Higher homotopy operations and Andr\'{e}-Quillen cohomology

TL;DR

This paper explores two approaches to realizing a Pi-algebra as the homotopy groups of a space, connecting algebraic cohomology obstructions with geometric higher homotopy operations, and providing explicit constructions and correspondences.

Contribution

It explicitly constructs cocycles for cohomology obstructions and minimal values of higher homotopy operations, linking algebraic and geometric obstruction theories.

Findings

Explicit cocycle constructions for cohomology obstructions

Construction of minimal higher homotopy operation values

Demonstration of correspondence between algebraic and geometric obstructions

Abstract

There are two main approaches to the problem of realizing a $\Pi$-algebra (a graded group $\Lambda$ 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 $G_\bullet$ of $\Lambda$ by a simplicial space $W_\bullet$ and proceed by induction on the simplicial dimension. The first provides a sequence of Andr\'{e}-Quillen cohomology classes in $H_{AQ}^{n+2}(\Lambda;\Omega^{n}\Lambda)$ for $n \geq 1$ as obstructions to the existence of successive Postnikov sections for $W_\bullet$ by work of Dwyer, Kan and Stover. The second gives a sequence of geometrically defined higher homotopy operations as the obstructions by earlier work of Blanc; these were identified with the obstruction theory of Dwyer, Kan and Smith in earlier work of the current authors. There are also (algebraic and geometric) obstructions for distinguishing between different realizations of $\Lambda$. In this paper we 1) provide an explicit construction of the cocycles representing the cohomology obstructions; 2) provide a similar explicit construction of certain minimal values of the higher homotopy operations (which reduce to "long Toda brackets"), and 3) show that these two constructions correspond under an evident map.