Simple Sphere Suspension Question

Question

I have heard it said that the suspension of an $n$-sphere is an $n+1$-sphere. This is stated without proof in chapter 0 of Hatcher's book on algebraic topology. More generally, it seems that the suspension of a simplex or cube is a simplex or cube (respectively) of one higher dimension. I can convince myself of this when the objects are of low enough dimension to visualize, but I can not see a way to prove this rigorously. How does one prove it?

Answer 1

The following is the way that I've found most natural, and easiest to think about.

First prove the following lemma (which is a good reason to think about smash products, for example!)

Lemma. Let $V$ and $W$ be vector spaces. Then there is a natural homeomorphism $S^V \wedge S^W \cong S^{V\oplus W}$, where $S^V$ denotes the one-point compactification of the vector space.

The result follows trivially, then, as soon as you prove

Lemma. The suspension, $\Sigma X$, of a pointed space is the same as $S^1 \wedge X$.

Happy learning!