Why is the infinite sphere contractible?

Question

Why is the infinite sphere contractible?

I know a proof from Hatcher p. 88, but I don't understand how this is possible. I really understand the statement and the proof, but in my imagination this is weird.

Thanks for your help.

Answer 1

I think one way of thinking about this is from the point of view of homotopy groups. The sphere $\mathbb{S}^n$ is $n$-connected, i.e., the connectivity of the $n$-sphere increases as $n$ increases. Of course, by definition $\mathbb{S}^{\infty}=\bigcup_{n=1}^{\infty} \mathbb{S}^n$ (more precisely, the direct limit should replace the union but you understand what I mean) so all homotopy groups of $\mathbb{S}^{\infty}$ vanish. Therefore, $\mathbb{S}^{\infty}$ is weakly contractible but by Whitehead's theorem, this is equivalent to contractibility.

I'd just like to emphasise the key point that high dimensional spheres are extremely connected and so you'd naturally expect that $\mathbb{S}^{\infty}$, the "highest dimension" sphere of them all, should be contractible!

I hope this helps!

Answer 2

I like to think of this in term of classifying spaces (I don't know which proof Hatcher presents). The category with two objects and an isomorphism between them (aka: the free-living isomorphism) is a contractible category, i.e., it is equivalent to a terminal category. Now, the classifying space of the free-living iso is easily seen to be the infinite dimensional sphere. So now the categorical equivalence between the free-living iso and a terminal category means the infinite dimension sphere is homotopic to the classifying space of a terminal category, namely a point.