A classifying space for principal G-bundles with connection, in the category of simplicial presheaves

Abstract

I denne masteroppgaven studerer vi hoved-G-bunter over glatte mangfoldigheter med kobling, og hvordan man kan lage universelle objekter for å klassifisere dem, tilsvarende Grassmannians for vektorbunter. Konstruksjonene tar oss inn i kategorien av simplisielle preknipper over glatte mangfoldigheter, og resulterer i et teorem som sier at de eneste naturlige differensialformene assosiert med koblingen på hoved-G-bunter er de man konstruerer med Chern-Weil-homomorfien. Dette ble først gjort av Freed og Hopkins i [7]. I siste kapittel tar vi en liten kikk på hva som ville skjedd hvis mangfoldighetene vi jobbet med var komplekse i steden for glatte.

In this master's thesis, we study principal G-bundles on smooth manifolds with connection, and how to make universal objects to classify them, similar to Grassmannians for vector bundles. The constructions take us into the category of simplicial presheaves on smooth manifolds, and result in a theorem that states that the only natural differential forms associated to the connection on principal G-bundles are the ones constructed in the Chern-Weil homomorphism. This was first obtained in this way by Freed and Hopkins in [7]. In the last chapter, we take a short look at what would happen if the manifolds considered were complex instead of smooth.