Some Examples of Homotopy ∏-Algebras

The homotopy ∏-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 ∏-algebras and look at some examples illustrating the depth of structure on homotopy groups; from graded group to graded Lie ring, to ∏-algebra and beyond. We also describe an abstract ∏-algebra and give three abstract ∏-algebra structures on the homotopy groups of the loop space of 'X' which can be realized as the homotopy ∏-algebras of three different spaces