A nonextendible Spin-c structure with nonzero d on a rational-ball boundary

The false claim is: if a closed oriented three-manifold $Y$ with $H_{\ast }(Y;\mathbb{Q})\cong H_{\ast }(S^3;\mathbb{Q})$ bounds a smooth compact four-manifold $B$ with $H_{\ast }(B;\mathbb{Q})\cong H_{\ast }(\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t};\mathbb{Q})$, then a nonzero correction term $d(Y,\mathfrak{t})$ for any boundary Spin${}^c$ structure $\mathfrak{t}$ obstructs the existence of $B$.

The actual vanishing theorem, which requires extension over the ball

Let $B_2$ be the rational homology ball bounded by $L(4,1)$ constructed as the complement of a tubular neighborhood of a conic in ${\mathbb{C}\mathbb{P}}^2$. The canonical Spin${}^c$ label $i=0$ has $d(L(4,1),0)=3\mathrm{/}4$. The extension set has exactly two elements and every extending structure has correction term zero. Therefore label $i=0$ is nonextendible, even though this rational-ball filling exists.

The explicit rational ball bounded by the four-surgery lens space

The extension-set size and metabolizer

The lens-space correction-term calculation

Vanishing on extending structures