PTFA subgroup extension preserves nonzero Blanchfield self-pairings

Intersect a PTFA series of $G$ with $H$. Each successive intersection is normal in the next, and its quotient embeds in the corresponding torsion-free abelian quotient of $G$. Subgroups of a torsion-free abelian group are torsion-free and abelian, directly from the definitions. Thus $H$ is PTFA, without choosing a global transversal. By [F2] both group rings have the asserted fractions and involutions. Apply [F1] to identify $F$ with its subring in $K$ and obtain the injective map $j$. The inclusions preserve involution because they agree on every group element and on $k$, and the inverse of a fraction denominator is forced in both rings.

On a connected component use the finite-atlas based paths in [F3] for compatible fiber frames of all four coefficient systems. Their singular complexes are literally related by tensor base change on those generators. Hence [F1] gives $H_1(M;B)=H_1(M;A){\otimes }_{A}B$, with the split injective unit asserted there. By [F2], vanishing over $F$ means that every $x\in H_1(M;A)$ has $xa=0$ for some nonzero $a\in A$. The composite functor $-{\otimes }_{A}B{\otimes }_{B}K$ is exact by [F1, F2], and therefore commutes with chain homology. Every such $x$ dies after tensoring with $K$, since $x\otimes z=(xa)\otimes a^{-1}z=0$. Thus $H_1(M;K)=0$. The same localization criterion says both first homology modules in the Statement are entirely torsion over their respective coefficient rings.

For either pair $(A,F)$ or $(B,K)$, the coefficient Bockstein $\beta :H^1(\mathrm{Hom}(C,F\mathrm{/}A))\to H^2(\mathrm{Hom}(C,A))$ is an isomorphism, with the analogous notation for $(B,K)$. Indeed degree-two cohomology with values in the division ring is zero by closed duality and the proved first-homology vanishing. Its degree-one cohomology is zero by the one-vertex model: over the division ring $d_1$ has image zero or the whole degree-zero term. In the first case every degree-one cycle is a boundary; in the second split $d_1$ using one preimage of $1$, and a dual cocycle is a coboundary because it vanishes on $\ker d_1=\mathrm{im}{d}_2$. The exact coefficient segment in [F3] then proves $\beta$ is an isomorphism. Define $B_A$ by inverse duality, then ${\beta }_A^{-1}$, then evaluation, and define $B_B$ by the same procedure. This requires no inverse evaluation map, and hence asserts no unjustified general nonsingularity.

Coefficient inclusion gives actual morphisms between the two cochain coefficient sequences $A\to F\to F\mathrm{/}A$ and $B\to K\to K\mathrm{/}B$. On dual cochains the extension is explicit: if $f:C_A\to F\mathrm{/}A$, set $f_B(c\otimes b)=j(f(c))b$, and use the identical formula for $A\to B$ and $F\to K$. Right-linearity of $f$ and the $A$-bimodule identity for $j$ make it balanced; the formula commutes with all differentials. The local cochain version is the same map on supplied simplex frames. The Bockstein is natural by [F3], so these maps commute with ${\beta }_A,{\beta }_B$. Cap duality also commutes with coefficient inclusion, by its finite simplex-and-transport formula using the same orientation class. Its unitary dual identifications commute because the base vector and standard form are the specified extensions of each other. No independent arbitrary gauge change is made.

If ${\nu }_A$ is the inverse-Bockstein class associated by duality to $\bar{x}$, step 4.1 sends it to the inverse-Bockstein class associated to $\overline{x\otimes 1}$, because ${\beta }_B$ is an isomorphism. Evaluating that cochain class on $y\otimes 1$ gives exactly $j({\nu }_A(y))$ by the displayed extension formula. This proves the stated value identity. Setting $y=x$ and using the injectivity of $j$ shows that a nonzero self-pairing cannot become zero. No nonsingularity theorem has entered this argument.

A compact manifold has finitely many connected components, so apply the construction to each with only finitely many base-fiber data. Chains, Bockstein maps and cap classes split over that finite list, and the linking pairings are their orthogonal sum. Empty $M$ gives zero modules and the unique zero pairing. All potentially infinite cochain constructions above are actual formula maps on supplied whole cochains; the only Bockstein lifting is the already proved finite-generator construction of [F3]. The Ore annihilator and denominator arguments use one finite expression at a time. Thus all stated subgroup, field and topological scopes are proved without a choice axiom.