The Snake lemma associates to a commutative diagram with exact rows an exact sequence

ker f_{1} \to ker f_{2} \to ker f_{3} δ coker f_{1} \to coker f_{2} \to coker f_{3} .

For

K = L_{1} \cup L_{2}

, apply it degree by degree to the short exact sequence of chain complexes

0 \to C_{*} (L_{1} \cap L_{2}) c \mapsto (c, - c) C_{*} (L_{1}) \oplus C_{*} (L_{2}) (a, b) \mapsto a + b C_{*} (K) \to 0.

(2)

The resulting connecting maps splice the kernels modulo boundaries into the Mayer--Vietoris sequence

\dots \to H_{n} (L_{1} \cap L_{2}) \to H_{n} (L_{1}) \oplus H_{n} (L_{2}) \to H_{n} (K) \partial H_{n - 1} (L_{1} \cap L_{2}) \to \dots .

(3)

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) ≅ 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}

. Then

\partial z_{1} = - \partial z_{2} \in C_{n - 1} (L_{1} \cap L_{2}),

(4)

and the Mayer--Vietoris boundary is

\partial x = [\partial z_{1}] = - [\partial z_{2}] .

(5)

It 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

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 Incoming links: Fundamental class of an orientable simplicial pseudomanifold