Exact sequence (Quotient space)
If is a space and is a nonempty closed subspace that is a deformation retract of some neighborhood in , then there is an exact sequence
where is the inclusion and is the quotient map .
Reduced homology of spheres (Proof)
and for .
For take so that . The terms in the long exact sequence are zero since is contractible.
Exactness of the sequence then implies that the maps are isomorphisms for and that . Starting with , for , the result follows by induction on .