The Bott periodicity theorem can be formulated as the existence of homotopy equivalences $\Omega^2(KU)\equiv KU$ and $\Omega^8(KO)=KO$. I always wondered whether this theorem could also be transferred to the other completions of $Q$, i. e. the $p$-adic fields. Of course, since the topology of $\mathbb Q^p$ behaves rather incompatibly with the topology of, say, $CW$ complexes, it's not really clear how to make sense of that. (The classifying space of the topological group should have the same homotopy groups as the classifying space constructed from the underlying discrete group, if I'm not mistaken). Therefore a more sophisticated approach (possibly via $p$-adic rigid geometry) might be necessary. So, does there exist anything which could be considered as the $p$-adic analogue of Bott periodicity?


As I understood it, the underlying algebraic reason for the lengths of the period in the real and complex case, respectively) is that the sequence of Clifford algebras $Cl(k,n)$ exhibits the same periodicity behaviour up to Morita equivalence (precisely, $Cl(k+8,\mathbb R )=M(Cl(k,\mathbb R),8)$ and $Cl(k+2,\mathbb R )=M(Cl(k,\mathbb R),2)$. A quick check [in the literature](http://archive.numdam.org/ARCHIVE/AMBP/AMBP_1994__1_1/AMBP_1994__1_1_85_0/AMBP_1994__1_1_85_0.pdf) shows that the clifford algebras over $\mathbb Q\_p$ are also periodic of length $2$ if $p\equiv 3 \mod 4$, $4$ if $p \equiv 1 \mod 4$ and $8$ for $p=2$, so at least at this algebraic level, Bott periodicity is present.



$p$-adic Bott periodicity?
The $p$-completed algebraic $K$-theory of the algebraic closure of $\mathbb{Q}\_p$, i.e., $K(\bar{\mathbb{Q}}\_p; \mathbb{Z}\_p)$, is equivalent to its second loop space, up to an issue about path components. This is due to Suslin. The descent to $\mathbb{Q}\_p$ is more subtle than the descent from $\mathbb{C}$ to $\mathbb{R}$, because the absolute Galois group of $\mathbb{Q}\_p$ is much more complicated than that of $\mathbb{R}$. Still, if you reduce to homotopy with $\mathbb{Z}/p$ coefficients, $K(\mathbb{Q}\_p; \mathbb{Z}/p)$ is equivalent to its $(2p-2)$-fold loop space, up to the same issue as before. Boekstedt and Madsen proved this using topological cyclic homology. I did the case $p=2$. Later it followed from the proof of the Lichtenbaum-Quillen conjectures by Voevodsky and others.


