A slice disk induces a Seifert metabolizer

Construct the compact oriented locally flat $R$ by [F5], and write $\mathrm{\Sigma }=\mathrm{\partial }R=F\cup D$ after matching collars. The Axiom of Choice [A1] is the hypothesis under which the supplied Seifert-pairing and duality interfaces are invoked. Capping the sole boundary circle of $F$ by a disk does not change $H_1$ because that circle is already the boundary of the oriented fundamental $2$-chain of $F$; by [F9] and exactness [F10], $H_1(F;\mathbb{Q})\cong H_1(\mathrm{\Sigma };\mathbb{Q})$ has dimension $2g$. In the pair sequence [F10], $H_0(\mathrm{\Sigma };\mathbb{Q})\to H_0(R;\mathbb{Q})$ is an isomorphism because both are connected. Exactness therefore makes $H_1(R;\mathbb{Q})\to H_1(R,\mathrm{\Sigma };\mathbb{Q})$ surjective. By [F2] and [F14], the target has dimension $b_2(R)$, so the image of $H_1(\mathrm{\Sigma };\mathbb{Q})\to H_1(R;\mathbb{Q})$ has dimension $b_1(R)-b_2(R)$.

Let $\widehat{R}$ be the double of $R$ along $\mathrm{\Sigma }$, with the second copy oppositely oriented so the orientations glue. Its two copies have open collar thickenings with intersection retracting to $\mathrm{\Sigma }$, so [F11] gives $\chi (\widehat{R})=2\chi (R)-\chi (\mathrm{\Sigma })$. By [F12] and [F13], $\chi (\widehat{R})=0$, and by [F9], $\chi (\mathrm{\Sigma })=2-2g$; hence $\chi (R)=1-g$. Also [F2] gives $H_3(R;\mathbb{Q})\cong H^0(R,\mathrm{\Sigma };\mathbb{Q})=0$: a relative zero-cocycle is locally constant on the connected $R$ and vanishes on the nonempty connected $\mathrm{\Sigma }$, so it is zero. With $b_0(R)=1$, the rational Euler-characteristic identity [F13] implies $b_1(R)-b_2(R)=g$. Thus the kernel $L=\ker (H_1(\mathrm{\Sigma };\mathbb{Q})\to H_1(R;\mathbb{Q}))$, identified with a subspace of $H_1(F;\mathbb{Q})$, has dimension $2g-g=g$.

Represent $x,y\in L$ by finite rational smooth cycles in the given $F$, away from $K$, and choose finite rational continuous bounding chains in $R$. Use its pre-collar-push model and the actual positive normal product of [F5]. Push the first chain into the positive complementary side, with boundary the actual positive Seifert push $x^{+}$; leave the second chain in $R$ with boundary $y$. An intrinsic boundary-collar push of their interiors and the common product near $F$ give the usual collared bounding chains in $B^4$, missing the corner $K$. Their supports are disjoint compact sets by the actual two-sided product. Choose disjoint relatively open smooth ambient neighborhoods $U,V\subset B^4$ of those supports, including their boundary cycles; these are smooth manifolds with boundary as open subsets of the standard smooth ball. In $U$, [F6] makes the smooth cycle $x^{+}$ bound a finite smooth real chain, since it bounds continuously there; likewise $y$ bounds in $V$. In each finite set of occurring simplices the equations for the boundary coefficients are rational, so rational Gaussian elimination replaces the real coefficients by rational ones without enlarging support. Near $S^3$, move the interiors inward and attach the standard product of the fixed boundary cycle with the collar interval. This puts both chains in common standard end collars, still in disjoint $U,V$. They therefore have empty interior intersection, and [F3] gives $V_{\mathbb{Q}}(x,y)=\mathrm{lk}(x^{+},y)=0$. The smoothing here concerns only chains in the disjoint smooth ambient regions; it assumes no smooth structure on $R$ or its TOP normal chart.

Apply the integral-saturation result [F4] to this $g$-dimensional rational isotropic subspace: $M=H_1(F;\mathbb{Z})\cap L$ is a primitive rank-$g$ direct summand with $V{\mathrm{\mid }}_{M\times M}=0$. By [F8] the pairing is on an integral Seifert lattice, so the metabolic-form definition makes $M$ a metabolizer. This proves the Statement for every locally flat slice disk, including every smooth slice disk.