Casson–Gordon signature-defect obstruction to sliceness

Given: Full AC, a smooth oriented knot $K$, a prime-power degree $q$, the linking form on $H_1({\mathrm{\Sigma }}_q(K);\mathbb{Z})$, and the hypothesis that every metabolizer admits a character vanishing on it with nonzero signature defect.

Full AC is assumed wherever the cover and orientation conventions use the choice axiom. The Axiom of Choice

For prime-power $q$, $H_1({\mathrm{\Sigma }}_q(K);\mathbb{Z})$ is finite, and its torsion linking form is nonsingular. Prime-power cyclic branched covers of knots have finite first homology

If $K$ is smoothly slice, a disk induces a metabolizer of this form. A slice disk induces a metabolizer on the branched-cover linking form

A knot is smoothly slice when it bounds a smooth proper disk in $B^4$. Smooth and topological slice genus

A metabolizer is a subgroup equal to its orthogonal complement for the finite nonsingular linking form. Finite nonsingular linking forms and metabolizers

A prime-power-order character is a finite-order homomorphism to $\mathbb{Q}\mathrm{/}\mathbb{Z}$ whose order is a prime power. Finite-order characters of a cyclic branched cover

For a smooth slice disk and any prime-power-order character vanishing on its induced metabolizer, the specified Casson–Gordon signature defect vanishes; its proof constructs the actual extending disk-exterior coefficient system and proves the full Witt comparison. Casson–Gordon signature defects vanish on the slice metabolizer