Is there a reasonably simple explanation for why Bott periodicity for $U$ and $O$ have periods 2 and 8, respectively? For example, in the $h$cobordism theorem the requirement that $n \geq 5$ has the intuitive explanation that you need enough space in the ambient manifold to use the Whitney trick to push a disk off of itself, which takes ambient dimension at least $2\cdot 2 + 1$. Is there a similar idea or fact that might convey to a (relative) layperson why the numbers 2 and 8 come up in Bott periodicity?

5$\begingroup$ I've always thought about these as coming from the "Bott periodicity" in the classification of real and complex Clifford algebras  which I tend to think as much less surprising as it just corresponds to the fact that $Cl_2(\mathbb{C})$ and $Cl_8(\mathbb{R})$ happen to be isomorphic to matrix algebras (which I'm happy to attribute to chances as there isn't that many option for what these algebras can be)  I'm not sure this answer your question, but maybe you'll want to comment on what sort of thing in that line of thought you would consider an answer to your question. $\endgroup$– Simon HenryCommented Aug 12 at 8:03

$\begingroup$ @Simon: I was under the impression that this does not actually give a proof of Bott periodicity, and one has to use Bott periodicity itself to relate Clifford algebras to Bott periodicity? $\endgroup$– Qiaochu YuanCommented Aug 12 at 18:06

$\begingroup$ @QiaochuYuan I might be wrong on this, but I am fairly sure I have seen proof of Bott periodicity relying on Clifford algebra in the context of the Ktheory of C* algebra (which immediately imply bott periodicity in topology Ktheory, and then the periodicity of these spectrum), but that was a long time ago, so I'd have to do some research to remind myself how this works... $\endgroup$– Simon HenryCommented Aug 12 at 18:43

2$\begingroup$ @SimonHenry AtiyahBottShapiro write in "Clifford Modules" (1963) that "one should look for a proof of the periodicity theorem using Clifford algebras. Since this paper was written a proof on these lines has in fact been found by R. Wood. See also the proof given in: J. Milnor, Morse Theory". Not sure if the proof made it into any of Wood's papers, but Milnor's is in section IV.24 of the Morse theory book. $\endgroup$– user509184Commented Aug 13 at 1:16

$\begingroup$ @SimonHenry Thanks! To be fair my question is somewhat vague; I was mainly wondering if there was some semimagical numerical fact that explains the 8 in particular, and your answer helps a lot. $\endgroup$– Andrew LeeCommented Aug 14 at 14:57
1 Answer
Here is one source of intuition for the 8fold and 2fold Bott periodicities, from the physics of topological states of matter. See this online lecture.
In physics, Bott periodicity governs the classification of the fundamental symmetries of free electrons: the antiunitary symmetries ${\cal T}_\pm$ (timereversal symmetry) and ${\cal P}_\pm$ (particlehole symmetry), and the unitary symmetry ${\cal C}={\cal PT}$ (chiral symmetry). The subscript $\pm$ distinguishes whether the antiunitary symmetry squares to $+1$ or $1$.
There are 8 symmetry classes with at least one antiunitary symmetry: $${\cal P}_+;\;\;{\cal P}_;\;\;{\cal T}_+;\;\;{\cal T}_;$$ $${\cal P}_+,{\cal T}_+;\;\;{\cal P}_+,{\cal T}_;\;\;{\cal P}_,{\cal T}_+;\;\;{\cal P}_,{\cal T}_;$$
There are 2 symmetry classes with no antiunitary symmetries, one with and one without chiral symmetry: $${\cal C};\;\;\times;$$
A gapped system on a $d$dimensional torus can be raised to dimension $d+1$ by adding one momentum component, in such a way that the gap does not close on the torus. This operation transforms the symmetry class in a way that reflects the Bott periodicity, as indicated in the diagram below (the "Bott clock").
The arrows indicate the change of symmetry if dimension $d$ is raised to dimension $d+1$. Without any antiunitary symmetry the cycle has period 2, with at least one antiunitary symmetry the cycle has period 8.

$\begingroup$ Thanks for the answer! I'm not a physicist but it's very interesting that there's a physical interpretation of Bott periodicity. $\endgroup$ Commented Aug 14 at 15:00