Manuel Rivera (Purdue University), An algebraic model for the free loop space.

Abstract:

I will describe a functorial algebraic construction that models the passage from a topological space to its free loop space, without imposing any restrictions on the fundamental group of the underlying space.

The input of the construction is a curved coalgebra with certain extra structure reminiscent of the linear dual of a B-infinity algebra. The construction to be discussed is a modified version of the coHochschild complex which takes this coalgebraic structure as input. This builds upon a framework for categorical Koszul duality recently proposed by Holstein and Lazarev. When the construction is applied to the coalgebra of chains, suitably interpreted, of an arbitrary simplicial set X one obtains a chain complex that is quasi-isomorphic to the chains on the free loop space of the geometric realization of X. This extends classical results regarding models for the free loop space of a simply connected space in terms of Hochschild homology.