
*** nonred ***

This is an example of two geometric figures which are unions of cubes 
which are difficult to reduce, although they are acyclic, that is, their 
homology is the homology of a one-point space.

The first example cannot be reduced by means of removing cubes, because 
the removal of any cube causes the change of the homology of the set. 
However, this set can be reduced to a single point by means of free face 
collapses. [Note: The new version of "homcubes" does not reduce it anymore.]

The second example is even more complicated. It has the property that 
the removal of any cube causes the change to its homology, and, moreover, 
by means of free face collapses it can be reduced to another set of dimension 
two, but not to a single point.
