Higher homotopy operations and André–Quillen cohomology

There are two main approaches to the problem of realizing a Π-algebra (a graded group Λ equipped with an action of the primary homotopy operations) as the homotopy groups of a space X. Both involve trying to realize an algebraic free simplicial resolution G• of Λ by a simplicial space W•, and proceed by induction on the simplicial dimension. The first provides a sequence of André–Quillen cohomology classes in Hn+2(Λ;ΩnΛ) (n⩾1) as obstructions to the existence of successive Postnikov sections for W• (cf. Dwyer et al. (1995) [27], ). The second gives a sequence of geometrically defined higher homotopy operations as the obstructions (cf. Blanc (1995) [8], ); these were identified in Blanc et al. (2010) [16], with the obstruction theory of Dwyer et al. (1989) [25]. There are also (algebraic and geometric) obstructions for distinguishing between different realizations of Λ.In this paper we(a)provide an explicit construction of the cocycles representing the cohomology obstructions;(b)provide a similar explicit construction of certain minimal values of the higher homotopy operations (which reduce to “long Toda brackets”); and(c)show that these two constructions correspond under an evident map.