Diagrams and the second homotopy group

Abstract

We use Klyachko's methods [A funny property of sphere and equations over groups, Comm. in Alg. 21 (1993) 2555--2575] (see also Fenn-Rourke, L'Enseignment Math. 42 (1996) 49--74 and math.GR/9810184 and Cohen-Rourke, math.GR/0009101) to prove that, if a 1-cell and a 2-cell are added to a complex with torsion-free fundamental group, and with the 2-cell attached by an amenable t-shape, then pi_2 changes by extension of scalars. It then follows using a result of Bogley and Pride, Proc. Edinburgh Math. Soc. 35 (1992) 1--39, that the resulting fundamental group is also torsion free. We also prove that the normal closure of the attaching word contains no words of smaller complexity.