On Homotopy Automorphisms of Koszul Spaces

PHD defense: Casper Guldberg

Abstract:

In this thesis we study the rational homotopy theory of the spaces of self-equivalences of Koszul spaces - that is, of simply connected spaces which are simultaneously formal and coformal in the language of rational homotopy theory. The primary tool to do so is the Homotopy Transfer Theorem for L1-algebras. We begin with a Lie model for the universal cover of B autX where X is a Koszul space, and construct a well-behaved contraction to a smaller chain complex using relations between the cohomology algebra and homotopy Lie algebra of a Koszul space. Then we study the transferred structure which retains all information about the rational homotopy type, and derive several structural properties. We establish criteria for coformality of the universal cover of B autX, improving on existing results, and provide examples: highly connected manifolds and two-stage spaces, among others. Our main example is that of ordered configurations in Rn, for which our model is small enough that we can compute several rational homotopy groups of the universal cover of B autX. Finally we study the group of components _0(autXQ) for a Koszul space X, and establish a sufficient condition for it to be isomorphic to the group of algebra automorphisms of the cohomology algebra of X, or equivalently the Lie algebra automorphisms of the homotopy Lie algebra of X.

Principal supervisor: 

Nathalie Wahl Professor, University of Copenhagen, Denmark

Co-supervisor:

Alexander Berglund Ass.Prof, Stockholms University, Sweden

Assessment Committee:

• Ib Madsen (chairman) Professor, University of Copenhagen, Denmark
• Kathryn Hess Professor, EPFL (Lausane), Swizerland
• Pascal Lambrechts  Professor, Université catholique de Louvain, Belgique