Abstract: We outline the proof of the following theorem, joint with Colin Rourke (University of Warwick).

THEOREM: Suppose that G is a torsion free group and (t) is the infinite cyclic group generated by an element t. Let w be an element of the free product Gx(t) and ((w)) the normal subgroup generated by w. Then the natural map G -> G*(t) -> (G*(t))/((w)) is onto iff w is conjugate to gt or gt^-1 for some g in G.

The proof uses Klyachko's ingenious car-crash method, used in the solution of the Kervaire conjecture. The theorem immediately applies topologically to the recognition of certain s-cobordisms.