In simplicial homology theory, the only homology theory I know so far, the Mayer-Vietoris Theorem says the following:
Let $X$ be a simplicial complex, let $X_1$ and $X_2$ be subcomplexes such that $X_1 \cup X_2 = X$. Then there exists a long exact sequence of homology groups:
$$\dots \to H_{n+1}(X) \to H_n(X_1 \cap X_2) \to H_n(X_1) \oplus H_n(X_2) \to H_n(X) \to \dots$$
I would like to know whether this can be generalized to a complex divided into more than two subcomplexes:
Let $X = \bigcup X_i$. Does there exist anything like the following exact sequence?
$$\dots \to H_{n+1}(X) \to H_n(\bigcap X_i) \to \bigoplus H_n(X_i) \to H_n(X) \to \dots$$