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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 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.

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    $\begingroup$ Your statements of Bott periodicity are slightly off; you should replace $BU$ and $BO$ with $KU = \mathbb{Z} \times BU$ and $KO = \mathbb{Z} \times BO$ respectively. These are the complex and real K-theory spectra, and so one possible $p$-adic replacement for them is the algebraic K-theory spectrum of $\mathbb{Q}_p$ (which completely ignores the $p$-adic topology). These are used by Dustin Clausen (arxiv.org/abs/1110.5851), for example, to give a $p$-adic J-homomorphism. $\endgroup$
    – Qiaochu Yuan
    Commented May 6, 2016 at 16:02
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    $\begingroup$ Also, I think the $\Omega_8$ statement is much cooler as two $\Omega_4$ statements, going between $KO$ and $KSp$. $\endgroup$
    – Allen Knutson
    Commented May 7, 2016 at 3:37
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    $\begingroup$ People generally call it the Friedlander-Milnor conjecture to distinguish it from other conjectures of Milnor. It is only for homology with finite coefficients and is false for rational coefficients. Suslin proved the relevant cases $G=U(\infty),O(\infty),Sp(\infty)$ using his rigidity theorem. $\endgroup$
    – Ben Wieland
    Commented May 8, 2016 at 14:49
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    $\begingroup$ @QiaochuYuan: it should be noted that the algebraic K-theory spectrum of $\mathbb{Q}_p$ is not periodic which follows from the localization sequence and the finite field case. $\endgroup$
    – Matthias Wendt
    Commented May 10, 2016 at 9:26
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    $\begingroup$ @user51223: the Friedlander-Milnor conjecture is a statement about algebraically closed fields (and of course finite coefficients as pointed out by Ben Wieland) or $\mathbb{R}$. For $\mathbb{R}$ and $\mathbb{C}$, the map $BG^\delta\to BG$ fails to be a homotopy equivalence because the fundamental groups are non-isomorphic. $\endgroup$
    – Matthias Wendt
    Commented May 10, 2016 at 9:31

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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.

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