Fiber Bundles in Analytic, Zariski, and Etale Topologies

Abstract

We compare the behavior of fiber bundles with structure group 𝐺 for 𝐺 the general linear group GL, the projective general linear group PGL, and the general affine group GA. We prove a criterion for when a GA-bundle is in fact a GL-bundle, i.e., a vector bundle, over the Riemann sphere ℙ1. We also discuss the behavior of fiber bundles under different topologies; specifically, we compare the analytic versus Zariski topologies over the Riemann sphere ℙ1,and compare the étale and Zariski topologies on schemes in general.