(Non)connectedness and (non)homogeneity

We discuss an approach to a problem posed by A.V. Arhangel'skii and E.K. van Douwen on a possibility to present a compact space as a continuous image of a homogeneous compact space. Then we suggest some ways of proving nonhomogeneity of τ-powers of a space X using points of local connectedness (or local contractibility) and components of path connectedness of X.