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:

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

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:

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