  We present some results supporting the Iwase-Sakai conjecture about
coincidence of the topological complexity $TC(X)$ and monoidal topological
complexity $TC^M(X)$. Using these results we provide lower and upper bounds for
the topological complexity of the wedge $X\vee Y$. We use these bounds to give
a counterexample to the conjecture asserting that $TC(X')\le TC(X)$ for any
covering map $p:X'\to X$.
  We discuss a possible reduction of the monoidal topological complexity to the
LS-category. Also we apply the LS-category to give a short proof of the
Arnold-Kuiper theorem.
