  Let $X$ be a two-cell complex with attaching map $\alpha\colon S^q\to S^p$,
and let $C_X$ be the cofiber of the diagonal inclusion $X\to X\times X$. It is
shown that the topological complexity (${\rm TC}$) of $X$ agrees with the
Lusternik-Schnirelmann category (${\rm cat}$) of $C_X$ in the (almost stable)
range $q\leq2p-1$. In addition, the equality ${\rm TC}(X)={\rm cat}(C_X)$ is
proved in the (strict) metastable range $2p-1<q\leq3(p-1)$ under fairly mild
conditions by making use of the Hopf invariant techniques recently developed by
the authors in their study of the sectional category of arbitrary maps.
