Canonical embeddings of S1×Δn−1 into orientable n-dimensional closed PL manifolds for n>4

For each embedded oriented circle c in an n-dimensional closed PL manifold M, we define a canonical embedding with respect to a triangulation of M, of S1×Δn−1 into a regular neighborhood of the embedded circle in M. For n>4, we give a necessary and sufficient condition for an embedding of a surface with boundary in M, such that the embedding together with the canonical embeddings on the boundary components of the surface, extends to an embedding of the regular neighborhood of the surface in Rn to a regular neighborhood of the embedded surface in M.