Regular Bundles on Orbifolds: A Short Proof of Presentability

Abstract

Let $\X$ be a Hausdorff second-countable smooth orbifold without boundary, possibly noncompact and ineffective. Suppose $\dim\X\leq n$ and $|G_x|\leq B$ for integers $n\geq0$ and $B\geq1$, 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 $\C[G_x]$. 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 $\X\simeq[M/U(R(n,B))]$ for a smooth manifold $M$ with a proper locally free action; $M$ is compact exactly when $|\X|$ is compact. Thus every compact smooth orbifold is presentable.