Boundary homology cokernels under bordism gluing

The supplied collars give an open cover of $Y$ by a neighborhood of $W$ and a neighborhood of $X$, retracting onto $W,X$, with intersection retracting onto $A$. In collar coordinates this is obtained by adding a short collar segment across the seam to each side; shrinking its height linearly gives the three retractions. Hence [F1] gives the exact window $H_2(A;R)\stackrel{(i_W,-i_X)}{\to }{H}_2(W;R)\oplus H_2(X;R)\stackrel{j_W+j_X}{\to }{H}_2(Y;R)\stackrel{\delta }{\to }{H}_1(A;R)\stackrel{(i_W,-i_X)}{\to }{H}_1(W;R)\oplus H_1(X;R)\mathrm{.}$ The last arrow is injective, since its $X$ coordinate is injective by hypothesis. Exactness therefore says that $j_W+j_X$ is surjective. This statement is elementwise and does not choose a simultaneous family of preimages.

Write $Q_Y=H_2(Y;R)\mathrm{/}\mathrm{im}{H}_2(D\bigsqcup B;R)$ and let $\pi$ be its quotient map. Every $x\in H_2(X;R)$ is the image of a class in $H_2(B;R)$, by hypothesis. Its image in $H_2(Y;R)$ therefore lies in the boundary image, so $\pi j_X=0$. For any $y\in H_2(Y;R)$ choose, for this one $y$, a preimage $(w,x)$ under the surjection of step 1.1. Then $\pi (y)=\pi j_W(w)$. Thus $\pi j_W:H_2(W;R)\to Q_Y$ is surjective.

A class from $H_2(D;R)$ maps to zero under $\pi j_W$ because $D$ is a component union of $\mathrm{\partial }Y$. A class $a\in H_2(A;R)$ has $j_W{i}_W(a)=j_X{i}_X(a)$ by the zero composite of the first two Mayer–Vietoris arrows. step 2.1 makes the latter class zero in $Q_Y$. Since $\mathrm{\partial }W=A\bigsqcup D$, every boundary class of $W$ is a sum of these two kinds, so $\pi j_W$ vanishes on the full boundary image. It therefore factors uniquely through the displayed source quotient, and the factored map is surjective by step 2.1. This proves the claimed canonical map, without any assumption on $H_2(A;R)\to H_2(X;R)$.

If the source quotient is a vector space with a finite basis of size $r$, its $r$ images span the target. Finite elimination deletes linearly dependent images until a basis remains, so the target has dimension at most $r$. Only finitely many operations and selections occur. The proof also covers an empty seam or empty outer boundary, zero homology groups and the zero coefficient ring. No step selects representatives for infinitely many classes; surjectivity is used for a single class at a time. Thus the result and the finite-dimensional consequence require no choice principle.