Homotopy types of reduced 2-nilpotent simplicial groups

TL;DR

This paper classifies the homotopy types of reduced 2-nilpotent simplicial groups using homology and boundary invariants, extending classical results and providing new insights into the structure of simply connected spaces and spectra.

Contribution

It introduces a classification of reduced 2-nilpotent simplicial groups based on homology invariants, generalizing previous results and analyzing homotopy groups of spheres in this context.

Findings

Classifies homotopy types of reduced 2-nilpotent simplicial groups.

Provides a new natural structure for the integral homology of simply connected spaces.

Computes homotopy groups of spheres for certain nilpotency classes.

Abstract

We classify the homotopy types of reduced 2-nilpotent simplicial groups in terms of the homology an d boundary invariants $b,\beta$. This contains as special cases results of J.H.C. Whitehead on 1-connected 4-dimensional complexes and of Quillen on reduced 2-nilpotent rational simplicial groups. Moreover it yields for 1-nilpotent (or abelian) simplicial groups a classification due to Dold-Kan. Our result describes a new natural structure of the integral homology of any simply connected space. We also classify the homotopy types of connective spectra in the category of 2-nilpotent simplicial groups. Moreover we compute homotopy groups of spheres in the category of $m$-nilpotent groups for $m=2,3$ and partially for $m=4,5$.