On the homotopy Pontryagin algebra of a connected space.

We present two chain level models of the Pontryagin algebra $H_*(\Omega|X|)$ of a reduced simplicial set $X$ with the perspective that they should incorporate much information about the homotopy type of $X$. The first is given by the chains on the Kan loop group $GX$, the second is an extended version of Adam's cobar construction on the chains of $X$. This second model owes a lot to Baues' "Geometry of the Cobar Construction". Recent work of Medina, Rivera et al. shows that both models are intimately related, and carry group-like $E_\infty$-bialgebra structures.