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What is this "long exact sequence in homology" thing?

Suppose we have a short exact sequence of chain complexes (i.e. three chain complexes, say ${A_n}$, ${B_n}$, and ${C_n}$ we say

\begin{displaymath}0 \rightarrow \{A_n\} \rightarrow \{B_n\} \rightarrow \{C_n\} \rightarrow 0 \end{displaymath}

is exact if

\begin{displaymath}0 \rightarrow A_n \rightarrow B_n \rightarrow C_n \rightarrow 0 \end{displaymath}

is exact for each $n$.) In such cases the snake lemma gives us a "boundary" map, connecting $H_n(X)$ to $H_{n-1}(A)$. This map fits into a long exact sequence (i.e. a sequence of arbitrary length that is exact at each position) in the way shown below:

\begin{displaymath}\cdots H_n(A) \rightarrow H_n(X) \rightarrow H_n(X,A) \rightarrow H_{n-1}(A) \rightarrow H_{n-1}(X) \cdots \end{displaymath}

Christine Devena 2003-07-25