Koszul duality for the little disk aperad

Specialeforsvar ved Lasse Bryder Nielsen

Titel: Koszul duality for the little disk operad

Abstract: In this thesis we will work with symmetric operads and their associated Koszul dual. The topological operad of little n-disks $D_n$ originates from May's investigation of iterated loop spaces. Our main interest is in the algebraic operad $H_*(D_n)$ defined as the homology operad of $D_n$. The main goal of the thesis is to prove that the operad $H_*(D_n)$ is Koszul. The first part of the thesis is devoted to introducing the notion of an operad, which is a structure encoding operations on certain objects of the underlying category. We will mostly work in the category of differentially graded vector spaces, however topological operads also play a crucial role. For simplicity we will only work over a ground field of characteristic zero. We make the differentially graded framework explicit in terms of how algebraic operads and their associated algebras are defined. Then we define several important notions such as the bar $B$ and cobar $\Omega$ constructions, twisting morphisms and the twisted composite complex associated to an algebraic operad. We prove the connection between a Koszul twisting morphism and the acyclicity of the twisted composite complexes. In chapter three we consider quadratic operads, which are operads generated by a set of operations satisfying some relations. These are the kind of operads where Koszul duality is defined. We define the Koszul dual cooperad $P^¡$. An operad $P$ is said to be Koszul when there is a quasi-isomorphism from the cobar of $P^¡$ to $P$. This gives a minimal model for the operad. In the last chapter we prove the main result that $H_*(D_n)$ is Koszul, using the proof given by Getzler and Jones in [GJ94]. This proof relies on a compactification of real configuration spaces due to Fulton and MacPherson.

Vejleder: Dan Erik Petersen
Censor:   Iver Mølgaard Ottosen, AAU