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Mayer-Vietoris sequence (Definition)

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:

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

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.



"Mayer-Vietoris sequence" is owned by bwebste.
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homology of $\mathbb{RP}^3$. (Example) by mathcam
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Cross-references: homology, level, well defined, class, represents, chain, sum, map, inclusions, induced, homology groups, exact sequence, topological space
There are 6 references to this entry.

This is version 3 of Mayer-Vietoris sequence, born on 2002-12-10, modified 2005-05-09.
Object id is 3724, canonical name is MayerVietorisSequence.
Accessed 6352 times total.

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AMS MSC: 55N10 (Algebraic topology :: Homology and cohomology theories :: Singular theory)
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