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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 $1 \leq i < k$)?

Michael Albanese
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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.

Qiaochu Yuan
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  • Do you know of any 'natural examples' other than the ones I mentioned in the question? – Michael Albanese Mar 12 '16 at 04:37
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    @Michael: there's a $4$-periodic spectrum called the L-theory spectrum that figures in surgery theory. There are also examples with periods like $2, 12, 24, 576$ coming from elliptic cohomology and related ideas. – Qiaochu Yuan Mar 12 '16 at 13:37
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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).

Eric Wofsey
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    What do you mean by $\Omega^{\infty - nk}Y$? – Michael Albanese Mar 12 '16 at 04:37
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    That refers the $(nk)$th space of the $\Omega$-spectrum. That is, if a spectrum $Y$ is a sequence of spaces $(Y_n){n\in\mathbb{Z}}$ together with equivalences $Y_n\to\Omega Y{n+1}$, $\Omega^{\infty-nk}Y$ refers to $ Y_{nk}$. – Eric Wofsey Mar 12 '16 at 04:55