Finite-index phenomena and the topology of bundle singularities

TL;DR

This paper explores the relationship between classical and quantum branched covers in topology and operator algebras, showing that certain topological conditions prevent dualization to quantum covers, and identifying spaces where all bundles are quantum branched.

Contribution

It demonstrates that for specific topological spaces, classical branched covers do not dualize to quantum ones, and characterizes spaces where all $C^*$ bundles are quantum branched, addressing open questions.

Findings

Classical branched covers do not dualize to quantum covers in certain topological spaces.

Quantum branched covers occur precisely when the space is metrizable under certain conditions.

Spaces like extremally disconnected or orderable have all $C^*$ bundles as quantum branched.

Abstract

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 $C^*$ 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 $C^*$ 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 $C^*$ bundles over compact metrizable spaces are quantum branched) and settles negatively a question of Blanchard-Gogi\'{c}. 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) $C^*$ bundles based thereon are quantum branched.