  We show that the Voronoi conjecture is true for parallelohedra with simply
connected $\delta$-surface. Namely, we show that if the boundary of
parallelohedron $P$ remains simply connected after removing closed
non-primitive faces of codimension 2, then $P$ is affinely equivalent to a
Dirichlet-Voronoi domain of some lattice. Also we construct the $\pi$-surface
associated with a parallelohedron and give another condition in terms of
homology group of the constructed surface. Every parallelohedron with simply
connected $\delta$-surface also satisfies the condition on homology group of
the $\pi$-surface.
