Unimodular intersection pairs force boundary kernel isotropy

All coefficient maps and pair maps in the definitions of ${\lambda }_R,{\lambda }_K$ commute with cap, conjugation and evaluation: on simplex representatives these are transport, the ring inclusion and finite sums. Thus scalar extension carries each displayed $R$ intersection identity to its $K$ counterpart. In a unitary frame a right-linear functional is conjugate to a right vector by the standard form, as in [F2]; this calculation is valid equally on simplices of a manifold with boundary and on its relative subcomplex. All cap and exact-sequence arguments below can therefore be performed on the intrinsic underlying abelian-group systems, retaining the left coefficient actions.

The pairing ${\lambda }_K$ descends in both variables to $V=H_2(W;K)\mathrm{/}\mathrm{im}{H}_2(M;K)$. For the first variable, take $b\in H_2(M;K)$. Closed duality represents $\bar{b}$ by a class of degree one on $M$ with dual coefficients. Its pair cohomology connector maps to the relative duality preimage of $\overline{i_{\ast }b}$: extending the cocycle to $W$, the identity $\delta a\cap [W,M]=a{\mathrm{\mid }}_M\cap [M]+\mathrm{\partial }(a\cap [W,M])$ in this degree proves the equality on homology. Its image in absolute cohomology is zero by exactness, so ${\lambda }_K(i_{\ast }b,q)=0$. For the second variable, the absolute cohomology class defining ${\lambda }_K(p,-)$ is the image of a relative class; its restriction to $M$ is zero. Naturality of evaluation gives ${\lambda }_K(p,i_{\ast }b)=0$. These also prove that the kernel of the map from relative degree-two cohomology to absolute cohomology corresponds under duality exactly to the boundary image in degree-two homology.

The images of the $2m$ given classes in $V$ are linearly independent. Indeed if $v={\sum }_j{\mathrm{\ell }}_j{r}_j+d_j{s}_j$ is zero there, pair it on the left with each ${\mathrm{\ell }}_i$ to obtain $s_i=0$, and then with each $d_i$ to obtain $r_i=0$. The dimension bound makes them a basis of $V$, including the empty basis if $m=0$. Write $D_{ij}={\lambda }_R(d_i,d_j)\in R$. If a class $z\in H_2(W;K)$ satisfies ${\lambda }_K(z,{\mathrm{\ell }}_i),{\lambda }_K(z,d_i)\in R$ for every $i$, write its class in that basis as ${\sum }_j{\mathrm{\ell }}_j{r}_j+d_j{s}_j$. Then ${\lambda }_K(z,{\mathrm{\ell }}_i)={\bar{s}}_i$ and ${\lambda }_K(z,d_i)={\bar{r}}_i+{\sum }_j{\bar{s}}_j{D}_{ji}$. Involution stability of $R$ first gives $s_i\in R$ and then $r_i\in R$. Consequently $z$ is congruent modulo the boundary image to an actual class $p={\sum }_j{\mathrm{\ell }}_j{r}_j+d_j{s}_j\in H_2(W;R)$, viewed in $K$ homology. This is finite matrix algebra, not a selection of an infinite basis.

Let $x$ belong to the displayed kernel and extend it by zero to $H_1(M;R)$. The homology pair sequence gives a relative class $a\in H_2(W,M;R)$ with $\mathrm{\partial }a=x$. Inverse absolute-to-relative cap duality sends $\bar{a}$ to $u\in H^2(W;{\mathcal{R}}^{\vee })$. By the cap boundary identity in degree two, its restriction to $M$ is the closed-duality preimage of $\bar{x}$. The image of $x$ in $K$ homology is zero, since it is supported on $M_0$ and $H_1(M_0;K)=0$. Hence $u_K$ restricts to zero on the whole of $M$, and the cohomology pair sequence gives a relative degree-two class mapping to $u_K$. Relative cap duality identifies it with $\bar{z}$ for some $z\in H_2(W;K)$. Therefore ${\lambda }_K(z,{\mathrm{\ell }}_i)=u_K({\mathrm{\ell }}_i)$ and ${\lambda }_K(z,d_i)=u_K(d_i)$, both in $R$ because $u$ was an $R$-cohomology class evaluated on $R$-cycles. Step 3.1 supplies an integral class $p$ congruent to $z$ modulo the boundary image. By the last assertion of step 2.1 its relative dual class has the same absolute $K$-image $u_K$.

Choose an $R$-valued cocycle representing $u$ and a relative $R$-valued cocycle representing the dual of $\bar{p}$. Their difference $a_2$ is an absolute $R$-valued two-cocycle whose image is zero in $K$ cohomology by step 4.1. By the definition of a zero cohomology class, there is one whole $K$-valued one-cochain $b$ with $\delta b=a_2$ in $K$ coefficients. No collection of quotient-fiber representatives is being lifted. Its quotient $\nu =b<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>R$ is an intrinsic $(K\mathrm{/}R)$-valued one-cocycle on $W$. The relative cocycle subtracted to form $a_2$ vanishes on $M$. Thus on $M_0$, the coefficient Bockstein sends $[\nu {\mathrm{\mid }}_{M_0}]$ to $[u{\mathrm{\mid }}_{M_0}]$, the duality preimage of $\bar{x}$, by the actual representative formula in [F3].

On $M_0$ the Bockstein in step 5.1 is an isomorphism. Its surrounding $H^2(K)$ term is zero by closed duality and $H_1(M_0;K)=0$. For its $H^1(K)$ term use [F3]'s one-vertex model over $K$. Its degree-zero term is $K$, and the image of $d_1$ is zero or $K$. In the first case $d_1=0$, and the homology vanishing forces every degree-one chain to be a boundary, so every dual one-cocycle is zero. In the second case choose one $z_1$ with $d_1{z}_1=1$; splitting off $z_{1}K$ and using $\ker d_1=\mathrm{im}{d}_2$ shows that each dual one-cocycle is a coboundary. The intrinsic dual identification [F2] therefore gives $H^1(K)=0$ too. Exactness [F3] proves the claimed Bockstein isomorphism. Hence the inverse-Bockstein–duality evaluation defining $B_0(x,y)$ is precisely evaluation of $\nu {\mathrm{\mid }}_{M_0}$ on a cycle for $y$.

If $y$ is also in the displayed kernel, its image cycle in $W$ is the boundary of one $R$-valued two-chain $c$. Cochain evaluation then gives $\nu (y)=\nu (\mathrm{\partial }c)=\delta \nu (c)=0$. By step 6.1 this is $B_0(x,y)$. The other boundary components were used only through the pair sequences and the quotient $V$; none was assumed $K$-acyclic. All bases, coefficients and lifts above belonged to finite matrix computations or one supplied whole cochain, and the coefficient systems carry the supplied parallel Hermitian structure throughout. Thus no choice principle or noncommutative universal-coefficient theorem is used. This proves the full claimed kernel isotropy.