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long exact sequence (of homology groups) (Definition)

If $ X$ is a topological space, and $ A$ and $ B$ are subspaces with $ X\supset A\supset B$, then there is a long exact sequence:

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

where $ i_*$ is induced by the inclusion $ i:(A,B)\hookrightarrow(X,B)$, $ j_*$ by the inclusion $ j:(X,B)\hookrightarrow(X,A)$, and $ \partial$ is the following map: given $ a\in H_n(X,A)$, choose a chain representing it. $ \partial a$ is an $ (n-1)$-chain of $ A$, so it represents an element of $ H_{n-1}(A,B)$. This is $ \partial_*a$.

When $ B$ is the empty set, we get the long exact sequence of the pair $ (X,A)$:

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

The existence of this long exact sequence follows from the short exact sequence

$\displaystyle \begin{CD} 0@>>>C_*(A,B)@>i_\sharp>> C_*(X,B) @>j_\sharp>> C_*(X,A)@>>>0 \end{CD}$
where $ i_\sharp$ and $ j_\sharp$ are the maps on chains induced by $ i$ and $ j$, by the Snake Lemma.



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See Also: $n$-chain, proof of snake lemma
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Cross-references: snake lemma, short exact sequence, empty set, represents, chain, map, inclusion, induced, exact sequence, subspaces, topological space
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This is version 5 of long exact sequence (of homology groups), born on 2002-12-10, modified 2003-07-18.
Object id is 3723, canonical name is LongExactSequenceOfHomologyGroups.
Accessed 3435 times total.

Classification:
AMS MSC: 55N10 (Algebraic topology :: Homology and cohomology theories :: Singular theory)
Pending Errata and Addenda
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