We know that topological K-theory is a generalized cohomology theory, and reduced K-theory is a reduced cohomology theory. Thus, both are representable with a sequence of pointed homotopy functors, via Brown's Representability Theorem. In particular, i found the following statement;
Let $X$ be a compact, Hausdorff topological space. Then, $$\tilde{K}(S^nX)=[X,U]_*$$ where $\tilde{K}$ is the reduced K-theory functor, $U$ is the direct limit of the unitary groups, and $S^nX$ denotes the (reduced) suspension of $X$.
I have a couple of questions:
- Why does not the functor on the right depend from $n$? I doubt that suspending a space as many times as I want I still get the same thing.
- Is this fact proved anywhere? I can't seem to find anything on the matter.
This proposition is instrumental in getting from the K-theoretic form of Bott's Periodicity Theorem to the topological/homotopical one, and I need it for a seminar I'm giving.
Thanks to everybody who will put some time and effort into answering me.
EDIT: In D.Husemoller's "Fibre Bundles", p.34, I found this:
There is an isomorphism of functors between $Vect_k(-)$ and $[-,G_k(\mathbb{C}^\infty)]$; here, $Vect_k(-)$ is the isomorphism classes of (in my case, complex) vector bundles, and $G_k(\mathbb{C}^\infty)$ is the Grassmann manifold of $k$-subspaces of the direct limit $\mathbb{C}^n\subseteq \mathbb{C}^{n+1}$.
Is there some way I can get to what I want from here, maybe using the group-Grassmann-Stiefel fibre bundle, $$U\rightarrow G_k(\mathbb{C}^\infty)\rightarrow V_k(\mathbb{C}^\infty)$$ or some variation thereof?