Casson–Gordon Witt class and signature defect

Assume AC. Let $K$ be a smooth oriented knot, let $q$ be a prime power, and let $\chi :H_1({\mathrm{\Sigma }}_q(K);\mathbb{Z})\to \mathbb{Q}\mathrm{/}\mathbb{Z}$ have prime-power order $d$. Choose a Casson–Gordon bordism datum $(M_{0,q},\eta ,r,W)$. Write $G=\mathbb{Z}\mathrm{/}d\oplus \mathbb{Z}$, let $Y\to W$ be the regular cover determined by $\eta$, and give $\mathbb{C}(t)$ the $G$-action $(a,n)z=\exp (2\pi ia\mathrm{/}d)t^{n}z$ and involution $\bar{t}=t^{-1}$ with complex conjugation on constants. Define $H_2^t(W;\mathbb{C})=H_2(Y;\mathbb{C}){\otimes }_{\mathbb{C}[G]}\mathbb{C}(t)\mathrm{.}$ The Casson–Gordon Witt class is $\tau ({\mathrm{\Sigma }}_q(K),\chi )=\frac{1}{r}([H_2^t(W;\mathbb{C})]-[Q_0(W){\otimes }_{\mathbb{C}}\mathbb{C}(t)])\in W(\mathbb{C}(t))\otimes \mathbb{Q},$ using the nonsingular twisted form and the ordinary nonsingular image form. Precisely, put $Q_0(W)=H_2(W;\mathbb{C})\mathrm{/}\mathrm{rad}({\lambda }_W)$ with its induced form; the untwisted term in the display is understood as $Q_0(W){\otimes }_{\mathbb{C}}\mathbb{C}(t)$. The twisted boundary complex is acyclic, so its algebraic Poincare cap preserves the absolute twisted module. This does not assert that an arbitrary ordinary absolute intersection form is nonsingular. Here $W(\mathbb{C}(t))$ is the Witt group of nonsingular Hermitian forms over the field with involution, modulo hyperbolic forms. Its signature defect is $\sigma (K,\chi )={\sigma }_1(\tau ({\mathrm{\Sigma }}_q(K),\chi ))$, where ${\sigma }_1$ evaluates a Hermitian representative at $t=e^{i\theta }$ and takes the one-sided limit as $\theta \to 0^{+}$. This is the specified Casson–Gordon invariant and signature homomorphism; it is not an arbitrary rho invariant or an unspecified difference of signatures.

The character pair and bounding data

Existence of a multiple bounding bordism

The capped twisted intersection form

Independence of the rationalized Witt class

The oriented bordism class convention

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

Axiom of Choice