long exact sequence (of homology groups)

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:

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

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 $(X,A)$:

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

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

 $\begin{CD}0@>{}>{}>C_{*}(A,B)@>{i_{\sharp}}>{}>C_{*}(X,B)@>{j_{\sharp}}>{}>C_{% *}(X,A)@>{}>{}>0\end{CD}$

where $i_{\sharp}$ and $j_{\sharp}$ are the maps on chains induced by $i$ and $j$, by the Snake Lemma.

Title long exact sequence (of homology groups) LongExactSequenceofHomologyGroups 2013-03-22 13:14:50 2013-03-22 13:14:50 mathcam (2727) mathcam (2727) 8 mathcam (2727) Definition msc 55N10 NChain ProofOfSnakeLemma