A note on coverings and Kervaire complexes

Abstract

In the combinatorial category, a two-complex X is said to be Kervaire if any set of equations modelled on X, over any group, has a solution in a larger group. The Kervaire-Laudenbach conjecture speculates that if H2(X) = 0 then X is Kervaire. We show that the validity of this conjecture would imply that all aspherical two-complexes are Kervaire. In particular, any two-complex homotopically equivalent to a two-manifold (≠ S2, RP2) would be Kervaire. We show that this is indeed the case for certain such two-complexes. We generalise this to staggered two-complexes, and, more generally, one-relator extensions of Kervaire complexes. We obtain similar results for diagrammatically reducible two-complexes. Our proofs make use of covering spaces.