Donald, Andrew
(2013)
Embedding 3manifolds in 4space and link concordance via double branched covers.
PhD thesis, University of Glasgow.
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Abstract
The double branched cover is a construction which provides a link between problems in knot theory and other questions in lowdimensional topology. Given a knot in a 3manifold, the double branched cover gives a natural way of associating a 3manifold to the knot. Similarly, the double branched cover of a properly embedded surface in a 4manifold is a 4manifold whose boundary is the double branched cover of the boundary link of the surface. Consequently, whenever a link in S^3 bounds certain types of surfaces, its double branched cover will bound a 4manifold of an appropriate type.
The most familiar situation in which this connection is used is the application to slice knots as the double branched cover of a smoothly slice knot is the boundary of a smooth rational ball. Examples of 3manifolds which bound rational balls can therefore easily be constructed by taking the double branched covers of slice knots while obstructions to a 3manifold bounding a rational ball can be interpreted as slicing obstructions. This thesis is primarily concerned with two different extensions of this idea.
Given a closed, orientable 3manifold, it is natural to ask whether it admits a smooth embedding in the foursphere $S^4$. Examples can be obtained by taking the double branched covers of doubly slice links. These are links which are crosssections of an unknotted embedding of a twosphere in S^4. Certain links can be shown to be doubly slice via ribbon diagrams with appropriate properties. Other embeddings can be obtained via Kirby calculus.
On the other hand, many obstructions to a 3manifold bounding a rational ball can be adapted to give stronger obstructions to embedding smoothly in S^4. Using an obstruction based on Donaldson's theorem on the intersection forms of definite 4manifolds, we determine precisely which connected sums of lens spaces smoothly embed. This method also gives strong constraints on the Seifert invariants of Seifert manifolds which embed when either the base orbifold is nonorientable or the first Betti number is odd. Other applicable methods, also based on obstructions to bounding a rational ball, include the d invariant from Ozsvath and Szabo's HeegaardFloer homology and the NeumannSiebenmann mubar invariant. These are used, in conjunction with some embedding results derived from doubly slice links, to examine the question of when the double branched cover of a 3 or 4 strand pretzel link embeds.
The fact that the double branched cover of a slice knot bounds a rational ball has a second interpretation in terms of knot concordance. In this viewpoint, the double branched cover gives a homomorphism from the concordance group of knots to the rational cobordism group of rational homology 3spheres. This can be extended to a concordance group of links using a notion of concordance based on Euler characteristic. This yields link concordance groups which contain the knot concordance group as a direct summand with an infinitely generated complement. The double branched cover homomorphism extends to large subgroups containing the knot concordance group.
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