Collared gluing has relative excision and evaluation maps

By [L1] choose a bicollar $M\times (-2a,2a)$ of the seam, with $W$ on the negative side and $W^{\mathrm{\prime }}$ on the positive side, and set $U=\mathrm{int}W\cup M\times (-2a,a)$ and $V=\mathrm{int}{W}^{\mathrm{\prime }}\cup M\times (-a,2a)$; then $U,V$ are open in $N$, they cover $N$, and $U\cap V=M\times (-a,a)$.

The collar-height map that sends $t\ge -a$ to zero and fixes $t\le -2a$, with monotone interpolation and the identity outside the collar, defines maps of pairs $(U,U\cap V)\to (W,M)$ and $(W,M)\to (U,U\cap V)$ that are inverse up to homotopy of pairs, by [L3] applied to the linear interpolation with the identity; hence the inclusion $(W,M)\to (U,U\cap V)$ is an equivalence of pairs, and the same construction on the other side gives $(W^{\mathrm{\prime }},M)\to (V,U\cap V)$.

Excision applies with $X=N$, $A=V$ and $Z=N\setminus U$: the set $Z$ is closed and contained in the open $V$, so by [L2] restriction gives isomorphisms $H^{\ast }(N,V;R)\to H^{\ast }(U,U\cap V;R)$ and, with the other cover, $H^{\ast }(N,U;R)\to H^{\ast }(V,U\cap V;R)$; composing with step 2.1 yields the isomorphisms $r_W:H^{\ast }(N,V;R)\to H^{\ast }(W,M;R)$ and $r_{W^{\mathrm{\prime }}}:H^{\ast }(N,U;R)\to H^{\ast }(W^{\mathrm{\prime }},M;R)$, and the same argument with [L2]'s homology clause gives the corresponding relative homology isomorphisms.

Naturality of the relative product under the maps of step 3.1 gives, for relative classes $a,b$ on $(W,M)$ represented through the inverse $E_W=r_W^{-1}$, the identity $r_W(E_W(a)⌣E_W(b))=a⌣b$, because the restriction maps preserve the quotient-cochain front/back products by [L6].

The image of $[N]$ under $H_8(N)\to H_8(N,V)$ is $[W,M]$: at every interior point of $W$ its local restriction is the prescribed orientation generator of $W$, and excision together with the collar homotopies of step 2.1 preserves that generator, so [L5]'s uniqueness identifies the class; on the second side the ambient orientation of $N$ is the reverse of that of $W^{\mathrm{\prime }}$, hence the image is $-[W^{\mathrm{\prime }},M]$.

Consequently, for $\eta \in H^8(N,V;R)$, naturality of Kronecker evaluation [L6] gives $⟨\eta ,[N]⟩=⟨r_W\eta ,[W,M]⟩$, and with the other cover the analogous identity holds with the negative sign on $-[W^{\mathrm{\prime }},M]$.

Components of the fillings that are closed or lie entirely on one side are treated by the same argument with the pair equal to the absolute pair; the comparison is componentwise and adds over the finitely many components.

Therefore the stated excision isomorphisms exist, the fundamental class restricts as asserted on the two sides, and the comparisons are compatible with relative products and Kronecker evaluation.