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Let $K$ be a field of characteristic zero, let $G$ be a connected linear$K$-algebraic group, and let $H$ be a connected closed subgroup of $G$. Let$X_c$ be a smooth compactification of $X=G/H$, and let$Y\overset{}{\longrightarrow}X_c$ be the universal $S$-torsor with trivialfibre over the class of the identity of $G$. Colliot-Th\'el\`ene andKunyavski\u{\i} have shown that $S$ is a flasque torus, and that the evaluationmap $X_c(K)\rightarrow \text{H}^1(K,S)$, induced by the universal torsor, issurjective when the field $K$ is 'good'; and the same is true when we restrictthe evaluation map to the $K$-points of $X$. In this article, we establish thatin cases where the field is not 'good', surjectivity may fail when the domainis $X(K)$. We provide two concrete examples: one over a field of cohomologicaldimension $2$ and the other over an arithmetic field, such as $\mathbbQ_7((t))$.

Non-Surjectivity of the Universal Torsor Evaluation Map for Homogeneous Spaces