ON L-2-HOMOLOGY AND ASPHERICITY

We use L-2 methods to show that if a group with a presentation of defi ciency one is an extension of Z by a finitely generated normal subgrou p then the 2-complex corresponding to any presentation of optimal defi ciency is aspherical and to prove a converse of the Cheeger-Gromov-Got tlieb theorem relating Euler characteristic and asphericity. These res ults are applied to the Whitehead conjecture, 4-manifolds and 2-knot g roups.