An extension of Rourke’s proof that Ω₃=0 to nonorientable manifolds

Abstract

A classical result in manifold theory states that every closed 3 3 -manifold bounds a compact 4 4 -manifold. In 1985 C. Rourke discovered a strikingly short and elementary proof of the orientable case of this theorem ( Ω 3 = 0 ) ({\Omega _3} = 0) . In this note we show that Rourke’s approach can be extended to include nonorientable 3 3 -manifolds.