A long exact sequence is an exact sequence with infinitely many terms, indexed by \(\ZZ\) or by an infinite interval of \(\ZZ\).
Let \(0\to A_\bullet\mapsvia{i}B_\bullet\mapsvia{j}C_\bullet\to0\) be a short exact sequence of chain complexes of abelian groups. There are homomorphisms \(\del\colon H_n(C_\bullet)\to H_{n-1}(A_\bullet)\), the connecting homomorphisms, such that \[ \cdots\to H_n(A_\bullet)\mapsvia{i_*}H_n(B_\bullet)\mapsvia{j_*}H_n(C_\bullet)\mapsvia{\del}H_{n-1}(A_\bullet)\to\cdots \] is exact [@Hat02].
For a topological space \(X\) and a subspace \(A\), the short exact sequence \(0\to C_\bullet(A)\to C_\bullet(X)\to C_\bullet(X,A)\to0\) of singular chain complexes gives the long exact sequence of the pair \[ \cdots\to H_n(A)\to H_n(X)\to H_n(X,A)\mapsvia{\del}H_{n-1}(A)\to\cdots\to H_0(X,A)\to0 \] [@Hat02].