The Voronoi conjecture for parallelohedra with simply connected $\delta$-surface

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.