Is it true that the loop space of a circle is contractible? Consider the long exact sequence in homotopy for the path fibration $\Omega S^1 \rightarrow \ast \rightarrow S^1$ shows all homotopy groups of the loop space to be zero, and then Whitehead's theorem kicks in and tells us that $\Omega S^1$ is contractible.
I can't see any flaw in this argument, but it's really rankling against my intuition of what a loop space "is". For instance, we know that $S^1$ has $\mathbb{Z}$ many loops up to homotopy, so surely it should have a fairly complicated loop space? So I suppose that there are two possible approaches to answering this question: either point out where I'm going wrong in my mathematics, or help rectify my intuition.
Secondly, does this generalise to the $n$-fold loop space of an $n$-sphere? I think it does; I've used the Serre spectral sequence to compute the cohomology rings of loop spaces of spheres and they all die at $\Omega^n S^n$.
Sorry for the "please check my working" nature of the question; it's sufficiently early in my mathematical life that I'm never convinced that there's no errors in my working (especially not with spectral sequences, which still feel a bit like doing magic to me). It was a lot easier to be (mathematically) confident as an undergraduate, when you could check your working against solutions sheets...