I'm reading Spanier, and I have understood that loop spaces are examples of H - groups.
However, I'm unable to explicitly work out an example of a non-trivial loop space.
My question: Give an example of a familiar topological space that is a loop space.
Every topological group $G$ is homotopy equivalent to a (based) loop space, namely the based loop space of its classifying space $BG$. So, for example, $S^1$ is homotopy equivalent to the based loop space of $BS^1 \cong \mathbb{CP}^{\infty}$. Whether that counts as "familiar" enough to you I don't know, because the price we pay for $G$ being "familiar" is that $BG$ is usually not "familiar." There are also sillier examples like the loop space $\Omega S^1$ being homotopy equivalent to $\mathbb{Z}$ which I don't think is very enlightening.
It's difficult to give a satisfying example here, where $X$ and $\Omega X$ are both "familiar" without being too trivial, because there is a kind of "uncertainty principle" guaranteeing that $X$ and $\Omega X$ cannot simultaneously be too simple. This can be made precise using the Serre spectral sequence to show that under some mild nontriviality hypotheses (I think it suffices to require that $\Omega X$ is simply connected but not weakly contractible, or something like that), either $X$ or $\Omega X$ or both must have nontrivial homology in arbitrarily high degrees (for example, all the loop spaces of the spheres $\Omega S^n, n \ge 2$ have homology in arbitrarily high degrees), and in particular they can't both be homotopy equivalent to manifolds, for example.