Tight Submanifolds, Smooth and Polyhedral

We begin by defining and studying tightness and the two-piece property for smooth and polyhedral surfaces in three-dimensional space. These results are then generalized to surfaces with boundary and with singularities, and to surfaces in higher dimensions. Later sections deal with generalizations to the case of smooth and polyhedral submanifolds of higher dimension and codimension, in particular highly connected submanifolds. Twenty-six open questions and a number of conjectures are included.

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