**Hatcher Exercise 2.2.31**

**Theorem**:
Let be the suspension of .
Then, for all .

*Proof*:
Let with and both identified to (distinct) points.
Let and (any overlap will do).
Then, is certainly .
and are both contractible, having for all , and deformation retracts onto , so .
Meyer-Vietoris gives:

for all , and the statement is immediate.