connecting homomorphism

The connecting homomorphism is the map that should not exist — and yet does. When you break a complex into a sub and a quotient, their homologies are linked by the obvious induced maps, but those alone leave gaps. The connecting homomorphism is the unexpected extra arrow that bridges those gaps, reaching across a degree to tie the homology of the quotient back to the homology of the sub, and thereby closing the chain into one long exact sequence.

It arises from the snake lemma applied to a short exact sequence of chain complexes 0 -> A -> B -> C -> 0. The connecting map ∂ : H_n(C) -> H_{n-1}(A) is defined by a diagram chase: take a cycle c in C_n, lift it to some b in B_n (possible by surjectivity), apply the differential to get d(b) in B_{n-1}, observe d(b) maps to zero in C so it comes from a unique a in A_{n-1}, and set ∂[c] = [a]. One checks a is a cycle and the class is independent of choices.

The defining property is that the resulting sequence ... -> H_n(A) -> H_n(B) -> H_n(C) -> H_{n-1}(A) -> ... is exact at every term — this is the whole reason long exact sequences exist. The connecting map is natural in the short exact sequence, which is what lets the five lemma compare different long exact sequences. In cohomology it raises degree, ∂ : H^n(C) -> H^{n+1}(A), and it underlies the connecting maps of every derived functor.