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

TL;DR

This paper proves the Voronoi conjecture for a class of parallelohedra with simply connected $\

Contribution

It establishes the conjecture for parallelohedra with simply connected $\\delta$-surface and introduces a homology-based condition involving the $\\pi$-surface.

Findings

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

Constructs the $\\pi$-surface and relates its homology to the conjecture.

Provides new topological conditions for parallelohedra to satisfy the conjecture.

Abstract

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.