Construction of the Connecting Homomorphism

It is easy to generalize the notions of submodule and quotient module to that of general chain complexes. 

Note that the boundary operator is well-defined on the quotient because of 

Clearly, the associated collections of inclusion maps i = in n, in An Bn, and projection mapsp = pn n,Pn Bn — Bn/An both describe chain maps. 

Given a subcomplex A of the chain complex B, there arises a natural question  

What is the relation between the homology groups of A and those of B  

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