The loop space of \(BU\) is the stable unitary group \(U = \mathop{\mathrm{colim}}_n U(n)\text{.}\) This is the stable analogue of \(\Omega BG \simeq G\) for a topological group \(G\text{,}\) and the real analogue is \(\Omega BO \simeq O = \mathop{\mathrm{colim}}_n O(n)\text{.}\) Computing \(\widetilde{K}\) of spheres or suspensions thus amounts to computing homotopy groups of \(U\text{.}\) The groups \(U(n)\) themselves have complicated homotopy groups, but in a fixed degree these stabilise as \(n\) grows.
arising from the transitive action of \(U(n+1)\) on \(S^{2n+1} \subset \mathbb{C}^{n+1}\) with stabiliser \(U(n)\text{.}\) Since \(\pi_i(S^{2n+1}) = 0\) for \(i \leq 2n\text{,}\) its long exact sequence shows that \(\pi_k(U(n)) \to \pi_k(U(n+1))\) is an isomorphism for \(k < 2n\text{.}\) As spheres are compact, \(\pi_k(U) = \mathop{\mathrm{colim}}_n \pi_k(U(n))\text{,}\) so \(\pi_k(U) \cong \pi_k(U(n))\) as soon as \(k < 2n\text{.}\) For example, \(\pi_3(U) \cong \pi_3(U(2)) \cong \pi_3(SU(2)) \cong \pi_3(S^3) \cong \mathbb{Z}\text{.}\)
\begin{equation*}
\mathbb{Z} \times BU \;\simeq\; \Omega^2(\mathbb{Z} \times BU).
\end{equation*}
In particular, since \(\Omega(\mathbb{Z} \times BU) \simeq \Omega BU \simeq U\text{,}\) the homotopy groups of the stable unitary group \(U\) are \(2\)-periodic:
\begin{equation*}
\pi_n(U) \;\cong\; \begin{cases} \mathbb{Z} & n \text{ odd}, \\ 0 & n \text{ even}. \end{cases}
\end{equation*}
There is a homotopy equivalence \(\mathbb{Z} \times BO \simeq \Omega^8(\mathbb{Z} \times BO)\text{.}\) In particular, the homotopy groups of the stable orthogonal group \(O\) are \(8\)-periodic:
The proofs are beyond the scope of this course. Bott’s original argument uses Morse theory on spaces of paths in unitary groups. Atiyah and Bott later gave an elementary proof of the complex case using clutching functions, which is the one presented in Hatcher, Vector Bundles and K-Theory. The hard step is the fundamental product theorem (Theorem 2.2, proved in §2.1), and periodicity follows formally in §2.2 (Theorem 2.11). A third proof, due to Atiyah, uses the index of families of elliptic operators.
