relative singular homology and excision theorem

The relative singular homology functor satisfies the Excision Axiom of Homology:

For a topological space and subspaces with (cf. closure, interior) it holds, that

is a quasi-isomorphism (even a chain homotopy equivalence)

Let be the image of the map of chain complexes

induced by on the elementary tensors.

Let furthermore be the the homology

Then

commutes and furthermore, the top left term is given by

By applying levelwise the first isomorphism theorem for groups, this induces an isomorphism on the horizontal cokernels

For any , the composition

is an isomorphism, where the first morphism is given by the isomorphism of the cokernels described above and the second by the 5-Lemma / 5-Lemma for quasi isomorphisms0

welches 5 lemma ** todo:

• represent elements in
• use barycentric subdivision