On an invariant of open manifolds

The combinatorial invariance of the obstruction $\Delta$ is proved and a Poincaré duality relation is derived for $\Delta$. It is shown that the invariant $\Delta$ is a meaningful concept, i.e., that there exist open manifolds for which $\Delta\ne0$. The results that are obtained are used for the construction of a boundary for an open manifold and for the fibering of a closed smooth manifold over a circle.