For which values of $k$ is there an $X$ with $\Omega^kX \cong X$?

Question

Bott periodicity can be formulated as $\Omega^2 U \cong U$ where $\Omega$ denotes the based loop space functor and $U$ is the direct limit of unitary groups. The real version can be formulated as $\Omega^8 O \cong O$ where $O$ is the direct limit of orthogonal groups.

Are there other spaces which are homotopy equivalent to an iterated loop space of themselves? For which values of $k$ is there an $X$ such that $\Omega^kX \cong X$ (but $\Omega^iX \not\cong X$ for $1 \leq i < k$)?

Answer 1

Sure, it is easy to construct such spaces. For instance, let $G$ be any nontrivial abelian group and let $X=\prod_{n=0}^\infty K(G,nk)$. More generally, given a spectrum $Y$, you could define $X=\prod_{n\in\mathbb{Z}} \Omega^{\infty-nk}Y$ (though for general $Y$ that $X$ might by chance have a smaller period).

Answer 2

There are always such spaces, for silly reasons: you can take $X$ to be a periodic product of Eilenberg-MacLane spaces.

All of the natural examples I know have even period, though.