Loop Groups and Twin Buildings Dedicated to John Stallings on the occasion of his 65th birthday

In these notes we describe some buildings related to complex Kac-Moody groups. First we describe the spherical building of SLnðCÞ (i.e. the projective geometry PGðC n Þ) and its Veronese representation. Next we recall the construction of the affine building associated to a discrete valuation on the rational function field CðzÞ. Then we describe the same building in terms of complex Laurent polynomials, and introduce the Veronese representation, which is an equivariant embedding of the building into an affine Kac-Moody algebra. Next, we intro- duce topological twin buildings. These buildings can be used for a proof which is a variant of the proof by Quillen and Mitchell, of Bott periodicity which uses only topological geometry. At the end we indicate very briefly that the whole process works also for affine real almost split Kac-Moody groups.