Monotone homogeneity and planar manifolds with boundary

It is known that the disk is homogeneous with respect to open maps; however, it was unknown whether or not the disk was homogeneous with respect to monotone open maps. Using techniques from algebraic topology we show that not only is the disk not homogeneous with respect to monotone maps, but no compact planar manifold with boundary is monotone homogeneous. We end with an example of a noncompact planar manifold with boundary that is monotone open homogeneous.