Corrigendum to: “Aspherical four-manifolds and the centres of two-knot groups”

Abstract

M. N. Dyer has pointed out that the proof of the key Lemma on hopfian rings in w of [1] is incorrect. As I have been unable to find a correct argument, the results on pages 465-469 are moot. (Corollaries 2 and 3 on page 470 are true as it is easy to see that the lemma holds for any commutative ring, while the results in w use only Kaplansky's original theorem, and not the lemma.) I hope that some ring-theorist may be able to prove the lemma. In the first line of page 469, the map from /-/~(C*) to Homr (H2(C,I, F) given by the universal coefficient spectral sequence is, a priori, only a monomorphism. However the theorem is still true (without any essential change in the argument), modulo the lemma. A. Suciu has pointed out that the map �9 in line 1 of page 471 should be replaced by its square 4 2, to ensure that the mapping torus M be orientable. I am grateful to Dyer and Suciu for their observations.