Equivariant intersections under collared bordism gluing

Put $T=X\cup D\subset Y$. This contains $\mathrm{\partial }Y$. Restriction induces an isomorphism $H^2(Y,T;{\mathcal{R}}^{\vee })\stackrel{\sim }{\to }{H}^2(W,A\cup D;{\mathcal{R}}^{\vee })\mathrm{.}$ To see the actual excision condition, remove from $X$ a closed core outside a short open collar of its seam $A$. That core is closed in $Y$ and lies in ${\mathrm{int}}_{Y}T$, because it misses the seam. Excision [F2] then leaves $W$ together with that short seam collar, and the remaining relative subspace is $D$ together with the collar. Pushing the collar to $A$ is a homotopy equivalence of pairs with $(W,A\cup D)$. Coefficient transports along the collar give its local-system comparison. This proves the isomorphism; it is restriction and commutes with forgetting relative conditions.

Let $u_p\in H^2(W,\mathrm{\partial }W;{\mathcal{R}}^{\vee })$ be the dual of $\bar{p}$, and let $v_p$ be its inverse image under step 1.1. Map $v_p$ into $H^2(Y,\mathrm{\partial }Y;{\mathcal{R}}^{\vee })$ since $\mathrm{\partial }Y\subset T$. Its cap with $[Y,\mathrm{\partial }Y]$ is $\overline{i_{\ast }p}$. Indeed the image of the orientation class in the excised pair restricts to $[W,\mathrm{\partial }W]$, since their pointwise orientation images on the interior of $W$ agree. The support/relative cap comparison of [F2] therefore identifies the cap with $u_p$ on $W$, followed by inclusion in $Y$. On representatives one may first apply the image-preserving small-chain operator: the portion in $X$ caps to zero because $v_p$ vanishes on $T$, and the portion on the remaining collar core gives precisely the cap on $W$. Changes of the orientation representatives are a boundary plus a chain in the relative subspace; the cap boundary identity makes their caps a boundary plus zero. Thus this is the actual cap map, not an unsupported geometric push of cycles.

Relative duality [F1] now says that the class in step 2.1 is the relative dual of $\overline{i_{\ast }p}$. Its absolute cohomology image restricts to the absolute image of $u_p$ on $W$, by the last sentence of step 1.1. Naturality of evaluation gives ${\lambda }_Y(i_{\ast }p,i_{\ast }q)=⟨j_Y{v}_p,i_{\ast }q⟩=⟨i^{\ast }{j}_Y{v}_p,q⟩=⟨j_W{u}_p,q⟩={\lambda }_W(p,q)\mathrm{.}$ Here each cochain system uses the restriction of the same ambient parallel Hermitian form, so no unrestricted frame change or independent coefficient identification is being inserted.

The displayed equality immediately preserves any finite list of zero self-pairings and unit dual-pairings. Empty seam or empty outer boundary causes no change in the restriction and relative maps; a disconnected gluing is treated on its finitely many compact components with the supplied coefficient systems. Every collar core and representative comparison is chosen for one finite calculation, and the small-chain operators are fixed formulas. Thus the full stated topological and coefficient scope uses no choice axiom.