**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**].

* 8/8/94 dpvc@geom.umn.edu -- *

*The Geometry Center*