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.
Cite this article
Vladimir Turaev, Sections of fiber bundles over surfaces and TQFTs. Quantum Topol. 1 (2010), no. 3, pp. 275–319