Torsion linking form and Casson–Gordon bordism data

Assume AC. Let $K$ be a smooth oriented knot, $q\ge 2$, and $M={\mathrm{\Sigma }}_q(K)$ oriented as the boundary of the positive-meridian branched cover. When $H_1(M;\mathbb{Z})$ is finite, its torsion linking form is the oriented pairing ${\lambda }_M:H_1(M;\mathbb{Z})\times H_1(M;\mathbb{Z})\to \mathbb{Q}\mathrm{/}\mathbb{Z}$ given by ${\lambda }_M(x,y)=(C\cdot b)\mathrm{/}n<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>\mathbb{Z}$, where integral $1$-cycles $a,b$ represent $x,y$, respectively, $n>0$ is an integer, and $C$ is an integral $2$-chain in $M$ with $\mathrm{\partial }C=na$, placed transversely to $b$; reversing the orientation negates ${\lambda }_M$. For a prime-power-order character $\chi$ of order $d$, the standard splitting $H_1(M_{0,q};\mathbb{Z})\cong H_1(M;\mathbb{Z})\oplus \mathbb{Z}⟨\tilde{\mu }⟩$ of the $q$-fold cyclic cover $M_{0,q}$ of zero surgery extends $\chi$ to $\eta :H_1(M_{0,q};\mathbb{Z})\to \mathbb{Z}\mathrm{/}d\oplus \mathbb{Z}$ by identifying its cyclic image with $\mathbb{Z}\mathrm{/}d$ and using the free-summand projection in the second factor. A Casson–Gordon bordism datum is a positive integer $r$ and a compact oriented $4$-manifold $W$ with a map to $K(\mathbb{Z}\mathrm{/}d\oplus \mathbb{Z},1)$ whose boundary represents $r$ copies of $(M_{0,q},\eta )$.

Finite nonsingular linking forms and metabolizers

The cyclic branched cover convention

Finite-order characters

Oriented bordism groups of maps

Well-definedness and nonsingularity of the torsion linking pairing

The cyclic cover of zero surgery

The asserted homology splitting

Charles Livingston and Swatee Naik, Introduction to Knot Concordance, §7.5.3, p. 139 and §8.1, pp. 164–166

Axiom of Choice