Local properties on the remainders of the topological groups

When does a topological group $G$ have a Hausdorff compactification $bG$ with a remainder belonging to a given class of spaces? In this paper, we mainly improve some results of A.V. Arhangel'ski\v{\i} and C. Liu's. Let $G$ be a non-locally compact topological group and $bG$ be a compactification of $G$. The following facts are established: (1) If $bG\setminus G$ has a locally a point-countable $p$-metabase and $\pi$-character of $bG\setminus G$ is countable, then $G$ and $bG$ are separable and metrizable; (2) If $bG\setminus G$ has locally a $\delta\theta$-base, then $G$ and $bG$ are separable and metrizable; (3) If $bG\setminus G$ has locally a quasi-$G_{\delta}$-diagonal, then $G$ and $bG$ are separable and metrizable. Finally, we give a partial answer for a question, which was posed by C. Liu in \cite{LC}.