An elementary approach to local combinatorial formulae for the Euler class of a PL spherical fiber bundle

TL;DR

This paper introduces a simple combinatorial method for computing the Euler class of PL spherical fiber bundles using elementary techniques and averaging sections, building on existing formulas by Mnev.

Contribution

It provides an accessible, elementary approach to local combinatorial formulas for the Euler class, avoiding complex advanced methods.

Findings

Derived new combinatorial formulas for Euler class

Utilized averaging sections technique in a simplicial context

Simplified computation of Euler classes in PL spherical bundles

Abstract

We present an elementary approach to local combinatorial formulae for the Euler class of a fiber-oriented triangulated spherical fiber bundle. The approach is based on sections averaging technique and very basic knowledge of simplicial (co)homology theory. Our formulae are close relatives of those by N. Mn\"{e}v.