Minimizing Coincidence in Positive Codimension

Let $f$ and $g$ be maps between smooth manifolds $M$ and $N$ of dimensions $n+m$ and $n$, respectively (where $m>0$ and $n>2$). Suppose that the image $(fxg)(M)$ intersects the diagonal $N\times N$ in finitely many points, whose preimages are smooth $m$-submanifolds in $M$. The problem of minimizing the coincidence set $\operatorname{Coin}(f,g)$ of the maps $f$ and $g$ with respect to these preimages and/or their components is considered. The author's earlier results are strengthened. Namely, sufficient conditions under which such a coincidence $m$-submanifold can be removed without additional dimensional constraints are obtained.