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We generalize the well known geometric characterizations of orientable n-dimensional manifolds, i.e., an n-dimensional closed PL manifold M is orientable if and only if each embedded circle in M has a regular neighborhood homeomorpic to the product of the circle with an (n – 1) ball.
Theorem. An n-dimensional, orientable, closed PL manifold M, n>4, is spin if and only if each embedded closed surface F in M has a a regular neighborhood homeomorpic to the product of the surface F with an (n – 2) ball.