Mayer-Vietoris homology sequence

Unreduced version
This statement holds, not just for homology with integer coefficients, but for homology with coefficients in any ring or module as long as the same ring or module is used consistently for all spaces.

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 $$X$$ (Note that in particular this condition is satisfied if $$U$$ and $$V$$ are open subsets whose union is $$X$$). 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:

$$(g,h) \mapsto H_n(k)g - H_n(l)h$$

The sequence ends at:

$$\ldots \to H_1(X) \to H_0(U \cap V) \to H_0(U) \oplus H_0(V) \to H_0(X) \to 0$$

Note that by adopting the convention that $$H_k$$s are zero for negative $$k$$ (a convention that can also be viewed as following from the definition) we can consider the sequence as going on forever.

This statement holds, not just for homology with integer coefficients, but for homology with coefficients in any ring or module.

Reduced version
This statement holds, not just for homology with integer coefficients, but for homology with coefficients in any ring or module as long as the same ring or module is used consistently for all spaces.

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 $$X$$ and the intersection of $$U$$ and $$V$$ is nonempty. Then we get a long exact sequence of reduced homology:

$$ \ldots \to H_n(U \cap V) \to H_n(U) \oplus H_n(V) \to H_n(X) \to H_{n-1}(U \cap V) \to \ldots \to H_1(X) \to \tilde{H}_0(U \cap V) \to \tilde{H}_0(U) \oplus \tilde{H}_0(V) \to \tilde{H}_0(X) \to 0$$

Note that for positive dimensions, reduced and unreduced homology coincide, so the sequence looks exactly the same in positive dimension. The difference in dimension zero can be attributed to the removal of a piece that looks like $$0 \to \mathbb{Z} \to \mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z} \to 0$$.

Note that by adopting the convention that $$\tilde{H}_k$$s are zero for negative $$k$$ (a convention that can also be viewed as following from the definition) we can consider the sequence as going on forever.

Formulation involving closed subsets
Suppose $$A$$ and $$B$$ are closed subsets of a topological space $$X$$ such that $$A \cap B$$ is empty and $$X \setminus (A \cup B)$$ is path-connected. Then, we apply the Mayer-Vietoris homology sequence setting $$U = X \setminus A$$ and $$V = X \setminus B$$.

The Seifert-van Kampen theorem
A fact that has a somewhat similar flavor to the Mayer-Vietoris sequence is the Seifert-van Kampen theorem, where we use open subsets $$U$$ and $$V$$ whose union is $$X$$, and whose intersection is a path-connected space. While the Mayer-Vietoris sequence is used to compute homology, the Seifert-van Kampen theorem is used to compute the fundamental group, which is the first homotopy group. These results are often used together, in conjunction with the Hurewicz theorem, to determine whether a topological space is $$n$$-connected for a given positive integer value of $$n$$.

General approach behind applications
The general approach is to select $$U$$ and $$V$$ as open subsets such that $$U,V,U \cap V$$ are all homotopy-equivalent to easier spaces. This is typically done by choosing $$U,V,U \cap V$$ so that they admit strong deformation retractions to the easier spaces. In the case of manifolds, for instance, we may want these to admit strong deformation retractions to manifolds of smaller dimensions.

For singular homology
For singular homology, the Mayer-Vietoris homology sequence can be viewed as the long exact sequence of homology of a short exact sequence of chain complexes, namely:

$$sd^mC_n(U \cap V) \to sd^mC_n(U) \oplus sd^mC_n(V) \to sd^m(C_n(X)$$

where $$sd$$ denotes the barycentric subdivision operator. Since $$sd$$ is homotopic to the identity map, the homologies of this are the homologies of the original chain complexes.

The rough idea is that by subdividing sufficiently, we can make sure that each simplex goes either entirely within $$U$$ or entirely within $$V$$.