# 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:

 $\begin{CD}\cdots @>{}>{}>H_{n}(C)@>{i_{*}\oplus-j_{*}}>{}>H_{n}(A)\oplus H_{n}% (B)@>{j_{*}+i_{*}}>{}>H_{n}(X)@>{\partial_{*}}>{}>H_{n-1}(C)@>{}>{}>\cdots\end% {CD}$

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 MayerVietorisSequence 2013-03-22 13:14:52 2013-03-22 13:14:52 bwebste (988) bwebste (988) 6 bwebste (988) Definition msc 55N10