On Adams cobar construction for non-simply connected spaces
We will describe a generalization of a celebrated classical result of Frank Adams, where he shows an algebraic construction which is defined for any differential coalgebra [connected in degree zero] and produces a differential algebra [not necessarily connected in degree zero] models the passage which starts from the chains on a connected space which form a coassociative differential coalgebra and goes to the chains on the model of the based loop space, the latter being a differential graded associative algebra using the loop space multiplication. It was important for Adams that the space in question be simply connected so that the based loop space would be connected as well.
We will show, for any pointed, connected topological space (X, b), not necessarily simply connected, this same algebraic construction, which Adams termed the cobar construction, applied to the differential graded coalgebra of normalized singular chains in X with vertices at b is homologically equivalent via (dga) maps to the differential graded associative algebra of singular chains on the Moore based loop space of X at b.
In this statement we may replace singular chains in X with a homologically equivalent combinatorial model such as the combinatorial object constructed by Dan Kan. We will deduce this result from more general categorical results which are of independent interest. This is joint work with Manuel Rivera.