Homotopy types of reduced 2-nilpotent simplicial groups

Details Homotopy types of reduced 2-nilpotent simplicial groups Homotopy types of reduced 2-nilpotent simplicial groups

2008-04-12

arxiv.org/abs/0804.2000v1

Ind. J. Pure Appl. Math. 40 (2009), 35-80

Mathematics

K-Theory and Homology

Scientific paper

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 $m=4,5$.

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