A low-dimensional counterexample to the HK-conjecture

Matui's HK-conjecture proposes an in-principle computation of the K-theory of the reduced C*-algebra of a (nice enough) groupoid in terms of the homology of the groupoid. While there are a number of positive results, the first counterexample is due to Scarparo in the case where the groupoid is essentially principal. Deeley gave the first principal counterexample using an action groupoid built from a flat manifold and an expansive self-cover. In this case, the invariants in the HK-conjecture are computed from the corresponding invariants of the flat manifold, and HK is a direct analogue of the following phenomenon in algebraic topology. For a CW-complex of dimension at most 3, the Atiyah-Hirzebruch spectral sequence computes K-theory explicitly as a direct sum of cohomology groups. However, this explicit computation fails in general for spaces of dimension 4 and greater. While Deeley's original counterexample uses a flat manifold of dimension at least 9, I will present a counterexample coming from a flat manifold of dimension 4.