Finite-index phenomena and the topology of bundle singularities

A classical branched cover is an open surjection of compact Hausdorff spaces with uniformly bounded finite fibers and analogously, a quantum branched cover is a unital $C^{\ast }$ embedding admitting a finite-index expectation. We show that whenever a compact Hausdorff space $Z$ contains a one-point compactification of an uncountable set, the incidence correspondence attached to the space of cardinality-$(\le n)$ subsets of $Z$ (for $n\ge 3$) is a classical branched cover that does not dualize to a quantum one. In particular, when $Z$ is dyadic, the resulting $C^{\ast }$ embeddings are quantum branched covers precisely when $Z$ is also metrizable. This provides a partial converse to an earlier result of the author’s (to the effect that continuous, unital, subhomogeneous $C^{\ast }$ bundles over compact metrizable spaces are quantum branched) and settles negatively a question of Blanchard-Gogić. There are also some positive results identifying classes of compact Hausdorff spaces (e.g. extremally disconnected or orderable) with the property that all (continuous, unital) $C^{\ast }$ bundles based thereon are quantum branched.