  The universal pairing for manifolds was defined and shown to lack positivity
in dimension 4 by Freedman et al. We prove an analogous result for 2-complexes,
and also show that the universal pairing does not detect the difference between
simple homotopy equivalence and 3-deformations. The question of whether these
two equivalence relations are different for 2-complexes is the subject of the
Andrews-Curtis conjecture. We also discuss the universal pairing for
higher-dimensional complexes and show that it is not positive.
