Maps into the torus and minimal coincidence sets for homotopies
Let $X,Y$ be manifolds of the same dimension. Given continuous mappings $f_i,g_i :X\to Y$, $i=0,1$, we consider the $1$-parameter coincidence problem of finding homotopies $f_t,g_t$, $0\leq t\leq 1$, such that the number of coincidence points for the pair $f_t,g_t$ is independent of $t$. When $Y$ is the torus and $f_0,g_0$ are coincidence free we produce coincidence free pairs $f_1,g_1$ such that no homotopy joining them is coincidence free at each level. When $X$ is also the torus we characterize the solution of the problem in terms of the Lefschetz coincidence number.