Question

Whats is the proof for the long exact sequence in cohomology?

Answer 1

Let (X, Y ) be a pair of spaces and let A be an abelian group. Then there is a
natural long exact sequence
0 → H0(X, Y ; A) → H0(X; A) → H0(Y ; A)→ H1(X, Y ; A) → · · · .

This sequence is called the long exact cohomology sequence of the pair (X, Y ).
Proof. By definition of the relative singular chain complex C∗(X, Y ) we have a short exact sequence
0 → C∗(Y ) → C∗(X) → C∗(X, Y ) → 0
which is levelwise split. Thus, Exercise 12 tells us that HomZ(−, A) induces a levelwise split short
exact sequence of cochain complexes
0 → C∗(X, Y ; A) → C∗(X; A) → C∗(Y ; A) → 0.
To conclude the proof it suffices to consider the associated long exact sequence in cohomology.As a next step we will show that singular cohomology is homotopy invariant. As in the case of
singular homology with coefficients, there is an ‘algebraic version’ of homotopies which will allow
us to establish the result. We ask the reader to introduce the notion of a cochain homotopy of maps
of cochain complexes which again consists of a collection of maps ‘against the differential’.