Abstract: 

We study the large rank limit of the moduli spaces of framed bundles on the
projective plane and the blown-up projective plane. These moduli spaces are
identified with various instanton moduli spaces on the 4-sphere and $\cpbar
$, the projective plane with the reverse orientation. We show that in the
direct limit topology, these moduli spaces are homotopic to classifying
spaces. For example, the moduli space of $Sp(\infty )$ or $SO(\infty )$
instantons on $\cpbar $ has the homotopy type of $BU(k)$ where $k$ is the
charge of the instantons. We use our results along with Taubes' result
concerning the $k\to \infty $ limit to obtain a novel proof of the homotopy
equivalences in the eight-fold Bott periodicity spectrum. We give explicit
constructions for these moduli spaces.