The Voronoi conjecture for parallelohedra with simply connected δ-surfaces
We show that the Voronoi conjecture is true for parallelohedra with simply connected δδ-surfaces. That is, we show that if the boundary of parallelohedron PP remains simply connected after removing closed nonprimitive faces of codimension 2, then PP is affinely equivalent to a Dirichlet–Voronoi domain of some lattice. Also, we construct the ππ-surface associated with a parallelohedron and give another condition in terms of a homology group of the constructed surface. Every parallelohedron with a simply connected δδ-surface also satisfies the condition on the homology group of the ππ-surface.