Unlink resolution covers and the Heegaard Floer TQFT

The topology assertion has a direct finite construction. Put the planar $k$-component unlink, $k\ge 1$, in its standard $k$-bridge position: the upper and lower balls each contain $k$ boundary-parallel arcs with the same noncrossing matching of their $2k$ endpoints. Choose their disjoint bridge disks. Cutting either ball along those disks trivializes the branch construction. Its double cover is two balls joined by $k$ one-handles; the first joins the two zero-handles, and each remaining handle adds one genus. Thus it is a genus-$k-1$ handlebody. The two bridge matchings and disk systems agree in the projection sphere, so the cover of the unlink is the double of this handlebody with matching meridian disks. Doubling each of its $k-1$ co-core disks gives disjoint nonseparating two-spheres. Cutting them removes those one-handles and leaves a sphere with paired ball holes; regluing the pairs gives ${\mathrm{\#}}^{k-1}(S^2\times S^1)$. For $k=1$ it is the double of a ball, hence $S^3$. The cover is constructed by crosswise gluing two labeled copies of the cut balls along the finitely many bridge disks, using $(s,z)\mapsto (s,z^2)$ along the branch arcs and the matching endpoint charts. These charts give its orientation and smooth structure directly. Thus this is a finite realization of the local branch convention in [F1], without invoking the generic AC-dependent covering classification or any Floer map.

Here is the complete finite algebra to which the external Floer map computation [F4]–[F5], applied below is applied. Order the unmarked circles of a based $k$-circle state, and assign to them generators $x_c$. The reduced state space is ${\mathrm{\Lambda }}_{{\mathbb{F}}_2}⟨x_c⟩$: the basis monomial on a subset labels those circles by $X$, the others by $1$, and the marked circle by the quotient label $1$. Exterior multiplication is commutative over ${\mathbb{F}}_2$ and each $x_c^2=0$. Thus this state space has dimension $2^{k-1}$, once an actual Floer vertex identification with it is supplied. This is a computation of the required state algebra, not yet of $\widehat{HF}$.

Write $A={\mathbb{F}}_2[X]\mathrm{/}(X^2)$. For an unmarked merge the required map is $m(1,1)=1$, $m(1,X)=m(X,1)=X$, $m(X,X)=0$. For an unmarked split it is $\mathrm{\Delta }(1)=1\otimes X+X\otimes 1$, $\mathrm{\Delta }(X)=X\otimes X$. If a marked circle merges with an unmarked one, reducing the output kills $X$ on the marked factor, so $1\otimes 1\mapsto 1$ and $1\otimes X\mapsto 0$. If a marked circle splits, reduce the child containing the original point: $\mathrm{\Delta }(1)$ then becomes $1_{\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}\mathrm{d}}\otimes X_{\mathrm{n}\mathrm{e}\mathrm{w}}$ (with factors exchanged when necessary). Hence marked split creates the unmarked label $X$, not $1$. All untouched factors are carried identically. These are literal quotient calculations and include both marked cases.

Equivalently, unmarked merge is the exterior-algebra quotient identifying $x_c,x_d$ with $x_e$. Unmarked split is $(x_d+x_e)\wedge \mathrm{\Lambda }\mathrm{ȷ}$, where $\mathrm{ȷ}(x_c)=x_d$ and other generators are unchanged; choosing $x_e$ instead gives the same map, since $(x_d+x_e{)}^2=0$. Marked merge is quotient by the disappearing $x_c$, and marked split is wedge with the newly created $x_e$. Thus these four precise quotient/wedge operations, in a common state basis, are the finite map targets for the analytic surgery-edge calculation.

Apply [F4] to the actual cover constructed in step1.1. Its rank-one exterior module gives the rank $2^{k-1}$, with generators represented by the arc lifts specified in [F5]. That same input supplies the actual Floer triangle maps and their common state bases. Under these bases the quotient/wedge operations of [F4], [F5] are precisely the four finite reduced tables proved in steps 1.2, 2.1, 3.1. In particular $[{\gamma }_0]=0$ kills the unwanted marked-circle factor and forces marked split to create the new $X$ label.

The vertex and map computations in the preceding step are externally imported, not inferred from analytic regularity [F3] or from a chosen common-top transfer. The topology and quotient algebra remain the local finite arguments above. All coefficients are ${\mathbb{F}}_2$, so no integral sign or instanton-saddle formula is used. The actual finite diagram has finitely many states and edges, with the analytic data in the stated AComega branch. This proves the full fixed-resolution Floer TQFT assertion with its compatible marked bases.