Here are some notes on Bott periodicity... taken from Milnor's excellent book Morse Theory.

He starts with O(n): the Lie group of orthogonal transformations of ℝⁿ. He lets Ω₁(𝑛) be the space of minimal geodesics from 1 to -1 in O(n). He then lets Ω₂(𝑛) be a space of minimal geodesics in Ω₁(𝑛), and so on.

By the end, he shows that Ω₈(n) is diffeomorphic to O(n/16), at least when n is a multiple of 16. So he's back where he started... roughly speaking.

(1/n)

First Milnor shows that Ω₁(𝑛) is the same as the space of 'complex structures' on ℝⁿ, meaning linear operators J: ℝⁿ → ℝⁿ with J² = −1. The basic idea is that any complex structure gives you a geodesic

g(t) = exp(Jt)

which goes from 1 when t = 0 to -1 when t = π. (Think rotations in the complex plane.)

(2/n)

Then Milnor lets Ω₂(n) be the space of minimal geodesics in Ω₁(𝑛) going from a chosen complex structure J₁ to -J₁. But he shows that Ω₂(n) is the same as the set of complex structures J₂ on ℝⁿ that anticommute with J₁:

J₁J₂ = -J₂J₁

The key is that

g(t) = J₁ exp(J₂t)

is a complex structure for each t∈ℝ iff J₁J₂ = -J₂J₁. And when this happens, g(t) is a minimal geodesic going from J₁ when t = 0 to -J₁ when t = π!

(3/n)

But a pair of anticommuting complex structures on ℝⁿ is the same as a 'quaternionic structure' on ℝⁿ: that is, a way to make the quaternions act ℍ as operators on ℝⁿ. The reason is that ℍ is the algebra freely generated by i, j such that i² = j² = -1 and ij = -ji.

So, Ω₂(n) is the space of 'quaternionic structures' on ℝⁿ.

Notice that ℝⁿ doesn't have any complex structures unless n is even! And it doesn't have any quaternionic structures unless n is a multiple of 4.

(4/n)

Keep going! Let Ω₃(n) be the space of minimal geodesics in Ω₂(n) going from some quaternionic structure (J₁,J₂) on ℝⁿ to the quaternionic structure (J₁,-J₂).

Surprise: Ω₃(n) is the same as the space of complex structures J₃ that anticommute with both J₁ and J₂!

Now, a triple of anticommuting complex structures on ℝⁿ is the same as an action of the Clifford algebra Cliff₃ on ℝⁿ. So Ω₃(n) is the space of Cliff₃ actions on ℝⁿ extending a chosen quaternionic structure on ℝⁿ.

(6/n)

We can keep doing this until the cows come home. But this will not let us see Bott periodicity.

Won't it? Even if we know the category of representations of Cliff₈ is equivalent to the category of representations of Cliff₀ = ℝ?

Anyway, Milnor goes in another direction, and describes Ω₃(n) in a seemingly different way, at least when n = 4k. It's the same as the space of all quaternionic subspaces of ℍᵏ.

But why? How is this connected to the Clifford algebra picture?

(7/n)

Say n = 4k and choose a quaternionic structure on ℝⁿ, so we can treat it as ℍᵏ.

Then we've seen Ω₃(n) is the space of Cliff₃ actions on ℍᵏ extending its quaternionic structure. But

Cliff₃ ≅ ℍ ⊕ ℍ

so such an action of Cliff₃ picks out a quaternionic subspace of ℍᵏ, namely the range of the idempotent (1,0) ∈ ℍ ⊕ ℍ.

So we can identify Ω₃(n) with the space of quaternionic subspaces of ℍᵏ when n = 4k. Nice!

(8/n)

Milnor points out that Ω₃(n), the space of all quaternionic subspaces of ℍᵏ where n = 4k, is not connected! Subspaces of different dimensions live in different components of Ω₃(n).

Which component should we use in the next step, when we'll define Ω₄(n) to be a space of minimal geodesics in Ω₃(n)? He picks the 'biggest' component, containing subspaces of dimension k/2.

For this k needs to be even. So now n needs to be a multiple of 8.

(9/n)