  Scarparo has constructed counterexamples to Matui's HK-conjecture. These
counterexample and other known counterexamples are essentially principal but
not principal. In the present paper, a counterexample to the HK-conjecture that
is principal is given. Like Scarparo's original counterexample, our
counterexample is the transformation groupoid associated to a particular
odometer. However, the relevant group is the fundamental group of a flat
manifold (and hence is torsion-free) and the associated odometer action is
free. The examples discussed here do satisfy the rational version of the
HK-conjecture.

K-Theory and Homology;Operator Algebras