Galois representations modulo $p$ that do not lift modulo $p^2$

Abstract: For every finite group $H$ and every finite $H$-module $A$, we determine the subgroup of negligible classes in $H^2(H,A)$, in the sense of Serre, over fields with enough roots of unity. As a consequence, we show that for every odd prime $p$, every integer $n\geq 3$, and every field $F$ containing a primitive $p$-th root of unity, there exists a continuous $n$-dimensional mod $p$ representation of the absolute Galois group of $F(x_1,\dots,x_p)$ which does not lift modulo $p^2$. This answers a question of Khare and Serre, and disproves a conjecture of Florence.