Generating varieties, Bott periodicity and instantons

Let G be the classical group and let M k ( G ) be the based moduli space of G-instantons on S 4 with instanton number k. It is known that M k ( G ) yields real and symplectic Bott periodicity, however an explicit geometric description of the homotopy equivalence has not been known. We consider certain orbit spaces in M k ( G ) and show that the restriction of the inclusion of M k ( G ) into the moduli space of connections, which, in turn, is explicitly described by the commutator map of G. We prove this restriction satisfies a triple loop space version of the generating variety argument of Bott (1958) [5], and it also gives real and symplectic Bott periodicity. This also gives a new proof of real and symplectic Bott periodicity.