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Ilya Nikokoshev
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One example would be a map induced by a morphism f: X \to Y in the long homology sequence.

E.g. suppose the top row is a cohomology of pair (X, A) and the bottom row is the cohomology of pair (Y, B). Then the theorem says that the H^n(X, A) can be squeezed between the n-th and n-1-th cohomology of X and A, because any morphism inducing isomorphism on those extends to H^n(X, A).

Ilya Nikokoshev
  • 15.1k
  • 12
  • 77
  • 129