Lifting of representations of Galois groups of local fields and of fundamental groups of surfaces

In a joint work with Andrea Conti and Mathieu Florence, we consider lifting questions for representations of profinite groups satisfying a two-dimensional Poincaré duality property: examples of such groups are absolute Galois groups of local fields, and (profinite completions of) topological fundamental groups of orientable compact surfaces. We prove in particular that any modulo \(p\) representation of such a group lifts to a representation modulo \(p^2\) (under an assumption about the roots of unity in the local field case), and that every unipotent representation modulo \(p\) lifts modulo \(p^r\), for all \(r\) (under a technical assumption about roots of unity in the arithmetic case). The strategy and the proof are as elementary as possible, and are based on Kummer theory and Poincaré duality.