Equality of arborescent and pseudo-collarable open manifolds for d ≥ 5

Establish that for every integer d ≥ 5, the class A_d of open d-manifolds that admit an arborescent triangulation equals the class PC_d of open contractible d-manifolds that are pseudo-collarable, have strongly semistable fundamental group at infinity, and have vanishing Chapman–Siebenmann obstruction.

Background

The authors define several classes of open d-manifolds: CAT_d^\Box (PL homeomorphic to CAT(0) cube complexes), CAT_d (admitting a complete polyhedral CAT(0) metric with finitely many shapes), A_d (admitting an arborescent triangulation), and PC_d (open contractible d-manifolds that are pseudo-collarable, have strongly semistable fundamental group at infinity, and have vanishing Chapman–Siebenmann obstruction).

They prove that for d ≥ 5, CAT_d^\Box = A_d ⊆ PC_d, and for d ≤ 4, all these classes reduce to R^d. The conjecture seeks to close the remaining gap by proving the reverse inclusion PC_d ⊆ A_d (equivalently, PC_d ⊆ CAT_d^\Box), which would show that every manifold in PC_d admits a CAT(0) cube structure and hence an arborescent triangulation.