Manuel Krannich: Top(d) and orthogonal calculus

Abstract: 

A central role in the study of manifolds and their automorphisms is played by the homotopy type of the topological group Top(d) of homeomorphisms of \(\mathbb{R}^d\), which is known to be related to algebraic K- and L-theory and, as discovered more recently, to operads and the study of graph complexes à la Kontsevich. The various pieces of information about Top(d) currently known can be packaged in terms of Weiss’ orthogonal calculus, which is a general machinery to study functors on the category of inner product spaces. This perspective does not only neatly clarify all that is currently known about Top(d), but also suggests a number of tempting tasks for the future study of Top(d). My plan for this talk is to give a gentle introduction to orthogonal calculus with a view towards Top(d), followed by explaining how joint work with O. Randal-Williams fits into the picture.