On π-surfaces of four-dimensional parallelohedra

Abstract

We show that every four-dimensional parallelohedron P satisfies a recently found condition of Garber, Gavrilyuk & Magazinov sufficient for the Voronoi conjecture being true for P. Namely we show that for every four-dimensional parallelohedron P the group of rational first homologies of its π-surface is generated by half-belt cycles.

16 pages, 7 figures

References in corpus (2)

• The Voronoi conjecture for parallelohedra with simply connected $\delta$-surface
• Belt diameter of $\Pi$-zonotopes

Cited by in corpus (3)

• The Voronoi conjecture for parallelohedra with simply connected $\delta$-surface
• On the Voronoi Conjecture for combinatorially Voronoi parallelohedra in dimension five
• On combinatorics of Voronoi polytopes for perturbations of the dual root lattices