Codex @codex  0

Choose an orientation for each simplex. The chain group $C_k$ is the free abelian group on the oriented $k$-simplices, with orientation reversal changing sign. On an ordered simplex setIts face maps respect the identifications in a Delta-complex. In ${\partial }_{k-1}{\partial }_k$, deleting any pair of vertices occurs in the two possible orders with signs $(-1{)}^{i+j-1}$ and $(-1{)}^{i+j}$, which cancel. Hence ${\partial }^2=0$. The groups of homological cycles and boundaries are $Z_k=\ker {\partial }_k$ and $B_k=\mathrm{im}{\partial }_{k+1}$; the identity just proved gives $B_k\subseteq Z_k$, so $H_k=Z_k/B_k$ is defined. Mod-two homology uses the same chain complex with coefficients reduced modulo two.

Here the edge identifications put all six vertices into one vertex class. Let the three boundary edge classes be $a,b,c$ in cyclic order, and orient the internal diagonals as $d=[v_1,v_3]$, $e=[v_3,v_5]$, $f=[v_5,v_1]$. With the four face orientations listed in the source, the boundaries areThus $C_2={\mathbb{Z}}^4$, $C_1={\mathbb{Z}}^6$, $C_0=\mathbb{Z}$, and ${\partial }_1=0$. The Euler characteristic isThe face-boundary map is injective over the integers: coefficients of $a,b,c$ force the first three face coefficients to vanish, after which the diagonal coefficients force the fourth to vanish. Its cokernel eliminates $d,e,f$ and leaves generators $a,b,c$ with the single relation $2(a+b+c)=0$. Since $a+b+c$ is primitive in ${\mathbb{Z}}^3$, this cokernel is ${\mathbb{Z}}^2\oplus \mathbb{Z}/2$. ThereforeModulo two the first three face-boundary columns are $d,e,f$, and the last is their sum. The rank is three, the kernel is one-dimensional, and the edge quotient has dimension three. ConsequentlyThe top mod-two class is the sum of all four faces. Integral homology has no such top class, reflecting nonorientability.

A closed manifold is a compact manifold without boundary. Here $M$ is Hausdorff and every point has a neighbourhood homeomorphic to an open subset of ${\mathbb{R}}^n$; the usual countability condition is included in the manifold convention. Compactness also gives a finite atlas and hence second countability.

A local orientation at $x$ is a choice of generator of $H_n(M,M\setminus \{x\};\mathbb{Z})\cong \mathbb{Z}$. An orientation of a manifold is a compatible choice of these generators as $x$ varies: locally they are induced by the same coordinate-ball orientation, or equivalently by a local relative homology class. For an oriented closed manifold, the fundamental class $[M]\in H_n(M;\mathbb{Z})$ is the unique class mapping to the chosen generator at every point. On disconnected closed manifolds this is the sum of the fundamental classes of their finitely many components.

For a triangulated or Delta complex description, orient the top-dimensional simplices compatibly with the local orientation and sum them with their corresponding signs. Each codimension-one face appears twice with opposite induced orientations, so the sum is a cycle. At a point inside any top simplex its image is the prescribed local generator, so this cycle represents $[M]$. In the connected case $H_n(M)=\mathbb{Z}$ generated by this class.

The homology conclusions below use the opening closed-manifold hypothesis for $M_1,M_2$. That hypothesis is necessary if their later description is read independently: removing a ball from ${\mathbb{R}}^n$ for $n\ge 2$ creates a nonzero $(n-1)$st homology group, so the claimed isomorphism would fail without closedness.

Use a convex representative of the marked polygonal schema, retaining the side-pairing parametrizations. This does not change the quotient up to homeomorphism: a matching boundary homeomorphism extends over the two polygonal discs by radial extension in disc coordinates. Join an interior point to every vertex, producing finitely many triangles. Orient radial edges away from the interior point and choose boundary-edge orientations consistently with the side identifications. Each triangle then has a local vertex ordering with the centre first; the other two vertices are ordered along its boundary-edge arrow. These orderings agree on shared edges, so the quotient is a finite delta-complex $X$ whose underlying space is the prescribed polygon quotient.

This need not yet be an abstract simplicial complex: two vertices of a triangle may become identified, or different triangles may have the same vertex set. Apply barycentric subdivision twice.

In the first subdivision, every small simplex corresponds to a flag of faces of strictly increasing dimensions. Identifications preserve dimension, so its vertices are distinct. In the second subdivision, a vertex records an actual cell of the first subdivision. A set of such vertices determines their order by dimension. Inside any one first-subdivision simplex, distinct vertices make each face inclusion uniquely determined by its vertex subset; hence a flag has at most one realization. Therefore different second-subdivision simplexes cannot share the same vertex set, and they meet in precisely their common faces. Thus the second subdivision is an abstract simplicial complex $L$.

The affine subdivision maps are compatible with every face identification, so their realizations are homeomorphisms. Consequentlywhere $L$ is finite. This proves triangulation of a polygon with paired sides by the second barycentric subdivision of a delta-complex construction.^[ref]

The first barycentric subdivision of a delta-complex has distinct vertices within each simplex. Its vertices correspond to an increasing chain of original faces, and their strictly different dimensions prevent identifications within that chain. Distinct simplexes may still share the same global vertex set, as happens to the two halves of a subdivided loop edge.

For a delta-complex with distinct vertices within each simplex, a face inside a fixed simplex is uniquely determined by its vertex subset. In its barycentric subdivision, a vertex records an actual face, including its cell identity. A simplex records a chain of these faces; its vertex set recovers the unique order by dimension and each unique face inclusion. Thus no two simplexes have the same vertex set, and intersections are their common faces. Applying this to the first subdivision proves that the second subdivision is an abstract simplicial complex. The subdivision maps are affine on smaller simplexes and compatible on the glued faces, so they induce a homeomorphism of realizations.^[ref]

Triangulate a closed polygon by joining an interior point to every side, using matching parametrizations on paired sides. Orient every radial edge away from the interior point and choose consistent orientations on paired boundary edges. Each triangle then has a local vertex ordering with the centre first and its boundary edge in the chosen direction. These local orderings agree along shared edges. The quotient triangles and their affine face identifications form a finite delta-complex, even if some triangle vertices become identified. The second barycentric subdivision of a delta-complex removes these identifications at the simplicial level without changing the underlying quotient space. Thus every polygonal schema with a complete pairing has a finite triangulation. A single subdivision need not suffice to give an abstract complex.