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Richard Hepworth: String Topology of Classifying Spaces

Let G be a compact Lie group. Form the classifying space BG. Then form the space LBG of all loops in the classifying space (the strings). Finally, take the homology H_*(LBG). String topology of classifying spaces asks:

What is the structure of H_*(LBG)?

The original answer, due to Chataur and Menichi, is that it is part of a "homological conformal field theory", which is an algebraic structure governed by surfaces and their diffeomorphisms. A more recent answer, due to Hepworth and Lahtinen, is that it is part of an "h-graph field theory" where the surfaces and diffeomorphisms are replaced by much looser homotopy-theoretical versions. With this in hand, it's natural to ask:

What is the use of such a field theory structure?

Lahtinen has used it to construct (many) non-zero classes in the homology of groups such as the "holomorph" Aut(F_n)-semidirect-F_n and the affine linear groups Z^n-semidirect-GL_n(Z). These groups are well understood in a certain "stable range", but the classes constructed all lie in the mysterious unstable part.

The first two lectures will discuss the h-graph field theory structure: what such a theory is, how to construct the one that appears in string topology, and how to do computations. The third lecture will then explain how to use the theory to construct nonzero classes in the homology of the holomorph and affine linear group.

Some references (the first should be particularly accessible):