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The Snake lemma associates to a commutative diagram with exact rows an exact sequenceFor $K=L_1\cup L_2$, apply it degree by degree to the short exact sequence of chain complexesThe resulting connecting maps splice the kernels modulo boundaries into the Mayer--Vietoris sequence

Now let $K$ satisfy the stated simplicial pseudomanifold conditions. In an $n$-cycle, cancellation at a common $(n-1)$-face determines the coefficient of either incident $n$-simplex from the other, up to the orientation sign. Connectivity through such faces therefore determines every top-simplex coefficient from one integer. If the signs are globally compatible, their oriented sum is a cycle and generates $H_n(K)\cong \mathbb{Z}$; if they are inconsistent, that integer must be zero and $H_n(K)=0$.

Let $x$ be the resulting fundamental class, represented by $z$. Since the intersection has dimension below $n$, split uniquely
$z=z_1+z_2$ into top chains in $L_1$ and $L_2$. Thenand the Mayer--Vietoris boundary isIt is nonzero exactly when both $L_1$ and $L_2$ contain top-dimensional simplices; if one side contains none, the fundamental cycle already lies in the other side, while if both do, connectedness forces a nonempty interface and its oriented boundary represents a nonzero class.

Finally take $K\cong S^3$, $L_1\cong S^1\times D^2$, and
$L_1\cap L_2\cong T^2$. The preceding boundaryis a nonzero map between copies of $\mathbb{Z}$ and sends the fundamental class to the oriented boundary torus, hence is an isomorphism. Exactness then givesThe next part of the sequence isSince $H_1(L_1)\cong \mathbb{Z}$, it follows that
$H_1(L_2)\cong \mathbb{Z}$. The degree-zero sequence makes $L_2$ connected. Therefore

A topological surface is a Hausdorff space with a countable basis of a topology in which every point has an open neighbourhood homeomorphic to an open subset of the Euclidean plane. We use the convention that a surface has no boundary unless a surface with boundary is explicitly specified. A closed surface is compact and has no boundary. Some conventions also include connectedness in the definition; we impose it on the closed combinatorial surface below.

A closed combinatorial surface is a finite connected simplicial complex $K$ with the following properties: every simplex is a face of a simplex of dimension two, and the link of a simplex at every vertex is a cycle graph. Explicitly, for a vertex $v$ its link is the simplicial subcomplexTo say this is a cycle graph means that there are distinct vertices $w_1,\dots ,w_m$, with $m\ge 3$, and exactly the link edges $\{w_i,w_{i+1}\}$, with indices modulo $m$. In particular, every edge of $K$ belongs to exactly two triangles. The vertex-link condition also requires that the triangles around each vertex form one cyclic fan; edge incidence alone would allow singular examples.

These conditions make $|K|$ a closed surface. A point inside a triangle has a disc neighbourhood, the two triangles at an edge give a disc neighbourhood, and the cyclic fan at a vertex gives a disc neighbourhood. Finiteness gives compactness, the Hausdorff property, and a countable basis of a topology from a realization in a finite-dimensional Euclidean space. If disconnected closed combinatorial surfaces are allowed, each connected component satisfies this definition separately.