Mayer-Vietoris homology sequence

This article defines a long exact sequence of homology groups, for topological spaces or pairs of topological spaces

Definition

Suppose $X$ is a topological space, and $U$ and $V$ are subsets of $X$ such that the union of the interiors of $U$ and $V$ cover $X$ . Then we get a long exact sequence of homology:

$\dots \to H_{n} (U \cap V) \to H_{n} (U) \oplus H_{n} (V) \to H_{n} (X) \to H_{n - 1} (U \cap V) \to \dots$

where the maps are as follows. Let $i, j$ be the inclusions from $U \cap V$ to $U$ and $k, l$ be the inclusions from $U, V$ into $X$ .

Then the map from the homology of $U \cap V$ is:

$f \mapsto (H_{n} (i) (f), H_{n} (j) (f))$

And the map from $H_{n} (U) \oplus H_{n} (V)$ is:

$(g, h) \mapsto H_{n} (k) g - H_{n} (l) h$