All (co)homology is with $\mathbb{Q}$ coefficients. All manifolds are compact, oriented, with boundary.
Let $M$ be a $4n$-dimensional manifold, then we have a symmetric bilinear pairing $H^{2n}(M,\partial M)\times H^{2n}(M,\partial M)\rightarrow\mathbb{Q},(\varphi,\psi)\mapsto\langle\varphi\cup\psi,\mu\rangle$ ($\mu$ the fundamental class). Currying makes this into a linear map $H^{2n}(M,\partial M)\rightarrow H^{2n}(M,\partial M)^{\ast}$, which fits into a triangle with the restriction map $H^{2n}(M,\partial M)\rightarrow H^{2n}(M)$ and the isomorphism $H^{2n}(M)\rightarrow H^{2n}(M,\parti…