Regular Bundles on Orbifolds: A Short Proof of Presentability

Let $\text{X}$ be a Hausdorff second-countable smooth orbifold without boundary, possibly noncompact and ineffective. Suppose $\dim \text{X}\le n$ and $\mathrm{\mid }{G}_x\mathrm{\mid }\le B$ for integers $n\ge 0$ and $B\ge 1$, where $G_x$ is the full stabilizer at $x$. We construct a smooth Hermitian bundle of explicit rank $R(n,B)$ whose fibre at $x$ is a positive multiple of $\text{C}[G_x]$. Adams operations cancel the Bott obstructions on the boundary spheres of a locally finite good triangulation, and the connectivity of Stiefel manifolds gives actual bundles representing the resulting virtual classes. Smoothing preserves all stabilizer representations. Unitary frames give $\text{X}\simeq [M\mathrm{/}U(R(n,B))]$ for a smooth manifold $M$ with a proper locally free action; $M$ is compact exactly when $\mathrm{\mid }\text{X}\mathrm{\mid }$ is compact. Thus every compact smooth orbifold is presentable.