Quasisections of circle bundles and Euler class

Abstract

Let $E\stackrel{\ddot{I}}{\to }€B$ 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 $\ddot{I}€\circ q(Q)=B$. 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 $E$.