Voronoi conjecture for five-dimensional parallelohedra

We prove the Voronoi conjecture for five-dimensional parallelohedra. Namely, we show that if a convex five-dimensional polytope $P$ tiles $\mathbb R^5$ with translations, then $P$ is an affine image of the Dirichlet-Voronoi polytope for a five-dimensional lattice. Our proof is based on an exhaustive combinatorial analysis of possible dual 3-cells and incident dual 4-cells encoding local structures around two-dimensional faces of five-dimensional parallelohedron $P$ and their edges aiming to prove existence of a free direction for $P$ paired with new properties established for parallelohedra (in any dimension) that have a free direction that guarantee the Voronoi conjecture for $P$.