On Residual Nilpotence of Projective Crossed Modules

Abstract

Let F be a free group, (M,∂, F) a non-aspherical projective F-crossed module. We prove that the action of Coker (∂) on Ker (∂) is faithful. Also we show that if (M,∂, F) is a residually nilpotent crossed module, then Coker (∂) is a residually nilpotent group. As a corollary, we get an alternative proof of Conduche's translation of Whitehead's asphericity conjecture in terms of residual nilpotence of certain crossed modules.

Keywords:

• Asphericity
• Crossed modules
• Residual nilpotence

Mathematics Subject Classification:

• Primary 20E26, 57M20
• Secondary 58B05

ACKNOWLEDGMENTS

Author thanks D. Conduche and I. B. S. Passi for useful discussions and the referee for important comments and suggestions on the paper. This research was partially supported by the Russian Foundation for Basic Research, grant No. 05-01-00993 and Russian Presidential grant No. MK-1487.2005.1.

Notes

Communicated by A. Olshanskiy.