A counterexample to the HK-conjecture that is principal

Abstract

Scarparo has constructed counterexamples to Matui’s HK-conjecture. These counterexamples 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.

Dynamic asymptotic dimension and Matui's HK conjecture

“…For $$k=4$$, the situation seems worse: one has $$$$\begin{equation*} \mu _4:E^4_{4,0}\rightarrow \frac{J_0:\mathcal {I}^5(C_0(G^0))}{J_0:\mathcal {I}^4(C_0(G^0))} \end{equation*}$$$$but $$E^4_{4,0}$$ could in principle be a proper subquotient of $$H_4(G)$$, so a priori one only gets a map from a subquotient of $$H_4(G)$$ to a subquotient of $$K_0(C^*_r(G))$$; the situation is similar to this for all $$k\geqslant 4$$. The recent principal counterexamples to the HK conjecture of Deeley [13] suggest that the a priori obstructions to the existence of the higher comparison maps discussed above really do pertain; however, we did not yet attempt the relevant computations.…”

“…The first, due to Scarparo [48] is the presence of torsion in isotropy groups. The second, due to Deeley [13] is due to torsion phenomena in $$K$$‐theory; however, Deeley's results do not contradict the “rational” HK conjecture one gets after tensoring with $$\mathbb {Q}$$, analogously to the classical fact that the Chern character is a rational isomorphism between $$K$$‐theory and cohomology. The third is exotic analytic phenomena connected to the failure of the Baum–Connes conjecture as discussed above (this is admittedly not exactly a failure of the HK conjecture, but it seems to us as evidence that the HK conjecture should sometimes fail when the Baum–Connes conjecture fails).…”

“…The conjecture has been confirmed in several interesting cases [17, 36, 39, 56] even beyond the originally postulated minimal setting. However, there are also counterexamples due to Scarparo [48] and Deeley [13]. Scarparo's counterexamples seem to be explained by the presence of torsion in isotropy groups, while Deeley's come from the presence of torsion phenomena in $$K$$‐theory.…”

Homology and K-theory of dynamical systems IV. Further structural results on groupoid homology

“…that follows from the Baum-Connes conjecture for coefficients for , see also [CCWD + 23,Dee23] for the case of flat manifolds.…”