Cockcroft properties of Thompson's group

In a study of the word problem for groups, R. J. Thompson considered a certain group F of self-homeomorphisms of the Cantor set and showed, among other things, that F is finitely presented. Using results of K. S. Brown and R. Geoghegan, M. N. Dyer showed that F is the fundamental group of a finite two-complex Z² having Euler characteristic one and which is Cockcroft, in the sense that each map of the two-sphere into Z² is homologically trivial. We show that no proper covering complex of Z² is Cockcroft. A general result on Cockcroft properties implies that no proper regular covering complex of any finite two-complex with fundamental group F is Cockcroft.