Monotone homogeneity and planar manifolds with boundary (Q1372690)

A space \(X\) is said to be homogeneous with respect to a given class \({\mathcal M}\) of mappings if for every two points \(x,y\in X\) there is a surjection \(f:X\to Y\) in \({\mathcal M}\) such that \(f(x)=y\). Using techniques from algebraic topology the author shows that no compact planar manifold with boundary (a disk, in particular) is homogeneous with respect to monotone mappings. An example is constructed of a noncompact planar manifold with boundary that is homogeneous with respect to monotone open mappings.