The cohomology of a manifold classifies geometric structures over it. One instance of this principle is the classification of line bundles via Chern classes. The classifying space BG associated to a (Lie) group G is a simplicial manifold which encodes the group structure. Its cohomology hence classifies geometric objects over G which play well with its multiplication. These are known as characteristic classes, and yield invariants of G-principal bundles.
I will introduce multiplicative gerbes and show how they realise classes in H^4(BG) when G is compact. Along the way, we will meet different versions of Lie group cohomology, smooth 2-groups and a few spectral sequences.
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- Junior Geometry and Topology Seminar