**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 .