Three-dimensional homology spheres and links with Alexander polynomial 1

Abstract

In this paper we establish a connection between two key problems of topological manifolds of dimension 3 and 4: the problem of the nontriviality of the kernel of the Rokhlin homomorphism R: θ→ℤ/2, where θ is the group of ℤ-homologically cobordant three-dimensional ℤ-homology spheres, and the problem of the existence of nonslice knots and links with Alexander polynomial 1. Namely, we show that if Kar R ≠ 0, then in the boundary of some four-dimensional compact homology ball V there exists a knot of genus 1 with Alexander polynomial 1, which does not bound a locally flat disk in any homology sphere with boundary ∂V (and in particular, in V). A similar result is established for links in S3. It is evident from the results of the paper that the obstruction to knots and links being slice may lie in Ker R. On the other hand, these results can be considered as steps in the direction of proving the triviality of the group Ker R.