A counterexample to the HK-conjecture that is principal

Abstract

Scarparo has constructed counterexamples to Matui's HK-conjecture and Ortega and Scarparo have constructed counterexamples to the rational HK-conjecture. These 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

“…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 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).…”

“…For 𝑘 = 4, the situation seems worse: one has 𝜇 4 ∶ 𝐸 4 4,0 → 𝐽 0 ∶  5 (𝐶 0 (𝐺 0 )) 𝐽 0 ∶  4 (𝐶 0 (𝐺 0 )) 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 ⩾ 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.…”

“…Due to similarities to Haslehurst's methods [25], we do not give a proof here: the interested reader can find a complete proof in the original arXiv version of this paper. With notation as in line (13), the map 𝜂 from line ( 14) above descends to a well-defined monoid isomorphism 𝜂 ∶ 𝐾 0 (𝐷(𝜙)) 𝑠 0 (𝐾 0 (𝐴)) → 𝑉 ∞ (𝐴, 𝐵; 𝜙) ≈ .…”

“…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.…”