Polygonal schemas and paired boundary edges

A polygonal schema is finite data consisting of a nonempty family of oriented, nondegenerate closed disks, each with its boundary divided into finitely many sides meeting only at their common endpoints, together with a partition of the sides into pairs. A disk with at least three sides is polygonal: its sides are the straight edges of that polygon. A disk with exactly two sides — a bigon — has its two sides realized as simple arcs meeting exactly at the two corners, and a disk with a single side — a monogon — has its sole side realized as the whole boundary arc from the corner to itself. Polygons are the general case; bigons and monogons occur only in the degenerate standard presentations (the digon $a{a}^{-1}$ for the sphere and the digon $aa$ for the projective plane) and as intermediate pieces of the cut-and-paste moves. Each paired pair of sides is identified by a specified homeomorphism that maps the marked corners of one side onto the marked corners of the other (in particular, a monogon corner maps to the other monogon corner). The map is required to be affine whenever both sides are straight edges. Give the disjoint union $X$ of the disks its usual topology and give the realization $Y$ the quotient topology (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection). The quotient vertices are the equivalence classes of disk corners, its edges are the classes of paired side interiors, and its faces are the images of disk interiors.

A schema is a connected surface schema when $Y$ is connected, every edge class has exactly two incident face-sides, and the link at each vertex class is a single cycle. Here the link has one arc for each disk corner in that class, with link endpoints joined according to the paired side germs. The cycle condition means that a sufficiently small vertex star is a disk. Each paired edge interior has two half-disk neighborhoods, which join to a disk; disk-interior points already have disk neighborhoods. Thus these local conditions give a boundaryless surface.

The topology has the required global properties as well. Each disk is a closed bounded subset of the plane and is compact by Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, so $X$ is a finite disjoint union of compact metric spaces. The pairing relation $R$ on $X$ is a finite union of the diagonal, the closed graphs of the side maps (each a continuous map of a compact side into a Hausdorff space, hence with closed graph), and the finitely many vertex-class pairs; hence $R$ is closed in $X\times X$. If $q:X\to Y$ is the quotient map and $F\subseteq X$ is closed, then

$q^{-1}(q(F))={\mathrm{pr}}_2(R\cap (F\times X))\mathrm{.}$

The product $X\times X$ is a finite disjoint union of products of closed bounded disks in ${\mathbb{R}}^4$, hence compact by Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line. This projection is compact and closed in the Hausdorff space $X$ (In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones). The quotient topology therefore makes $q(F)$ closed, so $q$ is a closed map. Two distinct fibers are disjoint compact subsets of $X$ and have disjoint open neighborhoods $U,V$ by the same compact-separation theorem. The open sets $Y\setminus q(X\setminus U)$ and $Y\setminus q(X\setminus V)$ then separate their quotient points. Hence $Y$ is Hausdorff.

The space $X$ has a countable basis, obtained by intersecting rational open rectangles with its finitely many disks. For each finite union $B$ of members of this basis, put $W_B=Y\setminus q(X\setminus B)$. The closed-map property makes every $W_B$ open. If $O\subseteq Y$ is open and $y\in O$, the compact fiber $q^{-1}(y)$ is covered by finitely many basis members whose union $B$ lies in $q^{-1}(O)$. Then $y\in W_B\subseteq O$. There are only countably many such finite unions, so the $W_B$ form a countable basis. Thus every connected surface schema realizes a nonempty compact connected Hausdorff, second-countable topological $2$-manifold, in the convention of Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces. These finite constructions use no full Axiom of Choice.

The quotient has a finite CW structure: the vertex classes are its $0$-cells, the paired side classes its $1$-cells, and the disk interiors its $2$-cells. If there are $V$ vertex classes, $E$ side pairs, and $F$ disks, then the cell counts are $(n_0,n_1,n_2)=(V,E,F)$.

A one-polygon schema is a connected surface schema with one face. Choose a reference direction on each paired edge and give it a letter. Read the face boundary cyclically: write $a$ when traversal agrees with the reference direction and $a^{-1}$ when it disagrees. Every letter occurs exactly twice. Changing letter names changes only labels; changing the starting side cyclically rotates the word; reversing the polygon orientation reverses the word and inverts each letter. These changes give homeomorphic quotient descriptions. For a one-polygon schema, each pair with opposite exponents is orientation compatible, while equal exponents give a twisted pairing. The sphere, orientable handles, and crosscaps below use this boundary-direction convention. The empty schema and schemas with unpaired boundary sides are excluded.