Minimal quasisections of circle bundles
Abstract
Let ExrightarrowπB be an oriented circle bundle over an oriented closed surface, and let ℰ denote its Euler number. A \textit{quasisection} is a smooth surface Q, possibly with boundary, together with a generic smooth map q:Q→ E such that π∘ q(Q)=B. The \textit{local formula} expresses the Euler number ℰ of a bundle as a sum of the weights of the \textit{singular vertices} of a quasisection. In the present paper determine the minimal number 𝒩 of singular vertices for closed and for bordered quasisections for a given circle bundle. For bordered quasisections, 𝒩b(E→ B) =⌈3|ℰ|/2⌉. For closed quasisections, If ℰ is even, 𝒩cl(E→ B) =3|ℰ|/2. If ℰ is odd, 𝒩cl(E→ B) =4|ℰ|. We also provide examples of quasisections that realize these minimal values.
Related
BibTeX
@misc{kiselev2026minimal,
title = {{Minimal quasisections of circle bundles}},
author = {Ivan Kiselev and Gaiane Panina and Timur Shamazov},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00679},
eprint = {2611.00679},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00679},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Minimal quasisections of circle bundles
AU - Ivan Kiselev
AU - Gaiane Panina
AU - Timur Shamazov
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00679
UR - https://arxiv.org/abs/2611.00679
AB - Let ExrightarrowπB be an oriented circle bundle over an oriented closed surface, and let ℰ denote its Euler number. A \textit{quasisection} is a smooth surface Q, possibly with boundary, together with a generic smooth map q:Q→ E such that π∘ q(Q)=B. The \textit{local formula} expresses the Euler number ℰ of a bundle as a sum of the weights of the \textit{singular vertices} of a quasisection. In the present paper determine the minimal number 𝒩 of singular vertices for closed and for bordered quasisections for a given circle bundle. For bordered quasisections, 𝒩_b(E→ B) =⌈3|ℰ|/2⌉. For closed quasisections, If ℰ is even, 𝒩_cl(E→ B) =3|ℰ|/2. If ℰ is odd, 𝒩_cl(E→ B) =4|ℰ|. We also provide examples of quasisections that realize these minimal values.
ER -