Tobias Lewerissa: Classifying spaces for fiber bundles

Abstract: A fiber bundle over a space \(X\) is a map \(E \to X\) that locally looks like a projection \(X \times F \to X\). On the overlap of any two trivial pieces, the "twisting" of the bundle is measured by transition maps \(F \to F\). If these all come from the action of a chosen group \(G\) on \(F\), the bundle is said to have structure group \(G\). In this talk, I will present the idea that such bundles are the same thing as homotopy classes of maps from \(X\) to a classifying space \(BG\), which depends only on the group \(G\). These spaces provide a new perspective on for example orientations and spin structures on manifolds, as well as characteristic classes of vector bundles.