Quasisections of circle bundles and Euler class

TL;DR

This work develops a local, combinatorial framework for the Euler class of an oriented circle bundle $E\to B$ over a closed surface by introducing and studying quasisections. The authors prove a local formula: the Euler number $\mathcal{E}(E\to B)$ equals a weighted count of essential singularities (types I, II, III) of a generic quasisection, with explicit weights derived from portrait data; a uniqueness theorem shows no other local assignment yields the Euler class. They extend to bordered quasisections, introducing additional vertex types and weights, and prove the corresponding local formula remains valid and unique. The approach connects to Kazarian’s multisection framework and offers a tractable, local method to compute global topological invariants from singularity data, with concrete constructions (pancake quasisections) illustrating the theory’s versatility.

Abstract

Let $ E \xrightarrow[\text{}]π 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 $π\circ q({Q})=B$. 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 $E$.

Figures (15)

• Figure 1: Singular curve and singular vertices, a fragment of a typical landscape.
• Figure 2: Left: The portrait of a regular point. Right: A portrait of a point $x$ lying on a fold.
• Figure 3: The list of all non-symm portraits, types $I$, $II$ right, and $III$ right.
• Figure 4: The portrait of a Whitney umbrella
• Figure 5: Extension of a partial section to a singular edge.

Theorems & Definitions (50)

• Definition 1
• Proposition 1
• Lemma 1
• Lemma 2
• proof
• Lemma 3
• Lemma 4
• Definition 2
• Theorem 1
• proof