Let $O_K$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ with perfect residue field. From the semi-stable conjecture ($C_{st}$) and the theory of slopes, we obtain isomorphisms between the maximal unramified quotients of certain Tate twists of $p$-adic étale cohomology groups and the cohomology groups of logarithmic Hodge-Witt sheaves for a proper semi-stable scheme over $O_K$. The object of this paper is to show that these isomorphisms are compatible with the symbol maps to the $p$-adic vanishing cycles and the logarithmic Hodge-Witt sheaves, and that they are compatible with the integral structures under certain restrictions. We also treats an open case and a proof of $C_{st}$ in such a case is given for that purpose. The results are used in the work of U. Jannsen and S. Saito [Doc. Math., J. DMV Extra Vol., 479--538 (2003; Zbl 1092.14503)].
