Homeomorphism-preserving polygonal schema moves

Disk representatives may be changed without changing the quotient. Choose a homeomorphism from each closed disk to the standard disk. A prescribed boundary-circle homeomorphism extends radially by $ru\mapsto r\phi (u)$, with inverse $ru\mapsto r{\phi }^{-1}(u)$; continuity at the center follows from preservation of radius. Thus marked boundary points and side parameters can be prescribed, and the pairings are conjugated by these disk homeomorphisms. Choose a parameter on one side of each pair and transport it to its mate; corner preservation makes the resulting boundary parametrizations consistent. Disks with at least three marked corners can consequently be represented by convex polygons with affine pairings, and bigons by round disks with two marked boundary arcs. After a cut or subdivision we again use such representatives. Inverse splits below glue distinct disks along proper boundary arcs, the copies of a split diagonal; gluing two entire monogon circles is not an inverse diagonal split. This ensures that the intermediate faces remain genuine disks even when a side is curved or is composed of several straight segments.

A diagonal is a polygonal arc $\delta \subseteq P$ whose relative interior lies in $\mathrm{int}P$ and whose two endpoints are boundary points of $P$; cutting $P$ along $\delta$ produces two closed polygonal disks $P_1,P_2$ and a continuous folding map $f:P_1\bigsqcup P_2\to P$ that is injective off the two copies of $\delta$ and identifies those two copies by the natural reversal, so that every point of $P$ is hit and the fibers of $f$ are exactly the classes of the equivalence relation $S$ generated by those two copies. The pasting $f$ is continuous by [L5], so the induced bijection $(P_1\bigsqcup P_2)\mathrm{/}S\to P$ is continuous by the final-topology property of the quotient [L2]; its source is compact as a continuous image of the compact disjoint union of the polygons [L3], its target is a Hausdorff subset of the plane, so it is a homeomorphism by [L4]. Consequently $(P_1\bigsqcup P_2)\mathrm{/}(S\cup R)\cong P\mathrm{/}R=Y$, because quotienting by a union of relations is the same as iterated quotienting; in particular the split adds the new paired pair of sides without changing the quotient. Conversely, if two faces of a schema have boundary words $ud$ and $d^{-1}v$ after the cyclic rotations that bring the glued occurrence to the front, then merging them along that paired pair produces a single face with boundary word $uv$ and the same quotient: the two disks quotiented by the identification of the $d$-sides form the same space as the one disk whose boundary is the concatenation of the two remaining arcs, the identification data being unchanged.

Edge subdivision. Replace one side $a$ of $P$ by two sides $b,c$ with $a=bc$ and $a^{-1}=c^{-1}{b}^{-1}$ in the boundary word, where the new vertex is inserted at an interior point of $a$ and the paired side $a^{\ast }$ is subdivided at the image point under the pairing; the point set of the polygon is unchanged, and the equivalence relation generated by the pairings is literally the same relation, since each pairing homeomorphism is only being described on two subarcs instead of on one arc.

Rotation and reversal. Reading the cyclic word from a different side does not change the disk, the pairings, or the relation, so the realization is identical; reversing the orientation of the polygon reverses the traversal of every side, so every exponent changes sign and the cyclic word is reversed, while the pairings themselves, and hence the quotient, are unchanged.

Cancellation of an adjacent inverse pair. If the word contains the consecutive string $a{a}^{-1}$, rotate its starting point so that it reads $Xa{a}^{-1}Z$ with both $X$ and $Z$ nonempty (the remaining paired letter supplies at least two side occurrences). Then split the single face by step 1.2 between the two occurrences using a new letter $d$, so that the two new faces have boundaries $Xad$ and $d^{-1}{a}^{-1}Z$; contract the string $ad$ of the first face to a single new side $c$ and $d^{-1}{a}^{-1}$ of the second face to $c^{-1}$, which is the inverse of the edge subdivision of step 1.3 and is legal because $a$ and $d$ are consecutive sides of one face and hence lead to a common vertex; the two faces now have boundaries $Xc$ and $c^{-1}Z$, and gluing them along the paired pair $c,c^{-1}$ by the inverse operation of step 1.2 produces a single face with boundary word $XZ$. Each of the three operations preserves the realization by steps 1.2 and 1.3, so $Xa{a}^{-1}Z$ and $XZ$ present homeomorphic quotients. If $X=Z=\epsilon$, the sole $a{a}^{-1}$ bigon already presents the sphere and is retained as the terminal geometric schema; writing the reduced word as empty is bookkeeping and does not assert an empty-face schema.

Separated crosscap form. Suppose the word reads $aXaY$ with strings $X,Y$, both occurrences of $a$ carrying the same exponent, and cut the single face between the two occurrences, along the diagonal from the vertex starting the first occurrence to the vertex starting the second; these are distinct boundary points because the arc from the first to the second occurrence contains the edge $a$, and the two faces have boundary words $aXb$ and $b^{-1}aY$ with a new letter $b$. The first face reads cyclically as $Xba$; the second occurrence of $a$ carries exponent $+1$, so the presentation of the second face used in the gluing is the inverse word $Y^{-1}{a}^{-1}b$, which reads cyclically as $a^{-1}b{Y}^{-1}$ and begins with $a^{-1}$. Gluing the two faces along the pair $a,a^{-1}$ by the inverse operation of step 1.2 concatenates $Xb$ with the remainder $b{Y}^{-1}$ and gives $Xbb{Y}^{-1}$, cyclically equal to $bb{Y}^{-1}X$. Thus $aXaY$ and $bb{Y}^{-1}X$ present homeomorphic quotients, and the two occurrences of the new letter $b$ are adjacent with equal exponents.

Adjacent crosscap form. Suppose the word contains two adjacent occurrences of a letter with equal exponents and reads $aaXY$ with $X$ nonempty. Cut the disk along a chord from the vertex between the two occurrences to the vertex between $X$ and $Y$, with the chord interior in the disk interior; this is a split as in step 1.2 whose two faces have boundary words $aXb$ and $Ya{b}^{-1}$, so the two $a$-occurrences lie in different faces, and the chord connects two distinct boundary points exactly because $X$ is nonempty. Both occurrences of $a$ carry exponent $+1$, so the presentation of the second face used in the gluing is the inverse word $b{a}^{-1}{Y}^{-1}$, which begins with $a^{-1}$; gluing the two faces along the pair $a,a^{-1}$ by the inverse operation of step 1.2 concatenates $Xb$ with the remainder $Y^{-1}b$ and gives $Xb{Y}^{-1}b$. Reversing the polygon orientation presents this same schema by the inverse word $b^{-1}Y{b}^{-1}{X}^{-1}$ by step 1.4, and renaming the two new sides with the opposite reference direction, which is legitimate because $b$ is a new pair of sides introduced by the cut, turns that word into $bYb{X}^{-1}$. Hence $aaXY$ and $bYb{X}^{-1}$ present homeomorphic quotients whenever $X$ is nonempty, and the two occurrences of the new letter in $bYb{X}^{-1}$ again carry equal exponents.

Conjugation rule. Suppose the cyclic word reads $aUV{a}^{-1}X$. Split the face between $U$ and $V$ along a new diagonal $c$ by step 1.2; the two face words are $aUc$ and $c^{-1}V{a}^{-1}X$. Rotate them to $Uca$ and $a^{-1}X{c}^{-1}V$. These already contain opposed occurrences $a,a^{-1}$, so merge along that pair by inverse $(P2)$ without reversing the second face. The new cyclic word is $UcX{c}^{-1}V$, which rotates to $c^{-1}VUcX$. Set $b=c^{-1}$ to get $bVU{b}^{-1}X$. The diagonal endpoints are distinct whenever this is an actual split; empty residues are interpreted by the same finite cut geometry. Each displayed cut or merge is a homeomorphism-preserving quotient operation, and the resulting one-face surface schema has the same realization.

Handle extraction. For the cyclic pattern $aUbV{a}^{-1}X{b}^{-1}Y$, apply the conjugation rule of step 2.4 successively to three opposed pairs, rotating between applications as displayed: $aUbV{a}^{-1}X{b}^{-1}Y\sim a_{1}bVU{a}_1^{-1}X{b}^{-1}Y\sim a_1{b}_1{a}_1^{-1}XVU{b}_1^{-1}Y=a_1^{-1}XVU{b}_1^{-1}Y{a}_1{b}_1\sim a_2^{-1}{b}_1^{-1}YXVU{a}_2{b}_1=a_2{b}_1{a}_2^{-1}{b}_1^{-1}YXVU$. Here each $\sim$ is one diagonal split followed by the inverse merge along a conjugate pair; each equality is cyclic rotation, so all intermediate quotients are homeomorphic. The final first four letters are a commutator block, and the residue is in the exact order $YXVU$.

One handle plus one crosscap gives three crosscaps. Begin with $aaXbc{b}^{-1}{c}^{-1}Y$. The adjacent-pair rule of step 2.3 applies to the nonempty separating string $Xbc$ even when $X=\epsilon$. Then apply the separated same-direction rule of step 2.2 three times, rotating as needed. The exact cyclic sequence is $aaXbc{b}^{-1}{c}^{-1}Y\sim a_1{b}^{-1}{c}^{-1}Y{a}_1{c}^{-1}{b}^{-1}{X}^{-1}=b^{-1}{c}^{-1}Y{a}_1{c}^{-1}{b}^{-1}{X}^{-1}{a}_1\sim b_1^{-1}{b}_1^{-1}{a}_1^{-1}X{c}^{-1}Y{a}_1{c}^{-1}=c^{-1}Y{a}_1{c}^{-1}{b}_1^{-1}{b}_1^{-1}{a}_1^{-1}X\sim c_1^{-1}{c}_1^{-1}{X}^{-1}{a}_1{b}_1{b}_{1}Y{a}_1=a_1{b}_1{b}_{1}Y{a}_1{c}_1^{-1}{c}_1^{-1}{X}^{-1}\sim a_2{a}_{2}X{c}_1{c}_1{b}_1{b}_{1}Y$. Every $\sim$ is a split/merge of genuine disks; the split faces have distinct diagonal endpoints because the displayed separator contains an edge. The residue strings may be empty, and the final three adjacent same-direction pairs can be relabelled by independently choosing their reference directions. All intermediate words have at least one paired letter and present the same surface.

Preservation of the surface link condition. Each move of steps 1.3, 1.4, 2.1, 2.2, 2.3 and 2.4 produces a new finite polygonal schema whose realization is homeomorphic to the given realization $Y$, as proved move by move above; since $Y$ is a surface, every point of the new realization has a disk neighborhood, and in the new schema an edge class with one incident face-side would give its points a half-disk neighborhood while a vertex class whose link is not a single cycle would give its points a cone or a multi-sector neighborhood, neither a disk. These local models depend only on the finite incidence data, so every edge class of the new schema has exactly two incident face-sides and every vertex class has one cyclic link: each move preserves the connected surface schema condition.

Every operation used above is a finite, explicitly described construction: one diagonal with two endpoints, one relabelling, or one sequence of finitely many splits and gluings of a single polygon; no family of nonempty sets is ever selected from, and the boundary words are finite. Hence none of the moves of the statement uses the Axiom of Choice, and the identities of steps 2.2, 2.3, 3.1 and 3.2 are explicit finite cut-and-paste derivations.