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):