Diagrams and the second homotopy group

摘要

We use Klyachko's methods [4, 6, 7, 10] 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 [2] 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.