Boundary middle form and boundary signature

Assume the Axiom of Choice as inherited from Poincare-Lefschetz duality. Let $W$ be a compact oriented smooth eight-manifold with boundary $M=\mathrm{\partial }W$, and consider real coefficients. Let $j:H^4(W,M;\mathbb{R})⟶H^4(W;\mathbb{R})$ be the forgetful map and put $I=\mathrm{im}j\subseteq H^4(W;\mathbb{R})$; by Compact oriented manifolds with boundary have finite-dimensional field cohomology and Poincare-Lefschetz duality (Poincaré–Lefschetz duality) the space $I$ is a finite-dimensional real vector space. For $x,y\in I$ choose relative lifts $\tilde{x},\tilde{y}\in H^4(W,M;\mathbb{R})$ with $j(\tilde{x})=x$, $j(\tilde{y})=y$ and define the boundary middle form $Q_W(x,y):=⟨\tilde{x}⌣\tilde{y},[W,M]⟩,$ where the product uses two relative factors and the evaluation is the relative Kronecker evaluation on the relative fundamental class Relative fundamental class and boundary orientation.

The following lemma The boundary middle form is well defined and glues ↗ proves that $Q_W$ is independent of the two relative lifts, is symmetric (it uses that both factors have degree four and Relative degree-four cup products are symmetric), and is nondegenerate on $I$: its radical is zero. Consequently $Q_W$ is a symmetric bilinear form on the finite-dimensional real vector space $I$, and its radical $\mathrm{rad}{Q}_W\subseteq I$ is the subspace of $x$ with $Q_W(x,y)=0$ for all $y\in I$. The boundary signature $\sigma (W):=\text{inertia signature of }(I,Q_W)$ is the inertia signature of the nondegenerate symmetric form induced on $I\mathrm{/}\mathrm{rad}{Q}_W$; when the radical is zero, as proved, no quotient is needed.

This is a boundary construction and is distinct from the closed middle-dimensional intersection form and closed signature of the signature-theorem page: the closed form is not applied to $W$ with boundary, because the relative fundamental class and the relative products are what make the displayed evaluation well defined. For closed $W$ the construction coincides with the closed middle form in degree four by the empty-boundary case of Poincare-Lefschetz duality. No orientation of $M$ is chosen separately: it is the induced boundary orientation.