A compact oriented rational homology four-ball admits a Spin-c structure

By [F1], $B$ has the homotopy type of a finite simplicial complex. Its integral chain groups are finite free abelian groups. Finite integer row and column operations, using the Euclidean algorithm, give Smith forms for their boundary maps, so each integral homology group is finitely generated. Tensoring these finite matrices with $\mathbb{Q}$ kills exactly their finite cyclic factors and preserves their free ranks. The assumed vanishing of positive rational homology therefore makes both $H_1(B;\mathbb{Z})$ and $H_2(B;\mathbb{Z})$ finite torsion groups. No assertion about a choice of infinitely many generators is needed.

Every $x\in H_2(B;\mathbb{Z})$ is torsion by step 1.1. Apply [F2]: its embedded oriented surface has integer normal Euler number zero. Hence $⟨w_2(TB),{\rho }_{2}x⟩=0$. Thus the evaluation homomorphism of [F3] with $G={\mathbb{F}}_2$ sends $w_2(TB)$ to zero on the entire integral group $H_2(B;\mathbb{Z})$. By exactness, $w_2(TB)$ is the image of a class $v\in {\mathrm{Ext}}^1(H_1(B;\mathbb{Z}),{\mathbb{F}}_2)$. This also covers zero $H_2$, without requiring mod-two $H_2$ to vanish.

The coefficient lift is onto, including two-torsion. Write the finite group $H_1(B;\mathbb{Z})$ as a finite direct sum of cyclic groups $\mathbb{Z}\mathrm{/}n$. From the free resolution $0\to \mathbb{Z}\stackrel{n}{\to }\mathbb{Z}\to \mathbb{Z}\mathrm{/}n\to 0$, applying $\mathrm{Hom}(-,G)$ gives ${\mathrm{Ext}}^1(\mathbb{Z}\mathrm{/}n,G)=G\mathrm{/}nG$. The coefficient reduction therefore induces the explicit map $\mathbb{Z}\mathrm{/}n\to {\mathbb{F}}_2\mathrm{/}n{\mathbb{F}}_2$. If $n$ is odd its target is zero; if $n$ is even its target is ${\mathbb{F}}_2$ and the reduction is onto. Finite direct sums give a surjection ${\mathrm{Ext}}^1(H_1(B),\mathbb{Z})\to {\mathrm{Ext}}^1(H_1(B),{\mathbb{F}}_2)$. When $H_1=0$, both groups are zero and the same conclusion holds. Choose a preimage $\tilde{v}$ of the one class $v$ in step 2.1.

Let $c\in H^2(B;\mathbb{Z})$ be the image of $\tilde{v}$ under the integral UCT injection. Naturality of [F3] under $\mathbb{Z}\to {\mathbb{F}}_2$ identifies its reduction with the mod-two UCT image of $v$, which is exactly $w_2(TB)$ by step 2.1. Thus $c$ is an actual integral lift. In fact it lies in the finite Ext subgroup; no assertion that distinct such lifts label distinct Spin${}^c$ structures is made.

Choose a smooth Riemannian metric by a finite trivializing atlas and a smooth partition, as in the construction of [F1]. Apply [F4] to $B$ with this metric and the integral lift $c$. Its transition-cocycle construction produces the required Spin${}^c(4)$ principal bundle and its equivariant map to the oriented tangent frame bundle. This proves the original existence claim. All possible finite cyclic two-primary factors were retained in step 3.1; neither a vanishing mod-two homology assumption nor a determinant-class bijection was used.