Sc toolkit van kampen existence

Let a diagram have a face. There is a face edge bordering the unbounded region of the union of faces: a generic ray from an interior point has a last crossing of that finite union. Deleting this open edge and its incident open face retracts that polygon onto its complementary boundary path, leaving a connected simply connected planar complex. At the chosen outer occurrence write the boundary as $AeB$ and the face word as $r=e{q}^{-1}$, so the new boundary is $AqB$. In the free group $AeB=(Ar{A}^{-1})(AqB)$, since the inserted $q^{-1}q$ and $A^{-1}A$ cancel. Repeat until every face has been removed; each removal contributes exactly one conjugate. The remaining connected simply connected graph is a tree, whose boundary freely cancels to the empty word by deleting end edges. Thus the original boundary is freely equal to a product of exactly $m$ conjugates when there are $m$ faces. This peeling is the algebraic unfolding of the diagram into face polygons and conjugating paths. It also treats $m=0$ directly.

Conversely suppose $w$ lies in the normal closure. First freely reduce it to $\bar{w}$. By [F2] express $\bar{w}$ as a product ${\prod }_{j=1}^m{u}_j{r}_j{u}_j^{-1}$ in the free group, using the least possible $m$. Draw $m$ disjoint polygons in planar order, joining their basepoints to a common point by separate whiskers labelled $u_j$. Its outer word is that literal product. With $m=0$, a sequence of inverse-pair insertions into the empty walk gives a tree reading any freely trivial word.

Fold consecutive outer edges whose letters cancel, identifying them with opposite traversals. If they already form an end spur, delete the spur. Otherwise the two edges bound a sector in the unbounded region and can be identified across that sector. Distinct other endpoints are merged; the sector closes to a slit, so the resulting complex remains planar and simply connected. If the other endpoints already agree, the two edges instead enclose a component. The fold would seal this component into a sphere attached at a point. Before sealing it, discard its interior and identify the two edges: its boundary is precisely the cancelling pair, so the outside boundary word undergoes the same free cancellation. Any positive-area discarded component would give a diagram with fewer than $m$ faces for the same freely reduced word; step 1.1 would give a product with fewer than $m$ factors, contrary to minimality. A component of area zero is a tree and is removed by spur deletions. This accounts for the possible spherical closure, including when the outside word is empty; then minimality already forces $m=0$.

Each fold decreases outer length by two, so the finite reduction terminates with a planar simply connected diagram reading exactly $\bar{w}$. To recover $w$, reverse its chosen free reduction: for each insertion of $a{a}^{-1}$ at a boundary occurrence attach a fresh edge labelled $a$ in that occurrence's exterior sector. This adds a spur, introduces no face or hole, and inserts exactly the required pair. Together with step 1.1 this proves both directions and the $m$-factor assertion.