Finite-index phenomena and the topology of bundle singularities
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909327159197696 |
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| author | Chirvasitu, Alexandru |
| author_facet | Chirvasitu, Alexandru |
| contents | 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-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) $C^*$ bundles based thereon are quantum branched. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_17807 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finite-index phenomena and the topology of bundle singularities Chirvasitu, Alexandru Functional Analysis General Topology Operator Algebras Quantum Algebra 54C65, 46L30, 46M20, 46A04, 54C60, 54B15, 46L85, 26E25 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-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) $C^*$ bundles based thereon are quantum branched. |
| title | Finite-index phenomena and the topology of bundle singularities |
| topic | Functional Analysis General Topology Operator Algebras Quantum Algebra 54C65, 46L30, 46M20, 46A04, 54C60, 54B15, 46L85, 26E25 |
| url | https://arxiv.org/abs/2409.17807 |