Mayer-Vietoris sequence

Let $X$ is a topological space, and $A,B\subset X$ are such that $X=\mathrm{int}(A)\cup \mathrm{int}(B)$, and $C=A\cap B$. Then there is an exact sequence of homology groups:

$$\begin{array}{ccccccccccc}\hfill \mathrm{\cdots }\hfill & \hfill \to \hfill & \hfill H_n(C)\hfill & \hfill \stackrel{i_{*}\oplus -j_{*}}{\to }\hfill & \hfill H_n(A)\oplus H_n(B)\hfill & \hfill \stackrel{j_{*}+i_{*}}{\to }\hfill & \hfill H_n(X)\hfill & \hfill \stackrel{{\partial }_{*}}{\to }\hfill & \hfill H_{n-1}(C)\hfill & \hfill \to \hfill & \hfill \mathrm{\cdots }\hfill \end{array}$$

Here, $i_{*}$ is induced by the inclusions $i:B\hookrightarrow X$ and $j_{*}$ by $j:A\hookrightarrow X$, and ${\partial }_{*}$ is the following map: if $x$ is in $H_n(X)$, then it can be written as the sum of a chain in $A$ and one in $B$, $x=a+b$. $\partial a=-\partial b$, since $\partial x=0$. Thus, $\partial a$ is a chain in $C$, and so represents a class in $H_{n-1}(C)$. This is ${\partial }_{*}x$. One can easily check (by standard diagram chasing) that this map is well defined on the level of homology.

Title	Mayer-Vietoris sequence
Canonical name	MayerVietorisSequence
Date of creation	2013-03-22 13:14:52
Last modified on	2013-03-22 13:14:52
Owner	bwebste (988)
Last modified by	bwebste (988)
Numerical id	6
Author	bwebste (988)
Entry type	Definition
Classification	msc 55N10