I am currently reading Bott's The Stable Homotopy of the Classical Groups (1959), which was his original proof towards Bott periodicity. As a consequence of his proof, the stable homotopy group of classical matrix Lie groups including the unitary group, the orthogonal group, and the sympletic group have periodicity going like:
Meanwhile, in $K$-theory people also call the periodicity of Grothendieck ring as Bott periodicity. More significantly, take the topological space $X$ to be a point, then the reduced $K$-ring of $X$ satisfies the periodicity relation:
which coincide precisely with the periodicity of the stable homotopy groups of the classical groups (this table was cropped from J.F. Adams' Vector Fields on Spheres). I searched hard on internet but could hardly find an explanation that is illuminating.
Can anyone give some insight on why the two periodicities coincide and suggest some reference for me regarding this topic?
Thanks in advance for answering.
This is an elaboration of my comment above. Let $G = U$ or $G = O$, depending on whether you're interested in the complex case or the real case.
A quick reference for this fact can be found in May's Concise, chapter 24, where this is essentially a definition of $K$-theory. You may have another definition in mind, but most textbook accounts of $K$-theory will describe this link between $K$-theory and classifying spaces (or perhaps equivalently, Grassmannians).
Taking $X$ to be a point $P$, we get $$K^{-q}(P) \cong \pi_q(\mathbb{Z} \times BG)$$ for $q \geq 0$. For $q \geq 1$, this reduces to $K^{-q}(P) \cong \pi_q(BG)$.
By construction/definition/theorem/etc., $BG$ is the classifying space for $G$-bundles, and comes equipped with a universal principal $G$-bundle $G \to EG \to BG$. The total space $EG$ is contractible, so by the long exact sequence in homotopy applied to this fibration, we get $\pi_q(BG) \cong \pi_{q-1}(G)$.
Putting this altogether, we get $$K^{-q}(P) \cong \pi_{q-1}(G)$$ for $q \geq 1$. Note that this argument does not really use Bott periodicity.
There is another more "concrete" way to think about this connection between the $K$-theory of vector bundles and the homotopy groups of Lie groups. The $K$-theory group $K^{-q}(P)$ is the Grothendieck group of vector bundles on the sphere $S^q$. Such vector bundles can be classified in terms of the clutching construction: if we think of $S^q$ as the union of its upper and lower hemispheres, we can trivialize the bundle on these contractible hemispheres, and the extra information needed to reconstruct the vector bundle is how these trivial bundles are glued at the equator $S^{q-1}$, which is given by a map $S^{q-1}$ into the transition group $G$. This gives another way to view the relationship between $K^{-q}(P)$ and $\pi_{q-1}(G)$.