Matui's HK Conjecture Overview

• Matui’s HK Conjecture is a framework that relates the homology of ample étale groupoids to the topological K-theory of their reduced C*-algebras.
• The conjecture has driven verifications in classes such as AF, Deaconu–Renault, and graph groupoids, illustrating both successful cases and the need for revised formulations due to torsion obstructions.
• Its rational formulation, which employs the Baum–Connes assembly map and Chern character techniques, supports the conjecture in torsion-free settings while highlighting failures in the integral case.

Matui’s HK Conjecture is a proposed identification between the homology of an ample étale groupoid and the topological $K$-theory of its reduced groupoid $C^*$-algebra. In its standard parity-preserving form, for a minimal, essentially principal, ample groupoid $G$, it predicts

$$K_i(C_r^*(G))\;\cong\;\bigoplus_{k\ge 0} H_{2k+i}(G),\qquad i=0,1,$$

or equivalently $K_*(C_r^*(G))\cong \bigoplus_{i\ge 0} H_{*+2i}(G)$. The conjecture emerged as part of a broader program relating étale groupoids, topological full groups, and $C^*$-classification, and it has since developed in three directions: concrete verifications in major classes of ample groupoids, structural comparison maps from homology to $K$-theory, and revisions forced by counterexamples in the presence of torsion or higher-dimensional torsion phenomena (Farsi et al., 2018).

1. Original statement and main variants

Matui’s original conjecture is usually formulated for second-countable, locally compact, Hausdorff, ample groupoids, with minimality and essential principality or effectiveness depending on the version under discussion. In one common form, if $G$ is minimal, effective, ample Hausdorff and $G^{(0)}$ is a Cantor set, then

$$K_{0}(C^*(G))\cong\bigoplus_{n=0}^\infty H_{2n}(G),\qquad K_{1}(C^*(G))\cong\bigoplus_{n=0}^\infty H_{2n+1}(G).$$

A closely related formulation asserts a natural isomorphism $C^*$0 in examples where higher homology vanishes, such as graph groupoids (Nyland et al., 2020).

A rational version replaces integral coefficients by $C^*$1, or equivalently tensors $C^*$2-theory with $C^*$3. For ample groupoids with torsion-free stabilizers, a recent theorem proves the periodic rational form

$C^*$4

assuming the rational Baum–Connes conjecture. Writing

$C^*$5

this becomes $C^*$6 (Proietti et al., 9 Sep 2025).

The distinction between integral and rational forms is now essential. Counterexamples show that the integral conjecture can fail even for principal ample groupoids, while the rational conjecture holds in broad torsion-free settings. This suggests that the conjecture is best regarded not as a single statement, but as a family of comparison problems whose exact formulation depends on isotropy and torsion phenomena (Deeley, 2021).

2. Homology, periodicity, and canonical comparison maps

For an ample Hausdorff groupoid $C^*$7, the homology groups are defined from the simplicial nerve. One sets

$C^*$8

with boundary

$C^*$9

Then

$G$0

For rational coefficients, the same construction uses $G$1, and periodicization groups even and odd degrees together (Proietti et al., 9 Sep 2025).

A central feature of the subject is the existence of canonical homology comparison maps

$G$2

The map $G$3 is characterized by compatibility with the identification $G$4. The map $G$5 sends the class of a compact open full bisection $G$6 to the $G$7-class of the corresponding unitary $G$8. These maps are functorial in $G$9 and play the role of the degree-$$K_i(C_r^*(G))\;\cong\;\bigoplus_{k\ge 0} H_{2k+i}(G),\qquad i=0,1,$$0 and degree-$$K_i(C_r^*(G))\;\cong\;\bigoplus_{k\ge 0} H_{2k+i}(G),\qquad i=0,1,$$1 edges of the expected HK isomorphism (Matui, 1 Jul 2026).

In low degree there are more explicit descriptions. If $$K_i(C_r^*(G))\;\cong\;\bigoplus_{k\ge 0} H_{2k+i}(G),\qquad i=0,1,$$2 is a compact open bisection, then $$K_i(C_r^*(G))\;\cong\;\bigoplus_{k\ge 0} H_{2k+i}(G),\qquad i=0,1,$$3 is a partial isometry, and the degree-one comparison map can be written

$$K_i(C_r^*(G))\;\cong\;\bigoplus_{k\ge 0} H_{2k+i}(G),\qquad i=0,1,$$4

The degree-zero map is induced by the inclusion $$K_i(C_r^*(G))\;\cong\;\bigoplus_{k\ge 0} H_{2k+i}(G),\qquad i=0,1,$$5, and can also be described via mapping-cone or spectral-sequence constructions (Bönicke et al., 2021).

These comparison maps are the concrete remnants of the conjecture even where full HK is unknown or false. In several positive results, proving HK amounts to showing that $$K_i(C_r^*(G))\;\cong\;\bigoplus_{k\ge 0} H_{2k+i}(G),\qquad i=0,1,$$6 and $$K_i(C_r^*(G))\;\cong\;\bigoplus_{k\ge 0} H_{2k+i}(G),\qquad i=0,1,$$7 are isomorphisms and that the higher homology terms either vanish or are recovered by auxiliary comparison maps.

3. Established cases and structural verification results

The conjecture is known in several major classes. For AF groupoids, one has $$K_i(C_r^*(G))\;\cong\;\bigoplus_{k\ge 0} H_{2k+i}(G),\qquad i=0,1,$$8 for $$K_i(C_r^*(G))\;\cong\;\bigoplus_{k\ge 0} H_{2k+i}(G),\qquad i=0,1,$$9, $K_*(C_r^*(G))\cong \bigoplus_{i\ge 0} H_{*+2i}(G)$0, and $K_*(C_r^*(G))\cong \bigoplus_{i\ge 0} H_{*+2i}(G)$1. In the AF case the $K_*(C_r^*(G))\cong \bigoplus_{i\ge 0} H_{*+2i}(G)$2-to-$K_*(C_r^*(G))\cong \bigoplus_{i\ge 0} H_{*+2i}(G)$3 isomorphism can be made explicit, and it is in fact an order isomorphism between $K_*(C_r^*(G))\cong \bigoplus_{i\ge 0} H_{*+2i}(G)$4 and $K_*(C_r^*(G))\cong \bigoplus_{i\ge 0} H_{*+2i}(G)$5 (Lima, 2024).

For Deaconu–Renault groupoids $K_*(C_r^*(G))\cong \bigoplus_{i\ge 0} H_{*+2i}(G)$6 associated to commuting surjective local homeomorphisms of a totally disconnected space, the homology is computed by a chain complex built from the maps $K_*(C_r^*(G))\cong \bigoplus_{i\ge 0} H_{*+2i}(G)$7. In ranks $K_*(C_r^*(G))\cong \bigoplus_{i\ge 0} H_{*+2i}(G)$8, comparison with Kasparov’s spectral sequence shows that HK holds. Specializing to $K_*(C_r^*(G))\cong \bigoplus_{i\ge 0} H_{*+2i}(G)$9-graph groupoids, this yields verification for all row-finite $C^*$0- and $C^*$1-graphs with no sources, and for certain higher-rank single-vertex cases satisfying a coprimality condition (Farsi et al., 2018).

Graph groupoids form another large verified class. If $C^*$2 is a countable directed graph with graph groupoid $C^*$3, then

$C^*$4

and these coincide with

$C^*$5

Hence every graph groupoid has the HK property (Nyland et al., 2020).

A more structural verification comes from dynamic asymptotic dimension. If $C^*$6 is a principal, $C^*$7-compact, ample groupoid with finite dynamic asymptotic dimension $C^*$8, then $C^*$9 for all $K$0, and $K$1 is torsion-free. In particular, if $K$2 is second countable, principal, ample, $K$3, and $K$4 is finitely generated, then

$K$5

so Matui’s conjecture holds in this low-dimensional regime (Bönicke et al., 2021).

For Cantor transformation groupoids of poly-$K$6 actions, exact-sequence comparison with the Pimsner–Voiculescu sequence yields further positive results. For free actions of $K$7, $K$8, and the Klein bottle group, HK is recovered; for several classes of Hirsch length three and four, one either obtains HK or explicit exact sequences expressing $K$9-theory in terms of homology and cohomology (Matui, 1 Jul 2026).

4. Rational HK and the Chern character for torsion-free ample groupoids

The most general positive theorem currently available is rational. Let $G$0 be a second-countable, locally compact, Hausdorff, ample groupoid with torsion-free stabilizers, and assume the rational Baum–Connes assembly map

$G$1

is an isomorphism. Then the Chern character

$G$2

is an isomorphism of $G$3-vector spaces. This is the rational HK theorem for torsion-free ample groupoids (Proietti et al., 9 Sep 2025).

The construction does not proceed through the usual Chern–Connes character or periodic cyclic homology. Instead, it uses the $G$4-categorical viewpoint on bivariant $G$5-theory. The domain of the Baum–Connes assembly map is modeled by a cellular approximation $G$6 in the stable $G$7-category $G$8, and one identifies

$G$9

Applying the functor $G^{(0)}$0, smashing with $G^{(0)}$1, and using the Bott–Chern equivalence

$G^{(0)}$2

one obtains a map from the simplicial $G^{(0)}$3-valued spectrum of $G^{(0)}$4 to a periodicized Eilenberg–Mac Lane spectrum whose homotopy groups recover groupoid homology (Proietti et al., 9 Sep 2025).

This theorem subsumes earlier transformation-groupoid rational results. Deeley–Willett proved the rational HK conjecture for a large class of transformation groupoids with torsion-free stabilizers using the rational Baum–Connes conjecture and Raven’s Chern character. In that setting, if $G^{(0)}$5 is a countable discrete group acting on a totally disconnected space $G^{(0)}$6 with torsion-free stabilizers and rational Baum–Connes holds for $G^{(0)}$7, then

$G^{(0)}$8

(Deeley et al., 2024).

A plausible implication is that rational HK is now structurally tied to assembly and to the passage from equivariant $G^{(0)}$9-theory to groupoid homology. The recent torsion-free ample-groupoid theorem turns this bridge into a general mechanism rather than a phenomenon restricted to transformation groupoids.

5. Counterexamples, obstructions, and revised conjectures

The conjecture is false in full integral generality. Scarparo produced counterexamples in the essentially principal setting, and these examples also motivated the failure of the original rational conjecture when torsion isotropy is present. Deeley–Willett emphasize that all known counterexamples to the original rational form arise from torsion in isotropy, and they introduce a revised rational HK conjecture based on a blow-up groupoid $$K_{0}(C^*(G))\cong\bigoplus_{n=0}^\infty H_{2n}(G),\qquad K_{1}(C^*(G))\cong\bigoplus_{n=0}^\infty H_{2n+1}(G).$$0 that removes finite-order isotropy. The replacement target is

$$K_{0}(C^*(G))\cong\bigoplus_{n=0}^\infty H_{2n}(G),\qquad K_{1}(C^*(G))\cong\bigoplus_{n=0}^\infty H_{2n+1}(G).$$1

and for transformation groupoids satisfying rational Baum–Connes they prove

$$K_{0}(C^*(G))\cong\bigoplus_{n=0}^\infty H_{2n}(G),\qquad K_{1}(C^*(G))\cong\bigoplus_{n=0}^\infty H_{2n+1}(G).$$2

This revised form covers Scarparo’s dihedral odometer examples (Deeley et al., 2024).

The integral conjecture can fail even without isotropy. Deeley constructed a principal counterexample using a free odometer action of the fundamental group of a flat manifold. The resulting transformation groupoid is principal, minimal, ample, and second countable, yet the integral HK isomorphism fails because the inductive-limit torsion in $$K_{0}(C^*(G))\cong\bigoplus_{n=0}^\infty H_{2n}(G),\qquad K_{1}(C^*(G))\cong\bigoplus_{n=0}^\infty H_{2n+1}(G).$$3-theory is strictly smaller than the torsion detected by groupoid homology. At the same time, these examples satisfy the rational version of HK (Deeley, 2021).

This phenomenon was sharpened in low dimensions. For each dimension $$K_{0}(C^*(G))\cong\bigoplus_{n=0}^\infty H_{2n}(G),\qquad K_{1}(C^*(G))\cong\bigoplus_{n=0}^\infty H_{2n+1}(G).$$4, there exists a counterexample to HK built from a flat manifold odometer of dimension $$K_{0}(C^*(G))\cong\bigoplus_{n=0}^\infty H_{2n}(G),\qquad K_{1}(C^*(G))\cong\bigoplus_{n=0}^\infty H_{2n+1}(G).$$5, and this dimension is minimal: if $$K_{0}(C^*(G))\cong\bigoplus_{n=0}^\infty H_{2n}(G),\qquad K_{1}(C^*(G))\cong\bigoplus_{n=0}^\infty H_{2n+1}(G).$$6, the HK conjecture holds for the associated odometer. In dimension $$K_{0}(C^*(G))\cong\bigoplus_{n=0}^\infty H_{2n}(G),\qquad K_{1}(C^*(G))\cong\bigoplus_{n=0}^\infty H_{2n+1}(G).$$7, the Proietti–Yamashita spectral sequence yields short exact sequences

$$K_{0}(C^*(G))\cong\bigoplus_{n=0}^\infty H_{2n}(G),\qquad K_{1}(C^*(G))\cong\bigoplus_{n=0}^\infty H_{2n+1}(G).$$8

and these split in the odometer setting. For $$K_{0}(C^*(G))\cong\bigoplus_{n=0}^\infty H_{2n}(G),\qquad K_{1}(C^*(G))\cong\bigoplus_{n=0}^\infty H_{2n+1}(G).$$9, torsion mismatch stemming from flat-manifold $C^*$00-theory forces failure of the integral HK prediction (Chaiser, 7 Jul 2025).

These counterexamples clarify two different obstructions. Torsion isotropy obstructs the classical rational conjecture and motivates the revised blow-up formulation. Independently, even principal torsion-free groupoids can violate integral HK because topological $C^*$01-theory may carry extension or torsion information not visible in the direct sum of homology groups.

6. Connections, applications, and current role

The HK framework now serves as a nexus between groupoid homology, assembly theory, and classification invariants. For Smale spaces with totally disconnected stable sets, the rational HK theorem identifies the $C^*$02-theory of the unstable $C^*$03-algebra with Putnam’s stable homology: $C^*$04 The same paper gives applications to the rational homology of topological full groups, the homotopy type of the algebraic $C^*$05-theory spectrum of ample groupoids, and the Elliott invariant of classifiable $C^*$06-algebras with torsion-free $C^*$07 (Proietti et al., 9 Sep 2025).

For free Cantor actions of poly-$C^*$08 groups, HK is intertwined with gap-labelling. The comparison maps $C^*$09 and their cohomological analogues are matched against Pimsner–Voiculescu exact sequences, and Poincaré duality is used to recover the higher homology terms entering the parity decomposition. In low Hirsch length, this produces explicit $C^*$10-theory formulas and, in several cases, full or half gap-labelling results (Matui, 1 Jul 2026).

In the AF setting, positivity results strengthen the connection to ordered $C^*$11-theory. The explicit order isomorphism $C^*$12 can be used to characterize AF embeddability of Deaconu–Renault groupoid $C^*$13-algebras via the condition

$C^*$14

showing that homological techniques can detect order-theoretic phenomena in $C^*$15-theory (Lima, 2024).

The present state of the subject is therefore mixed but structurally coherent. Integral HK is a genuine theorem in several foundational classes and a false statement in general. Rational HK has become substantially more robust, especially for torsion-free stabilizers and under rational Baum–Connes assumptions. Revised formulations using blow-ups address torsion isotropy, while recent $C^*$16-categorical Chern character constructions suggest that the enduring content of the conjecture lies in a systematic passage from equivariant $C^*$17-theory to periodicized groupoid homology rather than in a universally valid integral isomorphism (Proietti et al., 9 Sep 2025).