Knot concordance and homology sphere groups

We study two homomorphisms to the rational homology sphere group. If $\psi$ denotes the inclusion homomorphism from the integral homology sphere group, then using work of Lisca we show that the image of $\psi$ intersects trivially with the subgroup of the rational homology sphere group generated by lens spaces. As corollaries this gives a new proof that the cokernel of $\psi$ is infinitely generated, and implies that a connected sum $K$ of 2-bridge knots is concordant to a knot with determinant 1 if and only if $K$ is smoothly slice. Furthermore, if $\beta$ denotes the homomorphism from the knot concordance group defined by taking double branched covers of knots, we prove that the kernel of $\beta$ contains a $\mathbb{Z}^{\infty}$ summand by analyzing the Tristram-Levine signatures of a family of knots whose double branched covers all bound rational homology balls.Comment: 14 pages, 6 figures. Version 3 contains minor changes. This is essentially the version accepted for publication by International Mathematics Research Notices (IMRN

Brieskorn spheres bounding rational balls

Fintushel and Stern showed that the Brieskorn sphere $\Sigma(2,3,7)$ bounds a rational homology ball, while its non-trivial Rokhlin invariant obstructs it from bounding an integral homology ball. It is known that their argument can be modified to show that the figure-eight knot is rationally slice, and we use this fact to provide the first additional examples of Brieskorn spheres that bound rational homology balls but not integral homology balls: the families $\Sigma(2,4n+1,12n+5)$ and $\Sigma(3,3n+1,12n+5)$ for $n$ odd. We also provide handlebody diagrams for a rational homology ball containing a rationally slice disk for the figure-eight knot, as well as for a rational homology ball bounded by $\Sigma(2,3,7)$. These handle diagrams necessarily contain 3-handles.Comment: 8 pages, 8 figure

Fibered ribbon disks

We study the relationship between fibered ribbon 1-knots and fibered ribbon 2-knots by studying fibered slice disks with handlebody fibers. We give a characterization of fibered homotopy-ribbon disks and give analogues of the Stallings twist for fibered disks and 2-knots. As an application, we produce infinite families of distinct homotopy-ribbon disks with homotopy equivalent exteriors, with potential relevance to the Slice-Ribbon Conjecture. We show that any fibered ribbon 2-knot can be obtained by doubling infinitely many different slice disks (sometimes in different contractible 4-manifolds). Finally, we illustrate these ideas for the examples arising from spinning fibered 1-knots.Comment: 20 pages, 3 figures. Version two has improved exposition and incorporates referee suggestions. This version has been accepted for publicatio

Embedding 3-manifolds in spin 4-manifolds

An invariant of orientable 3-manifolds is defined by taking the minimum $n$ such that a given 3-manifold embeds in the connected sum of $n$ copies of $S^2 \times S^2$, and we call this $n$ the embedding number of the 3-manifold. We give some general properties of this invariant, and make calculations for families of lens spaces and Brieskorn spheres. We show how to construct rational and integral homology spheres whose embedding numbers grow arbitrarily large, and which can be calculated exactly if we assume the 11/8-Conjecture. In a different direction we show that any simply connected 4-manifold can be split along a rational homology sphere into a positive definite piece and a negative definite piece.Comment: 27 pages, 14 figures. This is the final version. We made several corrections and small improvements, some suggested by the referee. This paper has been accepted for publication by the Journal of Topolog