Some examples of homotopy Π-algebras

The homotopy Π-algebra of a pointed topological space, X, consists of the homotopy groups of X together with the additional structure of the primary homotopy operations. We extend two well-known results for homotopy groups to homotopy Π-algebras and look at some examples illustrating the depth of structure on homotopy groups; from graded group to graded Lie ring, to Π-algebra and beyond. We also describe an abstract Π-algebra and give three abstract Π-algebra structures on the homotopy groups of the loop space of X which can be realized as the homotopy Π-algebras of three different spaces.