AN IDENTITY PROPERTY FOR 2-COMPLEX PAIRS

Abstract

An identity property defined for a pair of 2-complexes (Y,X) first arose in 1993 within a strategy for constructing a counterexample of infinite type to Whitehead's Asphericity Conjecture. In this note we make use of the theory of pictures to characterize a more general right N-identity property, where N < \pi _1Y. We also define combinatorial asphericity (CA) for the pair (Y,X) and determine a test for (CA) in the case that Y is obtained from X by the addition of a single 2-cell. This test can be used to determine an explicit generating set for \pi _2Y.