Branched covers and rational homology balls

TL;DR

This paper investigates the structure of the knot concordance group, showing that certain elements generated by specific knots have infinite order two subgroups, with their branched covers bounding rational homology balls, revealing complex algebraic properties.

Contribution

It demonstrates the existence of an infinitely generated two-torsion subgroup within the kernel of a key homomorphism in knot concordance, using properties of branched covers and rational homology balls.

Findings

Infinite subgroup generated by order two elements in the concordance group.

Each knot's cyclic covers bound rational homology balls for all n > 0.

Kernel contains an infinitely generated two-torsion subgroup.

Abstract

The concordance group of knots in the three-sphere contains an infinite subgroup generated by elements of order two, each one of which is represented by a knot K with the property that for every n > 0, the n-fold cyclic cover of S^3 branched over K bounds a rational homology ball. This implies that the kernel of the canonical homomorphism from the knot concordance group to the infinite direct sum of rational homology cobordism groups (defined via prime-power branched covers) contains an infinitely generated two-torsion subgroup.