Advances in Mathematics
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 H n+2(Λ;Ω nΛ) (n≥1) as obstructions to the existence of successive Postnikov sections for W . (cf. Dwyer et al. (1995) ). The second gives a sequence of geometrically defined higher homotopy operations as the obstructions (cf. Blanc (1995) ); these were identified in Blanc et al. (2010)  with the obstruction theory of Dwyer et al. (1989) . 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. © 2012 Elsevier Inc.
Blanc, David; Johnson, Mark W.; and Turner, James M., "Higher homotopy operations and André-Quillen cohomology" (2012). University Faculty Publications. 267.