2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62223We describe some buildings related to complex Kac-Moody groups. First we describe the spherical building of SLn(C) (i.e. the projective geometry PG(Cn)) and its Veronese representation. Next we recall the construction of the affine building associated to a discrete valuation on the rational function field $C(z)$. 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 introduce 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.Dedicated to John Stallings on the occasion of his 65th birthday. To appear in Geometriae DedicataGeometric TopologyAlgebraic Topology51E24; 51H15; 22E67; 53C42text