David Carchedi (George Mason)

Title:

Haefliger structures, orbifolds, topoi, and classifying spaces

Abstract:

In 1958, Haefliger introduced the notion of a Γ-structure, where Γ is a groupoid of germs generated by a pseudogroup of diffeomorphisms. In particular, there is his famous groupoid Γ^q for which Γ-structures correspond to codimension q-foliations, and many variants which classify foliations with various transverse geometric structures. We will explain a natural one-to-one correspondence between geometric structures on R^q and q-dimensional orbifolds which provides a beautiful interpretation of Haefliger’s groupoids. We will then show how one can use higher category theory to prove Segal’s theorem identifying the homotopy type of the classifying space B Γ^q with the classifying space of the monoid of self-embeddings of R^q, and to derived many variants thereof.