Sections of fiber bundles over surfaces

TL;DR

This paper investigates the existence and enumeration of sections in Serre fibrations over surfaces, providing a complete solution for fibers with finite fundamental groups using cohomology and Topological Quantum Field Theory techniques.

Contribution

It offers a novel complete solution for sections over surfaces when the fiber's fundamental group is finite, connecting cohomology and TQFT methods.

Findings

Complete classification for finite fundamental group fibers

Use of 2D cohomology classes linked to irreducible representations

Application of Topological Quantum Field Theory in proofs

Abstract

We study the existence problem and the enumeration problem for sections of Serre fibrations over compact orientable surfaces. When the fundamental group of the fiber is finite, a complete solution is given in terms of 2-dimensional cohomology classes associated with certain irreducible representations of this group. The proofs are based on Topological Quantum Field Theory.