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One can naturally associate to any pointed topological space a topological monoid of continuous loops that start and end at base point with multiplication given by concatenation of loops. Every element in this topological monoid has an inverse up to homotopy, so it is “almost” a topological group. This passage does not loose any homotopical information and it is often useful to recast the homotopy theory of spaces in terms of topological monoids/groups. In algebraic tooology one studies spaces by means of algebraic invariants. For instance, one can replace a topological space by a chain complex with additional structure- more precisely a coalgebra structure - through the classical singular chains construction. In this series of talks, I will describe how to model algebraically the passage from a pointed space to its topological monoid of based loops as a natural construction -called the “cobar functor” and originally due to F. Adams- that takes a differential graded (dg) coalgebra and produces a dg algebra. When applied to the singular chains on a pointed space, the cobar functor produces a model for the chains on the based loop space. A slightly new perspective on this construction will allow us to generalize many existing results in the literature from simply connected spaces to spaces with arbitrary fundamental group. We will discuss several applications and consequences of this new perspective such as obtaining algebraic models for non-simply connected spaces (extending the work of Sullivan, Quillen, Goerss, Mandell, and others) and models for the free loop space functor suitable for studying string topology in the non-simply connected setting. |