Tightness and Homology: the Modern Definition

Definition: A map f :M^m -> R^n is k-tight if, for all directions z and heights c, the map { p in M | z . f (p) <= c } -> M induces a monomorphism in the i-th Cech homology for each i from 0 to k.

Note that this definition is valid for manifolds of arbitrary dimension, both smooth an polyhedral, with or without boundary, in spaces of any dimension.

In this context, tightness in the sense of total absolute curvature corresponds to m-tightness, while the two-piece property corresponds to 0-tightness. To see the latter, note that the dot product of z with f is simply the height function in the direction z, and so the set { p in M | z . f (p) <= c } is the preimage of a half-space; since 0-dimensional homology counts the number of connected components, the fact that inclusion induces a monomorphism implies that there is only one component in the preimage.

In the case of closed surfaces without boundary (m = 2), 0-tightness and m-tightness are equivalent [K4, Ku1], though this is no longer true for surfaces with boundary [Ku1].