Pachner move and affine volume-preserving geometry in .
We present a characterization of those open n-manifolds (n ≥ 5) whose products with the real line are homeomorphic to interiors of compact (n+1)-manifolds with boundary.
It is proved that two planes that are properly homotopic in a noncompact, orientable, irreducible 3-manifold that is not homeomorphic to are isotopic. The end-reduction techniques of E. M. Brown and C. D. Feustal and M. G. Brin and T. L. Thickstun are used.