  We give a new, purely topological construction of Eisenstein cohomology
classes for Hilbert-Blumenthal varieties using the polylogarithm for families
of topological tori and a decomposition with respect to the units in the center
of $GL_2$. These classes are explicitly calculated in de Rham chomology and
compared with Harder's Eisenstein classes. For non-trivial coefficient systems
the whole Eisenstein cohomology in positive degrees is generated by these
topological Eisenstein classes. This gives an alternative proof for the
rationality of Harder's Eisenstein operator without using any multiplicity one
arguments. The text constitutes my 2016 Regensburg PhD thesis.
