Localized Blanchfield pairings and higher-order rho-invariants

Let $M$ be a closed oriented three-manifold with a specified coefficient homomorphism $\varphi :{\pi }_1(M_j)\to \mathrm{\Gamma }$ on each component, where $\mathrm{\Gamma }$ is PTFA. The integral coefficient ring is $\mathrm{\Lambda }=\mathbb{Z}[\mathrm{\Gamma }]$ with its fraction division ring $K\mathrm{\Gamma }$. More generally let $k$ be $\mathbb{Z}$ or a commutative field, let $K$ be the fraction division ring of $k[\mathrm{\Gamma }]$, and let $k[\mathrm{\Gamma }]\subseteq R\subseteq K$ be an involution-stable unital intermediate ring. The involution fixes $k$ and sends $g$ to $g^{-1}$; on fractions it is the involutive anti-automorphism extending this formula. Taking $k=\mathbb{Z}$ and $R=\mathrm{\Lambda }$ recovers the integral convention.

If $H_1(M;K)=0$, the coefficient Bockstein, oriented Poincaré duality and evaluation define the Blanchfield linking pairing $B_{M,\varphi ,R}:T{H}_1(M;R)\times T{H}_1(M;R)⟶K\mathrm{/}R\mathrm{.}$ Here $T$ denotes the submodule of elements with a nonzero right annihilator, the first variable is conjugate-linear, and the second is right-linear. Nonsingularity means that the adjoint $\overline{T{H}_1(M;R)}⟶{\mathrm{Hom}}_R(T{H}_1(M;R),K\mathrm{/}R)$ is an isomorphism of left $R$-modules, where $r\cdot \bar{x}=\overline{x\stackrel{ˉ}{r}}$ on the conjugate module. This is equivalent to the conjugate-dual formulation in the right-module convention. For $\mathrm{\Gamma }=\mathbb{Z}$ with generator $t$, $R=\mathbb{Z}[t^{\pm 1}]$ and $K=\mathbb{Q}(t)$ give the classical coefficient convention. Nonsingularity is a separate property of the defined pairing, with its coefficient hypotheses supplied by the nonsingularity theorem.

For a homomorphism φ:π₁(M)→G and an extension ψ:π₁(W)→G over a compact oriented 4-manifold W with ∂W=M, define the signature defect by ρ(M,φ;W,ψ)=sign_G^(2)(W,ψ)−sign(W). It is an invariant of (M,φ) only when a separate bordism-independence theorem applies.

Maximal-abelian-cover Alexander module

Integral solutions, von Neumann dimension, L²-signature and signature defect

Finite-projective signatures for arbitrary discrete coefficient groups

PTFA group

Rational PTFA fraction fields

Integral and field coefficients, intermediate Ore rings and involutions

COT, Theorem 2.13 and §2.1

Cha, §2.2, printed p.5

In the signature-defect clause, coefficient groups are regarded as discrete, with the standing full AC convention of the signature definition. For an arbitrary target group use the finite-projective signature of Topological finite-projective L2 duality and whole-component boundary gluing. Its proof 13.1 identifies this signature with that of the countable coefficient image, so it agrees with the countable-group convention in Integral half-solutions, Strebel classes, and L²-signatures. This supplies the signature used in the displayed defect without asserting bordism independence.