Quasisections of circle bundles and Euler class

TL;DR

This paper introduces a local formula for the Euler number of oriented circle bundles over surfaces using quasisections and proves its uniqueness, linking it to singularities and Morse bifurcations.

Contribution

It derives a new local formula for the Euler number of circle bundles via quasisections and establishes its uniqueness, connecting topological invariants with singularity theory.

Findings

Derived a local formula for the Euler number using quasisections.

Proved the uniqueness of the local Euler number formula.

Connected the formula to Morse bifurcations and singularity theory.

Abstract

Let $ E \xrightarrow[\text{}]{\pi} B$ be an oriented circle bundle over an oriented closed surface $B$. A quasisection is a smooth surface ${Q}$ (either closed or bordered) mapped by a generic smooth mapping $q$ to $E$ such that $\pi\circ q({Q})=B$. In the paper we derive a local formula for the Euler number, that is, we show that Euler number (Euler class) of the bundle equals the sum of weights of (some of) singularities of a quasisection.We also prove the uniqueness of such a formula. The local formula is a close relative of M. Kazarian's formula which relates the Euler number and Morse bifurcations of a generic function defined on the total space $E$.