Étale equivalence relations with certain prescribed torsion in their homology

The paper gives a careful, self-contained construction on a single Cantor path space using a split Bratteli diagram and a specially controlled partial homeomorphism, proving an isomorphism S(R) ≅ {(0,0)} ∪ ((D_+\{0}) ⊕ (E/Z)) and hence H0(R) ≅ D ⊕ (E/Z) (Theorem 6.4), with all étaleness checks handled via explicit bisection criteria and ‘recipe’ conditions; it also discusses approximately inner flip. The candidate solution attains the same invariants by a product-and-stripes construction that combines an AF tail-equivalence part realizing D with a torsion gadget realizing E/Z, and then shows the same semigroup and homology identifications. However, the model does not address minimality (prominent in the paper’s abstract) nor does it verify the étale join condition with the same care; still, on the core invariants S(R) and H0(R), the constructions are compatible and correct, albeit by different methods.