The cut surface, primitives of closed forms, and their boundary jumps

Let $q:D\to X$ be the quotient model from [F1,F2]. The boundary image is $C$, and no interior points are identified, so $q^{-1}(F)=D^{\circ }$ and this restriction is a homeomorphism by the quotient topology. The interior of a disk is path connected and contracts to a point, hence is simply connected by [F8]; this proves the cut geometry.

For any continuous path $\gamma :[0,1]\to X$, cover its image by neighborhoods $U$ with local primitives $H_U$ from [F3]. Compactness of $[0,1]$ and a Lebesgue number for the pulled-back cover give a finite subdivision so that each subpath lies in one such $U$. Define ${\mathcal{I}}_{\alpha }(\gamma ):={\sum }_j(H_{U_j}(\gamma (t_j))-H_{U_j}(\gamma (t_{j-1})))$. This value is independent of the subdivision and primitives: on each segment of a common refinement, the two primitives differ by a locally constant function on their overlap, and the connected path image lies in one component of that overlap. The definition is additive under concatenation and changes sign under path reversal.

The path integral is invariant under homotopy with fixed endpoints. For a homotopy $H:[0,1{]}^2\to X$, pull back the local-primitive cover along $H$. By [F4], choose $n$ so that $\sqrt{2}\mathrm{/}n$ is below a Lebesgue number, divide the square into an $n\times n$ grid, and split each small square into two triangles; each triangle maps into one primitive neighborhood. Choose such a neighborhood for each triangle using finite choice. The integral around each triangle is zero because it is the sum of endpoint differences of one primitive. Summing cancels all interior edges and leaves the integral around the square boundary; for a fixed-endpoint homotopy the two vertical edges are constant paths and contribute zero, so the two endpoint paths have equal integrals. For any two paths from $x_0$ to $x$ in $F$, their concatenation with one path reversed is a loop; [F8] makes its class trivial, hence a null-homotopy gives a homotopy between the two paths with endpoints fixed. Applying the square argument inside $F$ shows their integrals agree, so $f$ is well defined. Near each point, a local primitive $H_U$ gives $f=H_U+\text{constant}$, proving smoothness and $df=\alpha {\mathrm{\mid }}_F$. If $\alpha$ is holomorphic, [F5] gives a holomorphic local primitive and the same local equality proves that $f$ is holomorphic.

For $z\in D$, define $\widehat{f}(z)$ by integrating $\alpha$ along the image under $q$ of any path in $D$ from the lift of $x_0$ to $z$. The disk is simply connected, so the homotopy argument of step 2.1 makes this independent of the path. Near each point of $D$, a local primitive on $X$ shows that $\widehat{f}$ is that primitive composed with $q$, plus a constant; hence $\widehat{f}$ is continuous up to every boundary side and corner and restricts to $f$ on $D^{\circ }$. For a matched point at parameter $t$ on $A_i^{+}$ and $A_i^{-}$, the positively oriented boundary path from $A_i^{+}(t)$ to $A_i^{-}(t)$ traverses the remaining part of $A_i^{+}$, all of $B_i^{+}$, and the oppositely oriented matching part of $A_i^{-}$. The two $a_i$ contributions cancel by additivity and reversal, leaving ${\mathrm{\Pi }}_{\alpha }(b_i)$. For a matched point on $B_i^{+}$ and $B_i^{-}$, the corresponding boundary path traverses the remaining part of $B_i^{+}$, the inverse-oriented $A_i^{-}$, and the inverse-oriented matching part of $B_i^{-}$. The $b_i$ contributions cancel, leaving $-{\mathrm{\Pi }}_{\alpha }(a_i)$. These differences are independent of $t$. When $\alpha$ is holomorphic, [F6] identifies these local-primitive path integrals with $P$ on the named homology classes.