groups of 4-manifolds and their topological structure: for instance, a continued investigation of the Hausmann-Weinberger invariant of a group, the minimal Euler characteristic of a closed 4-manifold having that group as its fundamental group, and its generalizations, will be pursued.The goal of this proposal is the study of knots. From a three-dimensional perspective, questions about knots have been studied for over a hundred years. More recently it has been recognized that the structure of a four-dimensional space, a four-manifold, is connected to properties of classical knots that can be associated to that four-manifold. A main focus of this proposal is the study of those properties of knots that are of relevant to four-dimensional topology; many of these properties are encapsulated in something called the concordance group of knots. Questions regarding this concordance group offer test cases and motivation for the further development of techniques from general four-manifold theory. In addition, the study of concordance brings to the fore new and fascinating questions in the realm of classical three-dimensional knot theory, relating for instance to questions of knot symmetry and genus.