Homotopy types of reduced 2-nilpotent simplicial groups (Q1022920)

A classical result of D. M. Kan shows that the homotopy type of a connected space is determined by a simplicial group and, by the Dold-Kan theorem, the 1-nilpotent (abelian) simplicial groups are equivalent to chain complexes. Thus there is a clear motivation to study the homotopy type of reduced 2-nilpotent simplicial groups. In this paper the authors obtain their complete classification in terms of graded abelian groups and two invariants denoted respectively \(b:=\{b_n\}\) and \(\beta= \{\beta_n\}\). The description of \(b_n\) and \(\beta_n\) requires the use of quadratic functors and quadratic torsion functors. The authors classify also the homotopy type of connective spectra in the model category of 2-nilpotent simplicial groups. Since any simplicial group \(G\) yields the 2-nilpotent group \(G/[G,[G,G]]\), using the invariants \(b\) and \(\beta\), the authors describe a new natural structure of the integral homology of any 1-connected space. This structure generalizes the well known action of the Steenrod algebra.